From b92be312e8966b9ecc04e51566e5635a592fa538 Mon Sep 17 00:00:00 2001 From: Vectornaut Date: Mon, 21 Oct 2024 03:18:47 +0000 Subject: [PATCH] Engine prototype (#13) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit This PR adds code for a Julia-language prototype of a configuration solver, in the `engine-proto` folder. It uses Julia version 1.10.0. ### Approaches Development of this PR tried two broad approaches to the constraint geometry problem. Each one suggested various solution techniques. The Gram matrix approach, with the low-rank factorization technique, seems the most promising. - **Algebraic** *(In the `alg-test` subfolder).* Write the constraints as polynomials in the inversive coordinates of the elements, and use computational algebraic geometry techniques to solve the resulting system. We tried the following techniques. - **Gröbner bases** *(`Engine.Algebraic.jl`).* Symbolic. Find a Gröbner basis for the ideal generated by the constraint equations. Information about the solution variety, like its codimension, is then relatively easy to extract. - **Homotopy continuation** *(`Engine.Numerical.jl`).* Numerical. Cut the solution set along a random hyperplane to get a generic zero-dimensional slice, and then use a fancy homotopy technique to approximate the points in that slice. A few notes about our experiences can be found on the [engine prototype](wiki/Engine-prototype) wiki page. - **Gram matrix** *(in the `gram-test` subfolder).* A construction is described completely, up to conformal transformations, by the Gram matrix of the vectors representing its elements. Express the constraints as fixed entries of the Gram matrix, and use numerical linear algebra techniques to find a list of vectors whose Gram matrix fits the bill. We tried the following techniques. - **LDL decomposition** *(`gram-test.sage`, `gram-test.jl`, `overlap-test.jl`).* Find a cluster of up to five elements whose Gram matrix is completely filled in by the constraints. Use LDL decomposition to find a list of vectors with that Gram matrix. This technique can be made algebraic, as seen in `overlap-test.jl`. - **Low-rank factorization** *(source files listed in findings section).* Write down a quadratic loss function that says how far a set of vectors is from meeting the Gram matrix constraints. Use a smooth optimization technique like Newton's method or gradient descent to find a zero of the loss function. In addition to the polished prototype described in the results section, we have an early prototype using an off-the-shelf factorization package (`low-rank-test.jl`) and an visualization of the loss function landscape near global minima (`basin-shapes.jl`). The [Gram matrix parameterization](wiki/Gram-matrix-parameterization) wiki page contains detailed notes on this approach. ### Findings With the algebraic approach, we hit a performance wall pretty quickly as our constructions grew. It was often hard to find real solutions of the polynomial system, since the techniques we use work most naturally in the complex world. With the Gram matrix approach, on the other hand, we could solve interesting problems in acceptably short times using the low-rank factorization technique. We put the optimization routine in its own module (`Engine.jl`) and used it to solve five example problems: - `overlapping-pyramids.jl` - `circles-in-triangle.jl` - `sphere-in-tetrahedron.jl` - `tetrahedron-radius-ratio.jl` - `irisawa-hexlet.jl` We plan to use low-rank factorization of the Gram matrix in our first app prototype. ### Visualizations We used the visualizer in the `ganja-test` folder to visually check our low-rank factorization results. The visualizer runs [Ganja.js](https://enkimute.github.io/ganja.js/) in an Electron app, made with [Blink](https://github.com/JuliaGizmos/Blink.jl). Although Ganja.js makes beautiful pictures under most circumstances, we found two obstacles to using it in production. - It seems to have precision problems with low-curvature spheres. - We couldn't figure out how to customize its clipping and transparency settings, and the default settings often obscure construction details. Co-authored-by: Aaron Fenyes Co-authored-by: Glen Whitney Reviewed-on: https://code.studioinfinity.org/glen/dyna3/pulls/13 Co-authored-by: Vectornaut Co-committed-by: Vectornaut --- engine-proto/alg-test/ConstructionViewer.jl | 223 ++ engine-proto/alg-test/Engine.Algebraic.jl | 203 ++ engine-proto/alg-test/Engine.Numerical.jl | 53 + engine-proto/alg-test/Engine.jl | 76 + engine-proto/alg-test/HittingSet.jl | 111 + engine-proto/ganja-test/ganja-test.html | 96 + engine-proto/ganja-test/ganja-test.jl | 127 ++ engine-proto/gram-test/Engine.jl | 450 ++++ engine-proto/gram-test/basin-shapes.jl | 99 + engine-proto/gram-test/circles-in-triangle.jl | 76 + engine-proto/gram-test/ganja-1.0.204.js | 1913 +++++++++++++++++ engine-proto/gram-test/gram-test.jl | 85 + engine-proto/gram-test/gram-test.sage | 27 + engine-proto/gram-test/irisawa-hexlet.jl | 77 + engine-proto/gram-test/low-rank-test.jl | 49 + engine-proto/gram-test/overlap-test.jl | 37 + .../gram-test/overlapping-pyramids.jl | 90 + .../gram-test/sphere-in-tetrahedron.jl | 67 + .../gram-test/tetrahedron-radius-ratio.jl | 96 + notes/inversive.md | 43 +- 20 files changed, 3977 insertions(+), 21 deletions(-) create mode 100644 engine-proto/alg-test/ConstructionViewer.jl create mode 100644 engine-proto/alg-test/Engine.Algebraic.jl create mode 100644 engine-proto/alg-test/Engine.Numerical.jl create mode 100644 engine-proto/alg-test/Engine.jl create mode 100644 engine-proto/alg-test/HittingSet.jl create mode 100644 engine-proto/ganja-test/ganja-test.html create mode 100644 engine-proto/ganja-test/ganja-test.jl create mode 100644 engine-proto/gram-test/Engine.jl create mode 100644 engine-proto/gram-test/basin-shapes.jl create mode 100644 engine-proto/gram-test/circles-in-triangle.jl create mode 100644 engine-proto/gram-test/ganja-1.0.204.js create mode 100644 engine-proto/gram-test/gram-test.jl create mode 100644 engine-proto/gram-test/gram-test.sage create mode 100644 engine-proto/gram-test/irisawa-hexlet.jl create mode 100644 engine-proto/gram-test/low-rank-test.jl create mode 100644 engine-proto/gram-test/overlap-test.jl create mode 100644 engine-proto/gram-test/overlapping-pyramids.jl create mode 100644 engine-proto/gram-test/sphere-in-tetrahedron.jl create mode 100644 engine-proto/gram-test/tetrahedron-radius-ratio.jl diff --git a/engine-proto/alg-test/ConstructionViewer.jl b/engine-proto/alg-test/ConstructionViewer.jl new file mode 100644 index 0000000..b9c8ffb --- /dev/null +++ b/engine-proto/alg-test/ConstructionViewer.jl @@ -0,0 +1,223 @@ +module Viewer + +using Blink +using Colors +using Printf + +using Main.Engine + +export ConstructionViewer, display!, opentools!, closetools! + +# === Blink utilities === + +append_to_head!(w, type, content) = @js w begin + @var element = document.createElement($type) + element.appendChild(document.createTextNode($content)) + document.head.appendChild(element) +end + +style!(w, stylesheet) = append_to_head!(w, "style", stylesheet) + +script!(w, code) = append_to_head!(w, "script", code) + +# === construction viewer === + +mutable struct ConstructionViewer + win::Window + + function ConstructionViewer() + # create window and open developer console + win = Window(Blink.Dict(:width => 620, :height => 830)) + + # set stylesheet + style!(win, """ + body { + background-color: #ccc; + } + + /* the maximum dimensions keep Ganja from blowing up the canvas */ + #view { + display: block; + width: 600px; + height: 600px; + margin-top: 10px; + margin-left: 10px; + border-radius: 10px; + background-color: #f0f0f0; + } + + #control-panel { + width: 600px; + height: 200px; + box-sizing: border-box; + padding: 5px 10px 5px 10px; + margin-top: 10px; + margin-left: 10px; + overflow-y: scroll; + border-radius: 10px; + background-color: #f0f0f0; + } + + #control-panel > div { + margin-top: 5px; + padding: 4px; + border-radius: 5px; + border: solid; + font-family: monospace; + } + """) + + # load Ganja.js. for an automatically updated web-hosted version, load from + # + # https://unpkg.com/ganja.js + # + # instead + loadjs!(win, "http://localhost:8000/ganja-1.0.204.js") + + # create global functions and variables + script!(win, """ + // create algebra + var CGA3 = Algebra(4, 1); + + // initialize element list and palette + var elements = []; + var palette = []; + + // declare handles for the view and its options + var view; + var viewOpt; + + // declare handles for the controls + var controlPanel; + var visToggles; + + // create scene function + function scene() { + commands = []; + for (let n = 0; n < elements.length; ++n) { + if (visToggles[n].checked) { + commands.push(palette[n], elements[n]); + } + } + return commands; + } + + function updateView() { + requestAnimationFrame(view.update.bind(view, scene)); + } + """) + + @js win begin + # create view + viewOpt = Dict( + :conformal => true, + :gl => true, + :devicePixelRatio => window.devicePixelRatio + ) + view = CGA3.graph(scene, viewOpt) + view.setAttribute(:id, "view") + view.removeAttribute(:style) + document.body.replaceChildren(view) + + # create control panel + controlPanel = document.createElement(:div) + controlPanel.setAttribute(:id, "control-panel") + document.body.appendChild(controlPanel) + end + + new(win) + end +end + +mprod(v, w) = + v[1]*w[1] + v[2]*w[2] + v[3]*w[3] + v[4]*w[4] - v[5]*w[5] + +function display!(viewer::ConstructionViewer, elements::Matrix) + # load elements + elements_full = [] + for elt in eachcol(Engine.unmix * elements) + if mprod(elt, elt) < 0.5 + elt_full = [0; elt; fill(0, 26)] + else + # `elt` is a spacelike vector, representing a generalized sphere, so we + # take its Hodge dual before passing it to Ganja.js. the dual represents + # the same generalized sphere, but Ganja.js only displays planes when + # they're represented by vectors in grade 4 rather than grade 1 + elt_full = [fill(0, 26); -elt[5]; -elt[4]; elt[3]; -elt[2]; elt[1]; 0] + end + push!(elements_full, elt_full) + end + @js viewer.win elements = $elements_full.map((elt) -> @new CGA3(elt)) + + # generate palette. this is Gadfly's `default_discrete_colors` palette, + # available under the MIT license + palette = distinguishable_colors( + length(elements_full), + [LCHab(70, 60, 240)], + transform = c -> deuteranopic(c, 0.5), + lchoices = Float64[65, 70, 75, 80], + cchoices = Float64[0, 50, 60, 70], + hchoices = range(0, stop=330, length=24) + ) + palette_packed = [RGB24(c).color for c in palette] + @js viewer.win palette = $palette_packed + + # create visibility toggles + @js viewer.win begin + controlPanel.replaceChildren() + visToggles = [] + end + for (elt, c) in zip(eachcol(elements), palette) + vec_str = join(map(t -> @sprintf("%.3f", t), elt), ", ") + color_str = "#$(hex(c))" + style_str = "background-color: $color_str; border-color: $color_str;" + @js viewer.win begin + @var toggle = document.createElement(:div) + toggle.setAttribute(:style, $style_str) + toggle.checked = true + toggle.addEventListener( + "click", + () -> begin + toggle.checked = !toggle.checked + toggle.style.backgroundColor = toggle.checked ? $color_str : "inherit"; + updateView() + end + ) + toggle.appendChild(document.createTextNode($vec_str)) + visToggles.push(toggle); + controlPanel.appendChild(toggle); + end + end + + # update view + @js viewer.win updateView() +end + +function opentools!(viewer::ConstructionViewer) + size(viewer.win, 1240, 830) + opentools(viewer.win) +end + +function closetools!(viewer::ConstructionViewer) + closetools(viewer.win) + size(viewer.win, 620, 830) +end + +end + +# ~~~ sandbox setup ~~~ + +elements = let + a = sqrt(BigFloat(3)/2) + sqrt(0.5) * BigFloat[ + 1 1 -1 -1 0 + 1 -1 1 -1 0 + 1 -1 -1 1 0 + 0.5 0.5 0.5 0.5 1+a + 0.5 0.5 0.5 0.5 1-a + ] +end + +# show construction +viewer = Viewer.ConstructionViewer() +Viewer.display!(viewer, elements) \ No newline at end of file diff --git a/engine-proto/alg-test/Engine.Algebraic.jl b/engine-proto/alg-test/Engine.Algebraic.jl new file mode 100644 index 0000000..a9b6667 --- /dev/null +++ b/engine-proto/alg-test/Engine.Algebraic.jl @@ -0,0 +1,203 @@ +module Algebraic + +export + codimension, dimension, + Construction, realize, + Element, Point, Sphere, + Relation, LiesOn, AlignsWithBy, mprod + +import Subscripts +using LinearAlgebra +using AbstractAlgebra +using Groebner +using ...HittingSet + +# --- commutative algebra --- + +# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate +# polynomial rings when the coefficients are integers. we use Groebner to extend +# support to rationals and to finite fields of prime order +Generic.reduce_gens(I::Generic.Ideal{U}) where {T <: FieldElement, U <: MPolyRingElem{T}} = + Generic.Ideal{U}(base_ring(I), groebner(gens(I))) + +function codimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} + leading = [exponent_vector(f, 1) for f in gens(I)] + targets = [Set(findall(.!iszero.(exp_vec))) for exp_vec in leading] + length(HittingSet.solve(HittingSetProblem(targets), maxdepth)) +end + +dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} = + length(gens(base_ring(I))) - codimension(I, maxdepth) + +# --- primitve elements --- + +abstract type Element{T} end + +mutable struct Point{T} <: Element{T} + coords::Vector{MPolyRingElem{T}} + vec::Union{Vector{MPolyRingElem{T}}, Nothing} + rel::Nothing + + ## [to do] constructor argument never needed? + Point{T}( + coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[], + vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing + ) where T = new(coords, vec, nothing) +end + +function buildvec!(pt::Point) + coordring = parent(pt.coords[1]) + pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...] +end + +mutable struct Sphere{T} <: Element{T} + coords::Vector{MPolyRingElem{T}} + vec::Union{Vector{MPolyRingElem{T}}, Nothing} + rel::Union{MPolyRingElem{T}, Nothing} + + ## [to do] constructor argument never needed? + Sphere{T}( + coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[], + vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing, + rel::Union{MPolyRingElem{T}, Nothing} = nothing + ) where T = new(coords, vec, rel) +end + +function buildvec!(sph::Sphere) + coordring = parent(sph.coords[1]) + sph.vec = sph.coords + sph.rel = mprod(sph.coords, sph.coords) + one(coordring) +end + +const coordnames = IdDict{Symbol, Vector{Union{Symbol, Nothing}}}( + nameof(Point) => [nothing, nothing, :xₚ, :yₚ, :zₚ], + nameof(Sphere) => [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ] +) + +coordname(elt::Element, index) = coordnames[nameof(typeof(elt))][index] + +function pushcoordname!(coordnamelist, indexed_elt::Tuple{Any, Element}, coordindex) + eltindex, elt = indexed_elt + name = coordname(elt, coordindex) + if !isnothing(name) + subscript = Subscripts.sub(string(eltindex)) + push!(coordnamelist, Symbol(name, subscript)) + end +end + +function takecoord!(coordlist, indexed_elt::Tuple{Any, Element}, coordindex) + elt = indexed_elt[2] + if !isnothing(coordname(elt, coordindex)) + push!(elt.coords, popfirst!(coordlist)) + end +end + +# --- primitive relations --- + +abstract type Relation{T} end + +mprod(v, w) = (v[1]*w[2] + w[1]*v[2]) / 2 - dot(v[3:end], w[3:end]) + +# elements: point, sphere +struct LiesOn{T} <: Relation{T} + elements::Vector{Element{T}} + + LiesOn{T}(pt::Point{T}, sph::Sphere{T}) where T = new{T}([pt, sph]) +end + +equation(rel::LiesOn) = mprod(rel.elements[1].vec, rel.elements[2].vec) + +# elements: sphere, sphere +struct AlignsWithBy{T} <: Relation{T} + elements::Vector{Element{T}} + cos_angle::T + + AlignsWithBy{T}(sph1::Sphere{T}, sph2::Sphere{T}, cos_angle::T) where T = new{T}([sph1, sph2], cos_angle) +end + +equation(rel::AlignsWithBy) = mprod(rel.elements[1].vec, rel.elements[2].vec) - rel.cos_angle + +# --- constructions --- + +mutable struct Construction{T} + points::Vector{Point{T}} + spheres::Vector{Sphere{T}} + relations::Vector{Relation{T}} + + function Construction{T}(; elements = Vector{Element{T}}(), relations = Vector{Relation{T}}()) where T + allelements = union(elements, (rel.elements for rel in relations)...) + new{T}( + filter(elt -> isa(elt, Point), allelements), + filter(elt -> isa(elt, Sphere), allelements), + relations + ) + end +end + +function Base.push!(ctx::Construction{T}, elt::Point{T}) where T + push!(ctx.points, elt) +end + +function Base.push!(ctx::Construction{T}, elt::Sphere{T}) where T + push!(ctx.spheres, elt) +end + +function Base.push!(ctx::Construction{T}, rel::Relation{T}) where T + push!(ctx.relations, rel) + for elt in rel.elements + push!(ctx, elt) + end +end + +function realize(ctx::Construction{T}) where T + # collect coordinate names + coordnamelist = Symbol[] + eltenum = enumerate(Iterators.flatten((ctx.spheres, ctx.points))) + for coordindex in 1:5 + for indexed_elt in eltenum + pushcoordname!(coordnamelist, indexed_elt, coordindex) + end + end + + # construct coordinate ring + coordring, coordqueue = polynomial_ring(parent_type(T)(), coordnamelist, ordering = :degrevlex) + + # retrieve coordinates + for (_, elt) in eltenum + empty!(elt.coords) + end + for coordindex in 1:5 + for indexed_elt in eltenum + takecoord!(coordqueue, indexed_elt, coordindex) + end + end + + # construct coordinate vectors + for (_, elt) in eltenum + buildvec!(elt) + end + + # turn relations into equations + eqns = vcat( + equation.(ctx.relations), + [elt.rel for (_, elt) in eltenum if !isnothing(elt.rel)] + ) + + # add relations to center, orient, and scale the construction + # [to do] the scaling constraint, as written, can be impossible to satisfy + # when all of the spheres have to go through the origin + if !isempty(ctx.points) + append!(eqns, [sum(pt.coords[k] for pt in ctx.points) for k in 1:3]) + end + if !isempty(ctx.spheres) + append!(eqns, [sum(sph.coords[k] for sph in ctx.spheres) for k in 3:4]) + end + n_elts = length(ctx.points) + length(ctx.spheres) + if n_elts > 0 + push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts) + end + + (Generic.Ideal(coordring, eqns), eqns) +end + +end \ No newline at end of file diff --git a/engine-proto/alg-test/Engine.Numerical.jl b/engine-proto/alg-test/Engine.Numerical.jl new file mode 100644 index 0000000..d1e14bd --- /dev/null +++ b/engine-proto/alg-test/Engine.Numerical.jl @@ -0,0 +1,53 @@ +module Numerical + +using Random: default_rng +using LinearAlgebra +using AbstractAlgebra +using HomotopyContinuation: + Variable, Expression, AbstractSystem, System, LinearSubspace, + nvariables, isreal, witness_set, results +import GLMakie +using ..Algebraic + +# --- polynomial conversion --- + +# hat tip Sascha Timme +# https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521 +function Base.convert(::Type{Expression}, f::MPolyRingElem) + variables = Variable.(symbols(parent(f))) + f_data = zip(coefficients(f), exponent_vectors(f)) + sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data) +end + +# create a ModelKit.System from an ideal in a multivariate polynomial ring. the +# variable ordering is taken from the polynomial ring +function System(I::Generic.Ideal) + eqns = Expression.(gens(I)) + variables = Variable.(symbols(base_ring(I))) + System(eqns, variables = variables) +end + +# --- sampling --- + +function real_samples(F::AbstractSystem, dim; rng = default_rng()) + # choose a random real hyperplane of codimension `dim` by intersecting + # hyperplanes whose normal vectors are uniformly distributed over the unit + # sphere + # [to do] guard against the unlikely event that one of the normals is zero + normals = transpose(hcat( + (normalize(randn(rng, nvariables(F))) for _ in 1:dim)... + )) + cut = LinearSubspace(normals, fill(0., dim)) + filter(isreal, results(witness_set(F, cut, seed = 0x1974abba))) +end + +AbstractAlgebra.evaluate(pt::Point, vals::Vector{<:RingElement}) = + GLMakie.Point3f([evaluate(u, vals) for u in pt.coords]) + +function AbstractAlgebra.evaluate(sph::Sphere, vals::Vector{<:RingElement}) + radius = 1 / evaluate(sph.coords[1], vals) + center = radius * [evaluate(u, vals) for u in sph.coords[3:end]] + GLMakie.Sphere(GLMakie.Point3f(center), radius) +end + +end \ No newline at end of file diff --git a/engine-proto/alg-test/Engine.jl b/engine-proto/alg-test/Engine.jl new file mode 100644 index 0000000..f6f92c5 --- /dev/null +++ b/engine-proto/alg-test/Engine.jl @@ -0,0 +1,76 @@ +include("HittingSet.jl") + +module Engine + +include("Engine.Algebraic.jl") +include("Engine.Numerical.jl") + +using .Algebraic +using .Numerical + +export Construction, mprod, codimension, dimension + +end + +# ~~~ sandbox setup ~~~ + +using Random +using Distributions +using LinearAlgebra +using AbstractAlgebra +using HomotopyContinuation +using GLMakie + +CoeffType = Rational{Int64} + +spheres = [Engine.Sphere{CoeffType}() for _ in 1:3] +tangencies = [ + Engine.AlignsWithBy{CoeffType}( + spheres[n], + spheres[mod1(n+1, length(spheres))], + CoeffType(1) + ) + for n in 1:3 +] +ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies) +ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph) +freedom = Engine.dimension(ideal_tan_sph) +println("Three mutually tangent spheres: $freedom degrees of freedom") + +# --- test rational cut --- + +coordring = base_ring(ideal_tan_sph) +vbls = Variable.(symbols(coordring)) + +# test a random witness set +system = CompiledSystem(System(eqns_tan_sph, variables = vbls)) +norm2 = vec -> real(dot(conj.(vec), vec)) +rng = MersenneTwister(6071) +n_planes = 6 +samples = [] +for _ in 1:n_planes + real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng)) + for soln in real_solns + if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples) + push!(samples, soln) + end + end +end +println("Found $(length(samples)) sample solutions") + +# show a sample solution +function show_solution(ctx, vals) + # evaluate elements + real_vals = real.(vals) + disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points] + disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres] + + # create scene + scene = Scene() + cam3d!(scene) + scatter!(scene, disp_points, color = :green) + for sph in disp_spheres + mesh!(scene, sph, color = :gray) + end + scene +end \ No newline at end of file diff --git a/engine-proto/alg-test/HittingSet.jl b/engine-proto/alg-test/HittingSet.jl new file mode 100644 index 0000000..347c4d2 --- /dev/null +++ b/engine-proto/alg-test/HittingSet.jl @@ -0,0 +1,111 @@ +module HittingSet + +export HittingSetProblem, solve + +HittingSetProblem{T} = Pair{Set{T}, Vector{Pair{T, Set{Set{T}}}}} + +# `targets` should be a collection of Set objects +function HittingSetProblem(targets, chosen = Set()) + wholeset = union(targets...) + T = eltype(wholeset) + unsorted_moves = [ + elt => Set(filter(s -> elt ∉ s, targets)) + for elt in wholeset + ] + moves = sort(unsorted_moves, by = pair -> length(pair.second)) + Set{T}(chosen) => moves +end + +function Base.display(problem::HittingSetProblem{T}) where T + println("HittingSetProblem{$T}") + + chosen = problem.first + println(" {", join(string.(chosen), ", "), "}") + + moves = problem.second + for (choice, missed) in moves + println(" | ", choice) + for s in missed + println(" | | {", join(string.(s), ", "), "}") + end + end + println() +end + +function solve(pblm::HittingSetProblem{T}, maxdepth = Inf) where T + problems = Dict(pblm) + while length(first(problems).first) < maxdepth + subproblems = typeof(problems)() + for (chosen, moves) in problems + if isempty(moves) + return chosen + else + for (choice, missed) in moves + to_be_chosen = union(chosen, Set([choice])) + if isempty(missed) + return to_be_chosen + elseif !haskey(subproblems, to_be_chosen) + push!(subproblems, HittingSetProblem(missed, to_be_chosen)) + end + end + end + end + problems = subproblems + end + problems +end + +function test(n = 1) + T = [Int64, Int64, Symbol, Symbol][n] + targets = Set{T}.([ + [ + [1, 3, 5], + [2, 3, 4], + [1, 4], + [2, 3, 4, 5], + [4, 5] + ], + # example from Amit Chakrabarti's graduate-level algorithms class (CS 105) + # notes by Valika K. Wan and Khanh Do Ba, Winter 2005 + # https://www.cs.dartmouth.edu/~ac/Teach/CS105-Winter05/ + [ + [1, 3], [1, 4], [1, 5], + [1, 3], [1, 2, 4], [1, 2, 5], + [4, 3], [ 2, 4], [ 2, 5], + [6, 3], [6, 4], [ 5] + ], + [ + [:w, :x, :y], + [:x, :y, :z], + [:w, :z], + [:x, :y] + ], + # Wikipedia showcases this as an example of a problem where the greedy + # algorithm performs especially poorly + [ + [:a, :x, :t1], + [:a, :y, :t2], + [:a, :y, :t3], + [:a, :z, :t4], + [:a, :z, :t5], + [:a, :z, :t6], + [:a, :z, :t7], + [:b, :x, :t8], + [:b, :y, :t9], + [:b, :y, :t10], + [:b, :z, :t11], + [:b, :z, :t12], + [:b, :z, :t13], + [:b, :z, :t14] + ] + ][n]) + problem = HittingSetProblem(targets) + if isa(problem, HittingSetProblem{T}) + println("Correct type") + else + println("Wrong type: ", typeof(problem)) + end + problem +end + +end \ No newline at end of file diff --git a/engine-proto/ganja-test/ganja-test.html b/engine-proto/ganja-test/ganja-test.html new file mode 100644 index 0000000..0207dcc --- /dev/null +++ b/engine-proto/ganja-test/ganja-test.html @@ -0,0 +1,96 @@ + + + + + + + +

+ + + diff --git a/engine-proto/ganja-test/ganja-test.jl b/engine-proto/ganja-test/ganja-test.jl new file mode 100644 index 0000000..6c55061 --- /dev/null +++ b/engine-proto/ganja-test/ganja-test.jl @@ -0,0 +1,127 @@ +using Blink +using Colors + +# === utilities === + +append_to_head!(w, type, content) = @js w begin + @var element = document.createElement($type) + element.appendChild(document.createTextNode($content)) + document.head.appendChild(element) +end + +style!(w, stylesheet) = append_to_head!(w, "style", stylesheet) + +script!(w, code) = append_to_head!(w, "script", code) + +function add_element!(vec) + # add element + full_vec = [0; vec; fill(0, 26)] + n = @js win elements.push(@new CGA3($full_vec)) + + # generate palette. this is Gadfly's `default_discrete_colors` palette, + # available under the MIT license + palette = distinguishable_colors( + n, + [LCHab(70, 60, 240)], + transform = c -> deuteranopic(c, 0.5), + lchoices = Float64[65, 70, 75, 80], + cchoices = Float64[0, 50, 60, 70], + hchoices = range(0, stop=330, length=24) + ) + palette_packed = [RGB24(c).color for c in palette] + @js win palette = $palette_packed +end + +# === build page === + +# create window and open developer console +win = Window() +opentools(win) + +# set stylesheet +style!(win, """ + body { + background-color: #ffe0f0; + } + + /* needed to keep Ganja canvas from blowing up */ + canvas { + min-width: 600px; + max-width: 600px; + min-height: 600px; + max-height: 600px; + } +""") + +# load Ganja.js +loadjs!(win, "https://unpkg.com/ganja.js") + +# create global functions and variables +script!(win, """ + // create algebra + var CGA3 = Algebra(4, 1); + + // initialize element list and palette + var elements = []; + var palette = []; + + // declare visualization handle + var graph; + + // create scene function + function scene() { + commands = []; + for (let n = 0; n < elements.length; ++n) { + commands.push(palette[n], elements[n]); + } + return commands; + } + + function flip() { + let last = elements.length - 1; + for (let n = 0; n < last; ++n) { + // reflect + elements[n] = CGA3.Mul(CGA3.Mul(elements[last], elements[n]), elements[last]); + + // de-noise + for (let k = 6; k < elements[n].length; ++k) { + elements[n][k] = 0; + } + } + requestAnimationFrame(graph.update.bind(graph, scene)); + } +""") + +# set up controls +body!(win, """ +

+""", async = false) + +# === set up visualization === + +# list elements. in the default view, e4 + e5 is the point at infinity +elements = sqrt(0.5) * BigFloat[ + 1 1 -1 -1 0; + 1 -1 1 -1 0; + 1 -1 -1 1 0; + 0 0 0 0 -sqrt(6); + 1 1 1 1 2 +] + +# load elements +for vec in eachcol(elements) + add_element!(vec) +end + +# initialize visualization +@js win begin + graph = CGA3.graph( + scene, + Dict( + "conformal" => true, + "gl" => true, + "grid" => true + ) + ) + document.body.appendChild(graph) +end \ No newline at end of file diff --git a/engine-proto/gram-test/Engine.jl b/engine-proto/gram-test/Engine.jl new file mode 100644 index 0000000..22f5914 --- /dev/null +++ b/engine-proto/gram-test/Engine.jl @@ -0,0 +1,450 @@ +module Engine + +using LinearAlgebra +using GenericLinearAlgebra +using SparseArrays +using Random +using Optim + +export + rand_on_shell, Q, DescentHistory, + realize_gram_gradient, realize_gram_newton, realize_gram_optim, realize_gram + +# === guessing === + +sconh(t, u) = 0.5*(exp(t) + u*exp(-t)) + +function rand_on_sphere(rng::AbstractRNG, ::Type{T}, n) where T + out = randn(rng, T, n) + tries_left = 2 + while dot(out, out) < 1e-6 && tries_left > 0 + out = randn(rng, T, n) + tries_left -= 1 + end + normalize(out) +end + +##[TO DO] write a test to confirm that the outputs are on the correct shells +function rand_on_shell(rng::AbstractRNG, shell::T) where T <: Number + space_part = rand_on_sphere(rng, T, 4) + rapidity = randn(rng, T) + sig = sign(shell) + nullmix * [sconh(rapidity, sig)*space_part; sconh(rapidity, -sig)] +end + +rand_on_shell(rng::AbstractRNG, shells::Array{T}) where T <: Number = + hcat([rand_on_shell(rng, sh) for sh in shells]...) + +rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), shells) + +# === elements === + +point(pos) = [pos; 0.5; 0.5 * dot(pos, pos)] + +plane(normal, offset) = [-normal; 0; -offset] + +function sphere(center, radius) + dist_sq = dot(center, center) + [ + center / radius; + 0.5 / radius; + 0.5 * (dist_sq / radius - radius) + ] +end + +# === Gram matrix realization === + +# basis changes +nullmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [-1 1; 1 1]//2] +unmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [-1 1; 1 1]] + +# the Lorentz form +Q = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [0 -2; -2 0]] + +# project a matrix onto the subspace of matrices whose entries vanish away from +# the given indices +function proj_to_entries(mat, indices) + result = zeros(size(mat)) + for (j, k) in indices + result[j, k] = mat[j, k] + end + result +end + +# the difference between the matrices `target` and `attempt`, projected onto the +# subspace of matrices whose entries vanish at each empty index of `target` +function proj_diff(target::SparseMatrixCSC{T, <:Any}, attempt::Matrix{T}) where T + J, K, values = findnz(target) + result = zeros(size(target)) + for (j, k, val) in zip(J, K, values) + result[j, k] = val - attempt[j, k] + end + result +end + +# a type for keeping track of gradient descent history +struct DescentHistory{T} + scaled_loss::Array{T} + neg_grad::Array{Matrix{T}} + base_step::Array{Matrix{T}} + hess::Array{Hermitian{T, Matrix{T}}} + slope::Array{T} + stepsize::Array{T} + positive::Array{Bool} + backoff_steps::Array{Int64} + last_line_L::Array{Matrix{T}} + last_line_loss::Array{T} + + function DescentHistory{T}( + scaled_loss = Array{T}(undef, 0), + neg_grad = Array{Matrix{T}}(undef, 0), + hess = Array{Hermitian{T, Matrix{T}}}(undef, 0), + base_step = Array{Matrix{T}}(undef, 0), + slope = Array{T}(undef, 0), + stepsize = Array{T}(undef, 0), + positive = Bool[], + backoff_steps = Int64[], + last_line_L = Array{Matrix{T}}(undef, 0), + last_line_loss = Array{T}(undef, 0) + ) where T + new(scaled_loss, neg_grad, hess, base_step, slope, stepsize, positive, backoff_steps, last_line_L, last_line_loss) + end +end + +# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every +# explicit entry of `gram`. use gradient descent starting from `guess` +function realize_gram_gradient( + gram::SparseMatrixCSC{T, <:Any}, + guess::Matrix{T}; + scaled_tol = 1e-30, + min_efficiency = 0.5, + init_stepsize = 1.0, + backoff = 0.9, + max_descent_steps = 600, + max_backoff_steps = 110 +) where T <: Number + # start history + history = DescentHistory{T}() + + # scale tolerance + scale_adjustment = sqrt(T(nnz(gram))) + tol = scale_adjustment * scaled_tol + + # initialize variables + stepsize = init_stepsize + L = copy(guess) + + # do gradient descent + Δ_proj = proj_diff(gram, L'*Q*L) + loss = dot(Δ_proj, Δ_proj) + for _ in 1:max_descent_steps + # stop if the loss is tolerably low + if loss < tol + break + end + + # find negative gradient of loss function + neg_grad = 4*Q*L*Δ_proj + slope = norm(neg_grad) + dir = neg_grad / slope + + # store current position, loss, and slope + L_last = L + loss_last = loss + push!(history.scaled_loss, loss / scale_adjustment) + push!(history.neg_grad, neg_grad) + push!(history.slope, slope) + + # find a good step size using backtracking line search + push!(history.stepsize, 0) + push!(history.backoff_steps, max_backoff_steps) + empty!(history.last_line_L) + empty!(history.last_line_loss) + for backoff_steps in 0:max_backoff_steps + history.stepsize[end] = stepsize + L = L_last + stepsize * dir + Δ_proj = proj_diff(gram, L'*Q*L) + loss = dot(Δ_proj, Δ_proj) + improvement = loss_last - loss + push!(history.last_line_L, L) + push!(history.last_line_loss, loss / scale_adjustment) + if improvement >= min_efficiency * stepsize * slope + history.backoff_steps[end] = backoff_steps + break + end + stepsize *= backoff + end + + # [DEBUG] if we've hit a wall, quit + if history.backoff_steps[end] == max_backoff_steps + break + end + end + + # return the factorization and its history + push!(history.scaled_loss, loss / scale_adjustment) + L, history +end + +function basis_matrix(::Type{T}, j, k, dims) where T + result = zeros(T, dims) + result[j, k] = one(T) + result +end + +# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every +# explicit entry of `gram`. use Newton's method starting from `guess` +function realize_gram_newton( + gram::SparseMatrixCSC{T, <:Any}, + guess::Matrix{T}; + scaled_tol = 1e-30, + rate = 1, + max_steps = 100 +) where T <: Number + # start history + history = DescentHistory{T}() + + # find the dimension of the search space + dims = size(guess) + element_dim, construction_dim = dims + total_dim = element_dim * construction_dim + + # list the constrained entries of the gram matrix + J, K, _ = findnz(gram) + constrained = zip(J, K) + + # scale the tolerance + scale_adjustment = sqrt(T(length(constrained))) + tol = scale_adjustment * scaled_tol + + # use Newton's method + L = copy(guess) + for step in 0:max_steps + # evaluate the loss function + Δ_proj = proj_diff(gram, L'*Q*L) + loss = dot(Δ_proj, Δ_proj) + + # store the current loss + push!(history.scaled_loss, loss / scale_adjustment) + + # stop if the loss is tolerably low + if loss < tol || step > max_steps + break + end + + # find the negative gradient of loss function + neg_grad = 4*Q*L*Δ_proj + + # find the negative Hessian of the loss function + hess = Matrix{T}(undef, total_dim, total_dim) + indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim] + for (j, k) in indices + basis_mat = basis_matrix(T, j, k, dims) + neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat + neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained) + deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj) + hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim) + end + hess = Hermitian(hess) + push!(history.hess, hess) + + # compute the Newton step + step = hess \ reshape(neg_grad, total_dim) + L += rate * reshape(step, dims) + end + + # return the factorization and its history + L, history +end + +LinearAlgebra.eigen!(A::Symmetric{BigFloat, Matrix{BigFloat}}; sortby::Nothing) = + eigen!(Hermitian(A)) + +function convertnz(type, mat) + J, K, values = findnz(mat) + sparse(J, K, type.(values)) +end + +function realize_gram_optim( + gram::SparseMatrixCSC{T, <:Any}, + guess::Matrix{T} +) where T <: Number + # find the dimension of the search space + dims = size(guess) + element_dim, construction_dim = dims + total_dim = element_dim * construction_dim + + # list the constrained entries of the gram matrix + J, K, _ = findnz(gram) + constrained = zip(J, K) + + # scale the loss function + scale_adjustment = length(constrained) + + function loss(L_vec) + L = reshape(L_vec, dims) + Δ_proj = proj_diff(gram, L'*Q*L) + dot(Δ_proj, Δ_proj) / scale_adjustment + end + + function loss_grad!(storage, L_vec) + L = reshape(L_vec, dims) + Δ_proj = proj_diff(gram, L'*Q*L) + storage .= reshape(-4*Q*L*Δ_proj, total_dim) / scale_adjustment + end + + function loss_hess!(storage, L_vec) + L = reshape(L_vec, dims) + Δ_proj = proj_diff(gram, L'*Q*L) + indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim] + for (j, k) in indices + basis_mat = basis_matrix(T, j, k, dims) + neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat + neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained) + deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj) / scale_adjustment + storage[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim) + end + end + + optimize( + loss, loss_grad!, loss_hess!, + reshape(guess, total_dim), + Newton() + ) +end + +# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every +# explicit entry of `gram`. use gradient descent starting from `guess` +function realize_gram( + gram::SparseMatrixCSC{T, <:Any}, + guess::Matrix{T}, + frozen = nothing; + scaled_tol = 1e-30, + min_efficiency = 0.5, + init_rate = 1.0, + backoff = 0.9, + reg_scale = 1.1, + max_descent_steps = 200, + max_backoff_steps = 110 +) where T <: Number + # start history + history = DescentHistory{T}() + + # find the dimension of the search space + dims = size(guess) + element_dim, construction_dim = dims + total_dim = element_dim * construction_dim + + # list the constrained entries of the gram matrix + J, K, _ = findnz(gram) + constrained = zip(J, K) + + # scale the tolerance + scale_adjustment = sqrt(T(length(constrained))) + tol = scale_adjustment * scaled_tol + + # list the un-frozen indices + has_frozen = !isnothing(frozen) + if has_frozen + is_unfrozen = fill(true, size(guess)) + is_unfrozen[frozen] .= false + unfrozen = findall(is_unfrozen) + unfrozen_stacked = reshape(is_unfrozen, total_dim) + end + + # initialize variables + grad_rate = init_rate + L = copy(guess) + + # use Newton's method with backtracking and gradient descent backup + Δ_proj = proj_diff(gram, L'*Q*L) + loss = dot(Δ_proj, Δ_proj) + for step in 1:max_descent_steps + # stop if the loss is tolerably low + if loss < tol + break + end + + # find the negative gradient of loss function + neg_grad = 4*Q*L*Δ_proj + + # find the negative Hessian of the loss function + hess = Matrix{T}(undef, total_dim, total_dim) + indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim] + for (j, k) in indices + basis_mat = basis_matrix(T, j, k, dims) + neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat + neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained) + deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj) + hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim) + end + hess = Hermitian(hess) + push!(history.hess, hess) + + # regularize the Hessian + min_eigval = minimum(eigvals(hess)) + push!(history.positive, min_eigval > 0) + if min_eigval <= 0 + hess -= reg_scale * min_eigval * I + end + + # compute the Newton step + neg_grad_stacked = reshape(neg_grad, total_dim) + if has_frozen + hess = hess[unfrozen_stacked, unfrozen_stacked] + neg_grad_compressed = neg_grad_stacked[unfrozen_stacked] + else + neg_grad_compressed = neg_grad_stacked + end + base_step_compressed = hess \ neg_grad_compressed + if has_frozen + base_step_stacked = zeros(total_dim) + base_step_stacked[unfrozen_stacked] .= base_step_compressed + else + base_step_stacked = base_step_compressed + end + base_step = reshape(base_step_stacked, dims) + push!(history.base_step, base_step) + + # store the current position, loss, and slope + L_last = L + loss_last = loss + push!(history.scaled_loss, loss / scale_adjustment) + push!(history.neg_grad, neg_grad) + push!(history.slope, norm(neg_grad)) + + # find a good step size using backtracking line search + push!(history.stepsize, 0) + push!(history.backoff_steps, max_backoff_steps) + empty!(history.last_line_L) + empty!(history.last_line_loss) + rate = one(T) + step_success = false + for backoff_steps in 0:max_backoff_steps + history.stepsize[end] = rate + L = L_last + rate * base_step + Δ_proj = proj_diff(gram, L'*Q*L) + loss = dot(Δ_proj, Δ_proj) + improvement = loss_last - loss + push!(history.last_line_L, L) + push!(history.last_line_loss, loss / scale_adjustment) + if improvement >= min_efficiency * rate * dot(neg_grad, base_step) + history.backoff_steps[end] = backoff_steps + step_success = true + break + end + rate *= backoff + end + + # if we've hit a wall, quit + if !step_success + return L_last, false, history + end + end + + # return the factorization and its history + push!(history.scaled_loss, loss / scale_adjustment) + L, loss < tol, history +end + +end \ No newline at end of file diff --git a/engine-proto/gram-test/basin-shapes.jl b/engine-proto/gram-test/basin-shapes.jl new file mode 100644 index 0000000..5c03c01 --- /dev/null +++ b/engine-proto/gram-test/basin-shapes.jl @@ -0,0 +1,99 @@ +include("Engine.jl") + +using LinearAlgebra +using SparseArrays + +function sphere_in_tetrahedron_shape() + # initialize the partial gram matrix for a sphere inscribed in a regular + # tetrahedron + J = Int64[] + K = Int64[] + values = BigFloat[] + for j in 1:5 + for k in 1:5 + push!(J, j) + push!(K, k) + if j == k + push!(values, 1) + elseif (j <= 4 && k <= 4) + push!(values, -1/BigFloat(3)) + else + push!(values, -1) + end + end + end + gram = sparse(J, K, values) + + # plot loss along a slice + loss_lin = [] + loss_sq = [] + mesh = range(0.9, 1.1, 101) + for t in mesh + L = hcat( + Engine.plane(normalize(BigFloat[ 1, 1, 1]), BigFloat(1)), + Engine.plane(normalize(BigFloat[ 1, -1, -1]), BigFloat(1)), + Engine.plane(normalize(BigFloat[-1, 1, -1]), BigFloat(1)), + Engine.plane(normalize(BigFloat[-1, -1, 1]), BigFloat(1)), + Engine.sphere(BigFloat[0, 0, 0], BigFloat(t)) + ) + Δ_proj = Engine.proj_diff(gram, L'*Engine.Q*L) + push!(loss_lin, norm(Δ_proj)) + push!(loss_sq, dot(Δ_proj, Δ_proj)) + end + mesh, loss_lin, loss_sq +end + +function circles_in_triangle_shape() + # initialize the partial gram matrix for a sphere inscribed in a regular + # tetrahedron + J = Int64[] + K = Int64[] + values = BigFloat[] + for j in 1:8 + for k in 1:8 + filled = false + if j == k + push!(values, 1) + filled = true + elseif (j == 1 || k == 1) + push!(values, 0) + filled = true + elseif (j == 2 || k == 2) + push!(values, -1) + filled = true + end + #=elseif (j <= 5 && j != 2 && k == 9 || k == 9 && k <= 5 && k != 2) + push!(values, 0) + filled = true + end=# + if filled + push!(J, j) + push!(K, k) + end + end + end + append!(J, [6, 4, 6, 5, 7, 5, 7, 3, 8, 3, 8, 4]) + append!(K, [4, 6, 5, 6, 5, 7, 3, 7, 3, 8, 4, 8]) + append!(values, fill(-1, 12)) + + # plot loss along a slice + loss_lin = [] + loss_sq = [] + mesh = range(0.99, 1.01, 101) + for t in mesh + L = hcat( + Engine.plane(BigFloat[0, 0, 1], BigFloat(0)), + Engine.sphere(BigFloat[0, 0, 0], BigFloat(t)), + Engine.plane(BigFloat[1, 0, 0], BigFloat(1)), + Engine.plane(BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(1)), + Engine.plane(BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(1)), + Engine.sphere(4//3*BigFloat[-1, 0, 0], BigFloat(1//3)), + Engine.sphere(4//3*BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//3)), + Engine.sphere(4//3*BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//3)) + ) + Δ_proj = Engine.proj_diff(gram, L'*Engine.Q*L) + push!(loss_lin, norm(Δ_proj)) + push!(loss_sq, dot(Δ_proj, Δ_proj)) + end + mesh, loss_lin, loss_sq +end \ No newline at end of file diff --git a/engine-proto/gram-test/circles-in-triangle.jl b/engine-proto/gram-test/circles-in-triangle.jl new file mode 100644 index 0000000..1bd22a7 --- /dev/null +++ b/engine-proto/gram-test/circles-in-triangle.jl @@ -0,0 +1,76 @@ +include("Engine.jl") + +using SparseArrays +using Random + +# initialize the partial gram matrix for a sphere inscribed in a regular +# tetrahedron +J = Int64[] +K = Int64[] +values = BigFloat[] +for j in 1:9 + for k in 1:9 + filled = false + if j == 9 + if k <= 5 && k != 2 + push!(values, 0) + filled = true + end + elseif k == 9 + if j <= 5 && j != 2 + push!(values, 0) + filled = true + end + elseif j == k + push!(values, 1) + filled = true + elseif j == 1 || k == 1 + push!(values, 0) + filled = true + elseif j == 2 || k == 2 + push!(values, -1) + filled = true + end + if filled + push!(J, j) + push!(K, k) + end + end +end +append!(J, [6, 4, 6, 5, 7, 5, 7, 3, 8, 3, 8, 4]) +append!(K, [4, 6, 5, 6, 5, 7, 3, 7, 3, 8, 4, 8]) +append!(values, fill(-1, 12)) +#= make construction rigid +append!(J, [3, 4, 4, 5]) +append!(K, [4, 3, 5, 4]) +append!(values, fill(-0.5, 4)) +=# +gram = sparse(J, K, values) + +# set initial guess +Random.seed!(58271) +guess = hcat( + Engine.plane(BigFloat[0, 0, 1], BigFloat(0)), + Engine.sphere(BigFloat[0, 0, 0], BigFloat(1//2)) + 0.1*Engine.rand_on_shell([BigFloat(-1)]), + Engine.plane(-BigFloat[1, 0, 0], BigFloat(-1)) + 0.1*Engine.rand_on_shell([BigFloat(-1)]), + Engine.plane(-BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(-1)) + 0.1*Engine.rand_on_shell([BigFloat(-1)]), + Engine.plane(-BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(-1)) + 0.1*Engine.rand_on_shell([BigFloat(-1)]), + Engine.sphere(BigFloat[-1, 0, 0], BigFloat(1//5)) + 0.1*Engine.rand_on_shell([BigFloat(-1)]), + Engine.sphere(BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//5)) + 0.1*Engine.rand_on_shell([BigFloat(-1)]), + Engine.sphere(BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//5)) + 0.1*Engine.rand_on_shell([BigFloat(-1)]), + BigFloat[0, 0, 0, 0, 1] +) +frozen = [CartesianIndex(j, 9) for j in 1:5] + +# complete the gram matrix using Newton's method with backtracking +L, success, history = Engine.realize_gram(gram, guess, frozen) +completed_gram = L'*Engine.Q*L +println("Completed Gram matrix:\n") +display(completed_gram) +if success + println("\nTarget accuracy achieved!") +else + println("\nFailed to reach target accuracy") +end +println("Steps: ", size(history.scaled_loss, 1)) +println("Loss: ", history.scaled_loss[end], "\n") \ No newline at end of file diff --git a/engine-proto/gram-test/ganja-1.0.204.js b/engine-proto/gram-test/ganja-1.0.204.js new file mode 100644 index 0000000..1e95e42 --- /dev/null +++ b/engine-proto/gram-test/ganja-1.0.204.js @@ -0,0 +1,1913 @@ +/** Ganja.js - Geometric Algebra - Not Just Algebra. + * @author Enki + * @link https://github.com/enkimute/ganja.js + */ + +/*********************************************************************************************************************/ +// +// Ganja.js is an Algebra generator for javascript. It generates a wide variety of Algebra's and supports operator +// overloading, algebraic literals and a variety of graphing options. +// +// Ganja.js is designed with prototyping and educational purposes in mind. Clean mathematical syntax is the primary +// target. +// +// Ganja.js exports only one function called *Algebra*. This function is used to generate Algebra classes. (say complex +// numbers, minkowski or 3D CGA). The returned class can be used to create, add, multiply etc, but also to upgrade +// javascript functions with algebraic literals, operator overloading, vectors, matrices and much more. +// +// As a simple example, multiplying two complex numbers 3+2i and 1+4i could be done like this : +// +// var complex = Algebra(0,1); +// var a = new complex([3,2]); +// var b = new complex([1,3]); +// var result = a.Mul(b); +// +// But the same can be written using operator overloading and algebraic literals. (where scientific notation with +// lowercase e is overloaded to directly specify generators (e1, e2, e12, ...)) +// +// var result = Algebra(0,1,()=>(3+2e1)*(1+4e1)); +// +// Please see github for user documentation and examples. +// +/*********************************************************************************************************************/ + +// Documentation below is for implementors. I'll assume you know about Clifford Algebra's, grades, its products, etc .. +// I'll also assume you are familiar with ES6. My style may feel a bith mathematical, advise is to read slow. + +(function (name, context, definition) { + if (typeof module != 'undefined' && module.exports) module.exports = definition(); + else if (typeof define == 'function' && define.amd) define(name, definition); + else context[name] = definition(); +}('Algebra', this, function () { + +/** Some helpers for eigenvalues for bivector split in high-d spaces **/ + function QR(M) { + // helpers + const {abs,sqrt} = Math; + const hyp = (a,b)=>abs(a)>abs(b)?abs(a)*sqrt(1+(b/a)**2):b==0?0:abs(b)*sqrt(1+(a/b)**2); + const [m,n] = [M.length, M[0].length]; + var qr = M.map(r=>r.map(c=>c)), Q = M.map(r=>r.map(c=>0)), R = M.map(r=>r.map(c=>0)), d = [], k, i, j, nrm; + // helper matrix + for (k=0; k=0; --k) { + for (i=0; i{ + var res = A.map(r=>r.map(c=>0)); + for(let i=0;iA[i][i]); + } + +/** The Algebra class generator. Possible calling signatures : + * Algebra([func]) => algebra with no dimensions, i.e. R. Optional function for the translator. + * Algebra(p,[func]) => 'p' positive dimensions and an optional function to pass to the translator. + * Algebra(p,q,[func]) => 'p' positive and 'q' negative dimensions and optional function. + * Algebra(p,q,r,[func]) => 'p' positive, 'q' negative and 'r' zero dimensions and optional function. + * Algebra({ => for custom basis, cayley, mixing, etc pass in an object as first parameter. + * [p:p], => optional 'p' for # of positive dimensions + * [q:q], => optional 'q' for # of negative dimensions + * [r:r], => optional 'r' for # of zero dimensions + * [metric:array], => alternative for p,q,r. e.g. ([1,1,1,-1] for spacetime) + * [basis:array], => array of strings with basis names. (e.g. ['1','e1','e2','e12']) + * [Cayley:Cayley], => optional custom Cayley table (strings). (e.g. [['1','e1'],['e1','-1']]) + * [mix:boolean], => Allows mixing of various algebras. (for space efficiency). + * [graded:boolean], => Use a graded algebra implementation. (automatic for +6D) + * [baseType:Float32Array] => optional basetype to use. (only for flat generator) + * },[func]) => optional function for the translator. + **/ + return function Algebra(p,q,r) { + // Resolve possible calling signatures so we know the numbers for p,q,r. Last argument can always be a function. + var fu=arguments[arguments.length-1],options=p; if (options instanceof Object) { + q = (p.q || (p.metric && p.metric.filter(x=>x==-1).length))| 0; + r = (p.r || (p.metric && p.metric.filter(x=>x==0).length)) | 0; + p = p.p === undefined ? (p.metric && p.metric.filter(x=>x==1).length) || 0 : p.p || 0; + } else { options={}; p=p|0; r=r|0; q=q|0; }; + + // Support for multi-dual-algebras + if (options.dual || (p==0 && q==0 && r<0)) { r=options.dual=options.dual||-r; // Create a dual number algebra if r<0 (old) or options.dual set(new) + options.basis = [...Array(r+1)].map((a,i)=>i?'e0'+i:'1'); options.metric = [1,...Array(r)]; options.tot=r+1; + options.Cayley = [...Array(r+1)].map((a,i)=>[...Array(r+1)].map((y,j)=>i*j==0?((i+j)?'e0'+(i+j):'1'):'0')); + } + if (options.over) options.baseType = Array; + + // Calculate the total number of dimensions. + var tot = options.tot = (options.tot||(p||0)+(q||0)+(r||0)||(options.basis&&options.basis.length))|0; + + // Unless specified, generate a full set of Clifford basis names. We generate them as an array of strings by starting + // from numbers in binary representation and changing the set bits into their relative position. + // Basis names are ordered first per grade, then lexically (not cyclic!). + // For 10 or more dimensions all names will be double digits ! 1e01 instead of 1e1 .. + var basis=(options.basis&&(options.basis.length==2**tot||r<0||options.Cayley)&&options.basis)||[...Array(2**tot)] // => [undefined, undefined, undefined, undefined, undefined, undefined, undefined, undefined] + .map((x,xi)=>(((1<<30)+xi).toString(2)).slice(-tot||-1) // => ["000", "001", "010", "011", "100", "101", "110", "111"] (index of array in base 2) + .replace(/./g,(a,ai)=>a=='0'?'':String.fromCharCode(66+ai-(r!=0)))) // => ["", "3", "2", "23", "1", "13", "12", "123"] (1 bits replaced with their positions, 0's removed) + .sort((a,b)=>(a.toString().length==b.toString().length)?(a>b?1:b>a?-1:0):a.toString().length-b.toString().length) // => ["", "1", "2", "3", "12", "13", "23", "123"] (sorted numerically) + .map(x=>x&&'e'+(x.replace(/./g,x=>('0'+(x.charCodeAt(0)-65)).slice(tot>9?-2:-1) ))||'1') // => ["1", "e1", "e2", "e3", "e12", "e13", "e23", "e123"] (converted to commonly used basis names) + + // See if the basis names start from 0 or 1, store grade per component and lowest component per grade. + var low=basis.length==1?1:basis[1].match(/\d+/g)[0]*1, + grades=options.grades||(options.dual&&basis.map((x,i)=>i?1:0))||basis.map(x=>tot>9?(x.length-1)/2:x.length-1), + grade_start=grades.map((a,b,c)=>c[b-1]!=a?b:-1).filter(x=>x+1).concat([basis.length]); + + // String-simplify a concatenation of two basis blades. (and supports custom basis names e.g. e21 instead of e12) + // This is the function that implements e1e1 = +1/-1/0 and e1e2=-e2e1. The brm function creates the remap dictionary. + var simplify = (s,p,q,r)=>{ + var sign=1,c,l,t=[],f=true,ss=s.match(tot>9?/(\d\d)/g:/(\d)/g);if (!ss) return s; s=ss; l=s.length; + while (f) { f=false; + // implement Ex*Ex = metric. + for (var i=0; i=(p+r)) sign*=-1; else if ((s[i]-low)t[i+1]) { c=t[i];t[i]=t[i+1];t[i+1]=c;sign*=-1;f=true; break;} if (f) { s=t;t=[];l=s.length; } + } + var ret=(sign==0)?'0':((sign==1)?'':'-')+(t.length?'e'+t.join(''):'1'); return (brm&&brm[ret])||(brm&&brm['-'+ret]&&'-'+brm['-'+ret])||ret; + }, + brm=(x=>{ var ret={}; for (var i in basis) ret[basis[i]=='1'?'1':simplify(basis[i],p,q,r)] = basis[i]; return ret; })(basis); + + // As an alternative to the string fiddling, one can also bit-fiddle. In this case the basisvectors are represented by integers with 1 bit per generator set. + var simplify_bits = (A,B,p2)=>{ var n=p2||(p+q+r),t=0,ab=A&B,res=A^B; if (ab&((1<>1); t&=B; t^=ab>>(p+r); t^=t>>16; t^=t>>8; t^=t>>4; return [1-2*(27030>>(t&15)&1),res]; }, + bc = (v)=>{ v=v-((v>>1)& 0x55555555); v=(v&0x33333333)+((v>>2)&0x33333333); var c=((v+(v>>4)&0xF0F0F0F)*0x1010101)>>24; return c }; + + if (!options.graded && tot <= 6 || options.graded===false || options.Cayley) { + // Faster and degenerate-metric-resistant dualization. (a remapping table that maps items into their duals). + var drm=basis.map((a,i)=>{ return {a:a,i:i} }) + .sort((a,b)=>a.a.length>b.a.length?1:a.a.lengthx.i).reverse(), + drms=drm.map((x,i)=>(x==0||i==0)?1:simplify(basis[x]+basis[i])[0]=='-'?-1:1); + + /// Store the full metric (also for bivectors etc ..) + var metric = options.Cayley&&options.Cayley.map((x,i)=>x[i]) || basis.map((x,xi)=>simplify(x+x,p,q,r)|0); metric[0]=1; + + /// Generate multiplication tables for the outer and geometric products. + var mulTable = options.Cayley||basis.map(x=>basis.map(y=>(x==1)?y:(y==1)?x:simplify(x+y,p,q,r))); + + // subalgebra support. (must be bit-order basis blades, does no error checking.) + if (options.even) options.basis = basis.filter(x=>x.length%2==1); + if (options.basis && !options.Cayley && r>=0 && options.basis.length != 2**tot) { + metric = metric.filter((x,i)=>options.basis.indexOf(basis[i])!=-1); + mulTable = mulTable.filter((x,i)=>options.basis.indexOf(basis[i])!=-1).map(x=>x.filter((x,i)=>options.basis.indexOf(basis[i])!=-1)); + basis = options.basis; + } + + /// Convert Cayley table to product matrices. The outer product selects the strict sum of the GP (but without metric), the inner product + /// is the left contraction. + var gp=basis.map(x=>basis.map(x=>'0')), cp=gp.map(x=>gp.map(x=>'0')), cps=gp.map(x=>gp.map(x=>'0')), op=gp.map(x=>gp.map(x=>'0')), gpo={}; // Storage for our product tables. + basis.forEach((x,xi)=>basis.forEach((y,yi)=>{ var n = mulTable[xi][yi].replace(/^-/,''); if (!gpo[n]) gpo[n]=[]; gpo[n].push([xi,yi]); })); + basis.forEach((o,oi)=>{ + gpo[o].forEach(([xi,yi])=>op[oi][xi]=(grades[oi]==grades[xi]+grades[yi])?((mulTable[xi][yi]=='0')?'0':((mulTable[xi][yi][0]!='-')?'':'-')+'b['+yi+']*this['+xi+']'):'0'); + gpo[o].forEach(([xi,yi])=>{ + gp[oi][xi] =((gp[oi][xi]=='0')?'':gp[oi][xi]+'+') + ((mulTable[xi][yi]=='0')?'0':((mulTable[xi][yi][0]!='-')?'':'-')+'b['+yi+']*this['+xi+']'); + cp[oi][xi] =((cp[oi][xi]=='0')?'':cp[oi][xi]+'+') + ((grades[oi]==grades[yi]-grades[xi])?gp[oi][xi]:'0'); + cps[oi][xi]=((cps[oi][xi]=='0')?'':cps[oi][xi]+'+') + ((grades[oi]==Math.abs(grades[yi]-grades[xi]))?gp[oi][xi]:'0'); + }); + }); + + /// Flat Algebra Multivector Base Class. + var generator = class MultiVector extends (options.baseType||Float32Array) { + /// constructor - create a floating point array with the correct number of coefficients. + constructor(a) { super(a||basis.length); return this; } + + /// grade selection - return a only the part of the input with the specified grade. + Grade(grade,res) { res=res||new this.constructor(); for (var i=0,l=res.length; i1e-10) res.push(((this[i]==1)&&i?'':((this[i]==-1)&&i)?'-':(this[i].toFixed(10)*1))+(i==0?'':tot==1&&q==1?'i':basis[i].replace('e','e_'))); return res.join('+').replace(/\+-/g,'-')||'0'; } + + /// Reversion, Involutions, Conjugation for any number of grades, component acces shortcuts. + get Negative (){ var res = new this.constructor(); for (var i=0; ia[drm[i]]*drms[i]); var res = new this.constructor(); res[res.length-1]=1; return this.Mul(res); }; + get UnDual (){ if (r) return this.map((x,i,a)=>a[drm[i]]*drms[a.length-i-1]); var res = new this.constructor(); res[res.length-1]=1; return this.Div(res); }; + get Length (){ return options.over?Math.sqrt(Math.abs(this.Mul(this.Conjugate).s.s)):Math.sqrt(Math.abs(this.Mul(this.Conjugate).s)); }; + get VLength (){ var res = 0; for (var i=0; i'res['+xi+']=b['+xi+']+this['+xi+']').join(';\n').replace(/(b|this)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';\nreturn res') + generator.prototype.Scale = new Function('b,res','res=res||new this.constructor();\n'+basis.map((x,xi)=>'res['+xi+']=b*this['+xi+']').join(';\n').replace(/(b|this)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';\nreturn res') + generator.prototype.Sub = new Function('b,res','res=res||new this.constructor();\n'+basis.map((x,xi)=>'res['+xi+']=this['+xi+']-b['+xi+']').join(';\n').replace(/(b|this)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';\nreturn res') + generator.prototype.Mul = new Function('b,res','res=res||new this.constructor();\n'+gp.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\+\-/g,'-').replace(/(\w*?)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a).replace(/\+0/g,'')+';').join('\n')+'\nreturn res;'); + generator.prototype.LDot = new Function('b,res','res=res||new this.constructor();\n'+cp.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\+\-/g,'-').replace(/\+0/g,'').replace(/(\w*?)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\n')+'\nreturn res;'); + generator.prototype.Dot = new Function('b,res','res=res||new this.constructor();\n'+cps.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\+\-/g,'-').replace(/\+0/g,'').replace(/(\w*?)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\n')+'\nreturn res;'); + generator.prototype.Wedge = new Function('b,res','res=res||new this.constructor();\n'+op.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\+\-/g,'-').replace(/\+0/g,'').replace(/(\w*?)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\n')+'\nreturn res;'); +// generator.prototype.Vee = new Function('b,res','res=res||new this.constructor();\n'+op.map((r,ri)=>'res['+drm[ri]+']='+r.map(x=>x.replace(/\[(.*?)\]/g,function(a,b){return '['+(drm[b|0])+']'})).join('+').replace(/\+\-/g,'-').replace(/\+0/g,'').replace(/(\w*?)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\n')+'\nreturn res;'); + /// Conforms to the new Chapter 11 now. + generator.prototype.Vee = new Function('b,res',('res=res||new this.constructor();\n'+op.map((r,ri)=>'res['+drm[ri]+']='+drms[ri]+'*('+r.map(x=>x.replace(/\[(.*?)\]/g,function(a,b){return '['+(drm[b|0])+']'+(drms[b|0]>0?"":"*-1")})).join('+').replace(/\+\-/g,'-').replace(/\+0/g,'').replace(/(\w*?)\[(.*?)\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+');').join('\n')+'\nreturn res;').replace(/(b\[)|(this\[)/g,a=>a=='b['?'this[':'b[')); + generator.prototype.eigenValues = eigenValues; + + /// Add getter and setters for the basis vectors/bivectors etc .. + basis.forEach((b,i)=>Object.defineProperty(generator.prototype, i?b:'s', { + configurable: true, get(){ return this[i] }, set(x){ this[i]=x; } + })); + + /// Graded generator for high-dimensional algebras. + } else { + + /// extra graded lookups. + var basisg = grade_start.slice(0,grade_start.length-1).map((x,i)=>basis.slice(x,grade_start[i+1])); + var counts = grade_start.map((x,i,a)=>i==a.length-1?0:a[i+1]-x).slice(0,tot+1); + var basis_bits = basis.map(x=>x=='1'?0:x.slice(1).match(tot>9?/\d\d/g:/\d/g).reduce((a,b)=>a+(1<<(b-low)),0)), + bits_basis = []; basis_bits.forEach((b,i)=>bits_basis[b]=i); + var metric = basisg.map((x,xi)=>x.map((y,yi)=>simplify_bits(basis_bits[grade_start[xi]+yi],basis_bits[grade_start[xi]+yi])[0])); + var drms = basisg.map((x,xi)=>x.map((y,yi)=>simplify_bits(basis_bits[grade_start[xi]+yi],(~basis_bits[grade_start[xi]+yi])&((1<(typeof x=="string")?"-"+x:-x):undefined):this[i]; + else { if (r[i]==undefined) r[i]=[]; for(var j=0,m=Math.max(this[i].length,b[i].length);jx&&x.map(y=>typeof y=="string"?y+"*"+s:y*s)); } + + // geometric product. + Mul(b,r) { + r=r||new this.constructor(); var gotstring=false; + for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],ig.map(e=>e&&(!(e+'').match(/-{0,1}\w+/))?'('+e+')':e)) + return r; + } + // outer product. + Wedge(b,r) { + r=r||new this.constructor(); + for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],ix).sort((a,b)=>((a.match(/\d+/)[0]|0)-(b.match(/\d+/)[0]|0))||((a.match(/\d+$/)[0]|0)-(b.match(/\d+$/)[0]|0))).map(x=>x.replace(/\/\/\d+$/,'')); + var r2 = 'float sum=0.0; float res=0.0;\n', g=0; + r.forEach(x=>{ + var cg = x.match(/\d+/)[0]|0; + if (cg != g) r2 += "sum += res*res;\nres = 0.0;\n"; + r2 += x.replace(/\[\d+\]/,'') + '\n'; + g=cg; + }); + r2+= "sum += res*res;\n"; + return r2; + } + // Inner product glsl output. + IPNS_GLSL(b,point_source) { + var r='',count=0,curg; + for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],ix).sort((a,b)=>((a.match(/\d+/)[0]|0)-(b.match(/\d+/)[0]|0))||((a.match(/\d+$/)[0]|0)-(b.match(/\d+$/)[0]|0))).map(x=>x.replace(/\/\/\d+$/,'')); + var r2 = 'float sum=0.0; float res=0.0;\n', g=0; + r.forEach(x=>{ + var cg = x.match(/\d+/)[0]|0; + if (cg != g) r2 += "sum += res*res;\nres = 0.0;\n"; + r2 += x.replace(/\[\d+\]/,'') + '\n'; + g=cg; + }); + r2+= "sum += res*res;\n"; + return r2; + } + // Left contraction. + LDot(b,r) { + r=r||new this.constructor(); + for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],ig&&g.map((c,ci)=>!c?undefined:((c+'').match(/[\+\-\*]/)?'('+c+')':c)+(gi==0?"":basisg[gi][ci])).filter(x=>x).join('+')).filter(x=>x).join('+').replace(/\+\-/g,'-')||"0"; } + get s () { if (this[0]) return this[0][0]||0; return 0; } + get Length () { var res=0; this.forEach((g,gi)=>g&&g.forEach((e,ei)=>res+=(e||0)**2*metric[gi][ei])); return Math.abs(res)**.5; } + get VLength () { var res=0; this.forEach((g,gi)=>g&&g.forEach((e,ei)=>res+=(e||0)**2)); return Math.abs(res)**.5; } + get Reverse () { var r=new this.constructor(); this.forEach((x,gi)=>x&&x.forEach((e,ei)=>{if(!r[gi])r[gi]=[]; r[gi][ei] = this[gi][ei]*[1,1,-1,-1][gi%4]; })); return r; } + get Involute () { var r=new this.constructor(); this.forEach((x,gi)=>x&&x.forEach((e,ei)=>{if(!r[gi])r[gi]=[]; r[gi][ei] = this[gi][ei]*[1,-1,1,-1][gi%4]; })); return r; } + get Conjugate () { var r=new this.constructor(); this.forEach((x,gi)=>x&&x.forEach((e,ei)=>{if(!r[gi])r[gi]=[]; r[gi][ei] = this[gi][ei]*[1,-1,-1,1][gi%4]; })); return r; } + get Dual() { var r=new this.constructor(); this.forEach((g,gi)=>{ if (!g) return; r[tot-gi]=[]; g.forEach((e,ei)=>r[tot-gi][counts[gi]-1-ei]=drms[gi][ei]*e); }); return r; } + get Normalized () { return this.Scale(1/this.Length); } + } + + + // This generator is UNDER DEVELOPMENT - I'm publishing it so I can test on observable. + } + + // Generate a new class for our algebra. It extends the javascript typed arrays (default float32 but can be specified in options). + var res = class Element extends generator { + + // constructor - create a floating point array with the correct number of coefficients. + constructor(a) { super(a); if (this.upgrade) this.upgrade(); return this; } + + // Grade selection. (implemented by parent class). + Grade(grade,res) { res=res||new Element(); return super.Grade(grade,res); } + + // Right and Left divide - Defined on the elements, shortcuts to multiplying with the inverse. + Div (b,res) { return this.Mul(b.Inverse,res); } + LDiv (b,res) { return b.Inverse.Mul(this,res); } + + + // Bivector split - we handle all real cases, still have to add the complex cases for those exception scenarios. + Split (iter=50) { + var k = Math.floor((p+q+r)/2), OB = this.map(x=>x), B = this.map(x=>x), m = 1; + var Wi = [...Array(k)].map((r,i)=>{ m = m*(i+1); var Wi = B.Scale(1/m); B = B.Wedge(OB); return Wi; }); + if (k<3) { // The quadratic case is easy to solve. (for spaces <6D) + var TDT = this.Dot(this).s, TWT = this.Wedge(this); + if (TWT.VLength < 1E-5) return [this]; // bivector was simple. + var D = 0.5*Math.sqrt( TDT**2 - TWT.Mul(TWT).s ); + var eigen = [0.5*TDT + D, 0.5*TDT - D].sort((a,b)=>Math.abs(a)6D, closed form solutions of the characteristic polyn. are impossible, use eigenvalues of companion matrix. + var Wis = Wi.map((W,i)=>W.Mul(W).s*(-1)**(k-i+(k%2)) ); + var matrix = [...Array(k)].map((r,i)=>[...Array(k)].map((c,j)=>(j == k-1)?Wis[k-i-1]:(i-1==j)?1:0)); + var eigen = eigenValues(matrix,iter).sort((a,b)=>Math.abs(a){ + var r = Math.floor(k/2), N = Element.Scalar(0), DN = Element.Scalar(0); + for (var i=0; i<=r; ++i) { N.Add( Wi[2*i+1].Scale(v**(r-i)), N); DN.Add( Wi[2*i].Scale(v**(r-i)), DN); } + if (DN.VLength == 0) return Element.Scalar(0); + var ret = N.Div(DN); sum.Add(ret, sum); return ret; + }); + return [this.Sub(sum),...res]; // Smallest eigvalue becomes B-rest + } + + // Factorize a motor + Factorize (iter=50) { + var S = this.Grade(2).Split(iter); + var P = this.Scale(1); + // if (P.s) { + var R = S.slice(0,S.length-1).map((Si,i)=>{ + var Mi = Element.Scalar(P.s).Add(Si); + var scale = Math.sqrt(Mi.Reverse.Mul(Mi).s); + return Mi.Scale(1/scale); + }); + R.push( R.reduce((tot,fact)=>tot.Mul(fact.Reverse), Element.Scalar(1)).Mul(P) ); + // } + return R; + } + + // exp - closed form exp. + Exp (taylor = false) { + if (r==1 && tot<=4 && Math.abs(this[0])<1E-9 && !options.over && !taylor) { + // https://www.researchgate.net/publication/360528787_Normalization_Square_Roots_and_the_Exponential_and_Logarithmic_Maps_in_Geometric_Algebras_of_Less_than_6D + // 0 1 2 3 4 5 + // 5 6 7 8 9 10 + var l = (this[8]*this[8] + this[9]*this[9] + this[10]*this[10]); + if (l==0) return new Element([1, 0,0,0,0, this[5], this[6], this[7], 0, 0, 0, 0,0,0,0, 0]); + var m = (this[5]*this[10] + this[6]*this[9] + this[7]*this[8]), a = Math.sqrt(l), c = Math.cos(a), s = Math.sin(a)/a, t = m/l*(c-s); + var test = Element.Element(c, 0,0,0,0, s*this[5] + t*this[10], s*this[6] + t*this[9], s*this[7] + t*this[8], s*this[8], s*this[9], s*this[10], 0,0,0,0, m*s); + //return test; // tbc .. investigate pss coeff?? + + var u = Math.sqrt(Math.abs(this.Dot(this).s)); if (Math.abs(u)<1E-5) return this.Add(Element.Scalar(1)); + var v = this.Wedge(this).Scale(-1/(2*u)); + var res2 = Element.Add(Element.Sub(Math.cos(u),v.Scale(Math.sin(u))),Element.Div(Element.Mul((Element.Add(Math.sin(u),v.Scale(Math.cos(u)))),this),(Element.Add(u,v)))); + //if ([...test].map(x=>x.toFixed(1))+'' != [...res2].map(x=>x.toFixed(1))+'') { console.log(test, res2); debugger } + + return res2; + } + if (!taylor && Math.abs(this[0])<1E-9 && !options.over) { + return this.Grade(2).Split().reduce((total,simple)=>{ + var square = simple.Mul(simple).s, len = Math.sqrt(Math.abs(square)); + if (len <= 1E-5) return total.Mul(Element.Scalar(1).Add(simple)); + if (square < 0) return total.Mul(Element.Scalar(Math.cos(len)).Add(simple.Scale(Math.sin(len)/len)) ); + return total.Mul(Element.Scalar(Math.cosh(len)).Add(simple.Scale(Math.sinh(len)/len)) ); + },Element.Scalar(1)); + } + if (options.dual) { var f=Math.exp(this.s); return this.map((x,i)=>i?x*f:f); } + var res = Element.Scalar(1), y=1, M= this.Scale(1), N=this.Scale(1); for (var x=1; x<15; x++) { res=res.Add(M.Scale(1/y)); M=M.Mul(N); y=y*(x+1); }; + return res; + } + + // Log - only for up to 3D PGA for now + Log (compat = false) { + if (options.over) return; + if (!compat) { + if (tot==4 && q==0 && r==1 && !options.over) { // https://www.researchgate.net/publication/360528787_Normalization_Square_Roots_and_the_Exponential_and_Logarithmic_Maps_in_Geometric_Algebras_of_Less_than_6D + if (Math.abs(this.s)>=.99999) return Element.Bivector(this[5],this[6],this[7],0,0,0).Scale(Math.sign(this.s)); + var a = 1/(1 - this[0]*this[0]), b = Math.acos(this[0])*Math.sqrt(a), c = a*this[15]*(1 - this[0]*b); + return Element.Bivector( c*this[10] + b*this[5], -c*this[9] + b*this[6], c*this[8] + b*this[7], b*this[8], b*this[9], b*this[10] ); + } + return this.Factorize().reduce((sum,bi)=>{ + var [ci,si] = [bi.s, bi.Grade(2)]; + var square = si.Mul(si).s; + var len = Math.sqrt(Math.abs(square)); + if (Math.abs(square) < 1E-5) return sum.Add(si); + if (square < 0) return sum.Add(si.Scale(Math.acos(ci)/len)); + return sum.Add(si.Scale(Math.acosh(ci)/len)); + },Element.Scalar(0)); + } + var b = this.Grade(2), bdb = Element.Dot(b,b).s; + if (Math.abs(bdb)<=1E-5) return this.s<0?b.Scale(-1):b; + var s = Math.sqrt(-bdb), bwb = Element.Wedge(b,b); + if (Math.abs(bwb[bwb.length-1])<=1E-5 || Math.abs(this.s)<=1E-5) return b.Scale(Math.atan2(s,this.s)/s); + var p = bwb.Scale(-1/(2*s)); + return Element.Mul(Element.Mul((Element.Add(Math.atan2(s,this.s),p.Scale(1/this.s))),b),Element.Sub(s,p)).Scale(1/(s*s)); + } + + // Helper for efficient inverses. (custom involutions - negates grades in arguments). + Map () { var res=new Element(); return super.Map(res,...arguments); } + + // Factories - Make it easy to generate vectors, bivectors, etc when using the functional API. None of the examples use this but + // users that have used other GA libraries will expect these calls. The Coeff() is used internally when translating algebraic literals. + static Element() { return new Element([...arguments]); }; + static Coeff() { return (new Element()).Coeff(...arguments); } + static Scalar(x) { return (new Element()).Coeff(0,x); } + static Vector() { return (new Element()).nVector(1,...arguments); } + static Bivector() { return (new Element()).nVector(2,...arguments); } + static Trivector() { return (new Element()).nVector(3,...arguments); } + static nVector(n) { return (new Element()).nVector(...arguments); } + + // Static operators. The parser will always translate operators to these static calls so that scalars, vectors, matrices and other non-multivectors can also be handled. + // The static operators typically handle functions and matrices, calling through to element methods for multivectors. They are intended to be flexible and allow as many + // types of arguments as possible. If performance is a consideration, one should use the generated element methods instead. (which only accept multivector arguments) + static toEl(x) { if (x instanceof Function) x=x(); if (!(x instanceof Element)) x=Element.Scalar(x); return x; } + + // Addition and subtraction. Subtraction with only one parameter is negation. + static Add(a,b,res) { + // Resolve expressions passed in. + while(a.call)a=a(); while(b.call)b=b(); if (a.Add && b.Add) return a.Add(b,res); + // If either is a string, the result is a string. + if ((typeof a=='string')||(typeof b=='string')) return a.toString()+b.toString(); + // If only one is an array, add the other element to each of the elements. + if ((a instanceof Array && !a.Add)^(b instanceof Array && !b.Add)) return (a instanceof Array)?a.map(x=>Element.Add(x,b)):b.map(x=>Element.Add(a,x)); + // If both are equal length arrays, add elements one-by-one + if ((a instanceof Array)&&(b instanceof Array)&&a.length==b.length) return a.map((x,xi)=>Element.Add(x,b[xi])); + // If they're both not elements let javascript resolve it. + if (!(a instanceof Element || b instanceof Element)) return a+b; + // Here we're left with scalars and multivectors, call through to generated code. + a=Element.toEl(a); b=Element.toEl(b); return a.Add(b,res); + } + + static Sub(a,b,res) { + // Resolve expressions passed in. + while(a.call)a=a(); while(b&&b.call) b=b(); if (a.Sub && b && b.Sub) return a.Sub(b,res); + // If only one is an array, add the other element to each of the elements. + if (b&&((a instanceof Array)^(b instanceof Array))) return (a instanceof Array)?a.map(x=>Element.Sub(x,b)):b.map(x=>Element.Sub(a,x)); + // If both are equal length arrays, add elements one-by-one + if (b&&(a instanceof Array)&&(b instanceof Array)&&a.length==b.length) return a.map((x,xi)=>Element.Sub(x,b[xi])); + // Negation + if (arguments.length==1) return Element.Mul(a,-1); + // If none are elements here, let js do it. + if (!(a instanceof Element || b instanceof Element)) return a-b; + // Here we're left with scalars and multivectors, call through to generated code. + a=Element.toEl(a); b=Element.toEl(b); return a.Sub(b,res); + } + + // The geometric product. (or matrix*matrix, matrix*vector, vector*vector product if called with 1D and 2D arrays) + static Mul(a,b,res) { + // Resolve expressions + while(a.call&&!a.length)a=a(); while(b.call&&!b.length)b=b(); if (a.Mul && b.Mul) return a.Mul(b,res); + // still functions -> experimental curry style (dont use this.) + if (a.call && b.call) return (ai,bi)=>Element.Mul(a(ai),b(bi)); + // scalar mul. + if (Number.isFinite(a) && b.Scale) return b.Scale(a); else if (Number.isFinite(b) && a.Scale) return a.Scale(b); + // Handle matrices and vectors. + if ((a instanceof Array)&&(b instanceof Array)) { + // vector times vector performs a dot product. (which internally uses the GP on each component) + if((!(a[0] instanceof Array) || (a[0] instanceof Element)) &&(!(b[0] instanceof Array) || (b[0] instanceof Element))) { var r=tot?Element.Scalar(0):0; a.forEach((x,i)=>r=Element.Add(r,Element.Mul(x,b[i]),r)); return r; } + // Array times vector + if(!(b[0] instanceof Array)) return a.map((x,i)=>Element.Mul(a[i],b)); + // Array times Array + var r=a.map((x,i)=>b[0].map((y,j)=>{ var r=tot?Element.Scalar(0):0; x.forEach((xa,k)=>r=Element.Add(r,Element.Mul(xa,b[k][j]))); return r; })); + // Return resulting array or scalar if 1 by 1. + if (r.length==1 && r[0].length==1) return r[0][0]; else return r; + } + // Only one is an array multiply each of its elements with the other. + if ((a instanceof Array)^(b instanceof Array)) return (a instanceof Array)?a.map(x=>Element.Mul(x,b)):b.map(x=>Element.Mul(a,x)); + // Try js multiplication, else call through to geometric product. + var r=a*b; if (!isNaN(r)) return r; + a=Element.toEl(a); b=Element.toEl(b); return a.Mul(b,res); + } + + // The inner product. (default is left contraction). + static LDot(a,b,res) { + // Expressions + while(a.call)a=a(); while(b.call)b=b(); //if (a.LDot) return a.LDot(b,res); + // Map elements in array + if (b instanceof Array && !(a instanceof Array)) return b.map(x=>Element.LDot(a,x)); + if (a instanceof Array && !(b instanceof Array)) return a.map(x=>Element.LDot(x,b)); + // js if numbers, else contraction product. + if (!(a instanceof Element || b instanceof Element)) return a*b; + a=Element.toEl(a);b=Element.toEl(b); return a.LDot(b,res); + } + + // The symmetric inner product. (default is left contraction). + static Dot(a,b,res) { + // Expressions + while(a.call)a=a(); while(b.call)b=b(); //if (a.LDot) return a.LDot(b,res); + // js if numbers, else contraction product. + if (!(a instanceof Element || b instanceof Element)) return a|b; + a=Element.toEl(a);b=Element.toEl(b); return a.Dot(b,res); + } + + // The outer product. (Grassman product - no use of metric) + static Wedge(a,b,res) { + // normal behavior for booleans/numbers + if (typeof a in {boolean:1,number:1} && typeof b in {boolean:1,number:1}) return a^b; + // Expressions + while(a.call)a=a(); while(b.call)b=b(); if (a.Wedge) return a.Wedge(Element.toEl(b),res); + // The outer product of two vectors is a matrix .. internally Mul not Wedge ! + if (a instanceof Array && b instanceof Array) return a.map(xa=>b.map(xb=>Element.Mul(xa,xb))); + // js, else generated wedge product. + if (!(a instanceof Element || b instanceof Element)) return a*b; + a=Element.toEl(a);b=Element.toEl(b); return a.Wedge(b,res); + } + + // The regressive product. (Dual of the outer product of the duals). + static Vee(a,b,res) { + // normal behavior for booleans/numbers + if (typeof a in {boolean:1,number:1} && typeof b in {boolean:1,number:1}) return a&b; + // Expressions + while(a.call)a=a(); while(b.call)b=b(); if (a.Vee) return a.Vee(Element.toEl(b),res); + // js, else generated vee product. (shortcut for dual of wedge of duals) + if (!(a instanceof Element || b instanceof Element)) return 0; + a=Element.toEl(a);b=Element.toEl(b); return a.Vee(b,res); + } + + // The sandwich product. Provided for convenience (>>> operator) + static sw(a,b) { + // Skip strings/colors + if (typeof b == "string" || typeof b =="number") return b; + // Expressions + while(a.call)a=a(); while(b.call)b=b(); if (a.sw) return a.sw(b); + // Map elements in array + if (b instanceof Array && !b.Add) return b.map(x=>Element.sw(a,x)); + // Call through. no specific generated code for it so just perform the muls. + a=Element.toEl(a); b=Element.toEl(b); return a.Mul(b).Mul(a.Reverse); + } + + // Division - scalars or cal through to element method. + static Div(a,b,res) { + // Expressions + while(a.call)a=a(); while(b.call)b=b(); + // For DDG experiments, I'll include a quick cholesky on matrices here. (vector/matrix) + if ((a instanceof Array) && (b instanceof Array) && (b[0] instanceof Array)) { + // factor + var R = b.flat(), i, j, k, sum, i_n, j_n, n=b[0].length, s=new Array(n), x=new Array(n), yi; + for (i=0;i=0; i--) for (x[i] /= R[i*n+i], j=i+1; ja.map((r,ri)=>Element.Conjugate(a[ri][ci]))); return Element.toEl(a).Conjugate; } + static Normalize(a) { return Element.toEl(a).Normalized; }; + static Length(a) { return Element.toEl(a).Length }; + + // Comparison operators always use length. Handle expressions, then js or length comparison + static eq(a,b) { if (!(a instanceof Element)||!(b instanceof Element)) return a==b; while(a.call)a=a(); while(b.call)b=b(); for (var i=0; i(b instanceof Element?b.Length:b); } + static lte(a,b) { while(a.call)a=a(); while(b.call)b=b(); return (a instanceof Element?a.Length:a)<=(b instanceof Element?b.Length:b); } + static gte(a,b) { while(a.call)a=a(); while(b.call)b=b(); return (a instanceof Element?a.Length:a)>=(b instanceof Element?b.Length:b); } + + // Debug output and printing multivectors. + static describe(x) { if (x===true) console.log(`Basis\n${basis}\nMetric\n${metric.slice(1,1+tot)}\nCayley\n${mulTable.map(x=>(x.map(x=>(' '+x).slice(-2-tot)))).join('\n')}\nMatrix Form:\n`+gp.map(x=>x.map(x=>x.match(/(-*b\[\d+\])/)).map(x=>x&&((x[1].match(/-/)||' ')+String.fromCharCode(65+1*x[1].match(/\d+/)))||' 0')).join('\n')); return {basis:basisg||basis,metric,mulTable,matrix:gp.map(x=>x.map(x=>x.replace(/\*this\[.+\]/,'').replace(/b\[(\d+)\]/,(a,x)=>(metric[x]==-1||metric[x]==0&&grades[x]>1&&(-1)**grades[x]==(metric[basis.indexOf(basis[x].replace('0',''))]||(-1)**grades[x])?'-':'')+basis[x]).replace('--','')))} } + + // Direct sum of algebras - experimental + static sum(B){ + var A = Element; + // Get the multiplication tabe and basis. + var T1 = A.describe().mulTable, T2 = B.describe().mulTable; + var B1 = A.describe().basis, B2 = B.describe().basis; + // Get the maximum index of T1, minimum of T2 and rename T2 if needed. + var max_T1 = B1.filter(x=>x.match(/e/)).map(x=>x.match(/\d/g)).flat().map(x=>x|0).sort((a,b)=>b-a)[0]; + var max_T2 = B2.filter(x=>x.match(/e/)).map(x=>x.match(/\d/g)).flat().map(x=>x|0).sort((a,b)=>b-a)[0]; + var min_T2 = B2.filter(x=>x.match(/e/)).map(x=>x.match(/\d/g)).flat().map(x=>x|0).sort((a,b)=>a-b)[0]; + // remapping .. + T2 = T2.map(x=>x.map(y=>y.match(/e/)?y.replace(/(\d)/g,(x)=>(x|0)+max_T1):y.replace("1","e"+(1+max_T2+max_T1)))); + B2 = B2.map((y,i)=>i==0?y.replace("1","e"+(1+max_T2+max_T1)):y.replace(/(\d)/g,(x)=>(x|0)+max_T1)); + // Build the new basis and multable.. + var basis = [...B1,...B2]; + var Cayley = T1.map((x,i)=>[...x,...T2[0].map(x=>"0")]).concat(T2.map((x,i)=>[...T1[0].map(x=>"0"),...x])) + // Build the new algebra. + var grades = [...B1.map(x=>x=="1"?0:x.length-1),...B2.map((x,i)=>i?x.length-1:0)]; + var a = Algebra({basis,Cayley,grades,tot:Math.log2(B1.length)+Math.log2(B2.length)}) + // And patch up .. + a.Scalar = function(x) { + var res = new a(); + for (var i=0; i function of 1 parameter will be called with that parameter from -1 to 1 and graphed on a canvas. Returned values should also be in the [-1 1] range + // graph(function(x,y)) => functions of 2 parameters will be called from -1 to 1 on both arguments. Returned values can be 0-1 for greyscale or an array of three RGB values. + // graph(array) => array of algebraic elements (points, lines, circles, segments, texts, colors, ..) is graphed. + // graph(function=>array) => same as above, for animation scenario's this function is called each frame. + // An optional second parameter is an options object { width, height, animate, camera, scale, grid, canvas } + static graph(f,options) { + // Store the original input + if (!f) return; var origf=f; + // generate default options. + options=options||{}; options.scale=options.scale||1; options.camera=options.camera||(tot!=4?Element.Scalar(1): ( Element.Bivector(0,0,0,0,0,options.p||0).Exp() ).Mul( Element.Bivector(0,0,0,0,options.h||0,0).Exp()) ); + if (options.conformal && tot==4) var ni = options.ni||this.Coeff(4,1,3,1), no = options.no||this.Coeff(4,0.5,3,-0.5), minus_no = no.Scale(-1); + var ww=options.width, hh=options.height, cvs=options.canvas, tpcam=new Element([0,0,0,0,0,0,0,0,0,0,0,-5,0,0,1,0]),tpy=this.Coeff(4,1),tp=new Element(), + // project 3D to 2D. This allows to render 3D and 2D PGA with the same code. + project=(o)=>{ if (!o) return o; while (o.call) o=o(); +// if (o instanceof Element && o.length == 32) o = new Element([o[0],o[1],o[2],o[3],o[4],o[6],o[7],o[8],o[10],o[11],o[13],o[16],o[17],o[19],o[22],o[26]]); + // Clip 3D lines so they don't go past infinity. + if (o instanceof Element && o.length == 16 && o[8]**2+o[9]**2+o[10]**2>0.0001) { + o = [[options.clip||2,1,0,0],[-(options.clip||2),1,0,0],[options.clip||2,0,1,0],[-(options.clip||2),0,1,0],[options.clip||2,0,0,1],[-(options.clip||2),0,0,1]].map(v=>{ + var r = Element.Vector(...v).Wedge(o); return r[14]?r.Scale(1/r[14], r):undefined; + }).filter(x=>x && Math.abs(x[13])<= (options.clip||2)+0.001 && Math.abs(x[12]) <= (options.clip||2)+0.001 && Math.abs(x[11]) <= (options.clip||2) + 0.001).slice(0,2); + return o.map(o=>(tpcam).Vee(options.camera.Mul(o).Mul(options.camera.Conjugate)).Wedge(tpy)); + } + // Convert 3D planes to polies. + if (o instanceof Element && o.length == 16 && o.Grade(1).Length>0.01) { + var m = Element.Add(1, Element.Mul(o.Normalized, Element.Coeff(3,1))).Normalized, e0 = 0; + o=Element.sw(m,[[-1,-1],[-1,1],[1,1],[-1,-1],[1,1],[1,-1]].map(([x,z])=>Element.Trivector(x*o.Length,e0,z*o.Length,1))); + return o.map(o=>(tpcam).Vee(options.camera.Mul(o).Mul(options.camera.Conjugate)).Wedge(tpy)); + } + return (tot==4 && o instanceof Element && o.length==16)?(tpcam).Vee(options.camera.Mul(o).Mul(options.camera.Conjugate)).Wedge(tpy):(o.length==2**tot)?Element.sw(options.camera,o):o; + }; + // gl escape. + if (options.gl && !(tot==4 && options.conformal)) return Element.graphGL(f,options); if (options.up) return Element.graphGL2(f,options); + // if we get an array or function without parameters, we render c2d or p2d SVG points/lines/circles/etc + if (!(f instanceof Function) || f.length===0) { + // Our current cursor, color, animation state and 2D mapping. + var lx,ly,lr,color,res,anim=false,to2d=(tot==5)?[0, 8, 11, 13, 19, 17, 22, 26]:(tot==3)?[0,1,2,3,4,5,6,7]:[0,7,9,10,13,12,14,15]; + // Make sure we have an array of elements. (if its an object, convert to array with elements and names.) + if (f instanceof Function) f=f(); if (!(f instanceof Array)) f=[].concat.apply([],Object.keys(f).map((k)=>typeof f[k]=='number'?[f[k]]:[f[k],k])); + // The build function generates the actual SVG. It will be called everytime the user interacts or the anim flag is set. + function build(f,or) { + // Make sure we have an aray. + if (or && f && f instanceof Function) f=f(); + // Reset position and color for cursor. + lx=-2;ly=options.conformal?-1.85:1.85;lr=0;color='#444'; + // Create the svg element. (master template string till end of function) + var svg=new DOMParser().parseFromString(` + ${// Add a grid (option) + options.grid?(()=>{ + if (tot==4 && !options.conformal) { + const lines3d = (n,from,to,j,l=0, ox=0, oy=0, alpha=1)=>[``,...[...Array(n+1)].map((x,i)=>{ + var f=from.map((x,i)=>x*(i==3?1:(options.gridSize||1))), t=to.map((x,i)=>x*(i==3?1:(options.gridSize||1))); f[j] = t[j] = (i-(n/2))/(n/2) * (options.gridSize||1); + var D3a = Element.Trivector(...f), D2a = project(D3a), D3b = Element.Trivector(...t), D2b = project(D3b); + var lx=options.scale*D2a[drm[2]]/D2a[drm[1]]; if (drm[1]==6||drm[1]==14) lx*=-1; var ly=-options.scale*D2a[drm[3]]/D2a[drm[1]]; + var lx2=options.scale*D2b[drm[2]]/D2b[drm[1]]; if (drm[1]==6||drm[1]==14) lx2*=-1; var ly2=-options.scale*D2b[drm[3]]/D2b[drm[1]]; + var r = ``; + if (l && i && i!= n) r += `${((from[j]<0?-1:1)*(i-(n/2))/(n/2)*(options.gridSize||1)).toFixed(1)}` + return r; + }),'']; + var front = Element.sw(options.camera,Element.Trivector(1,0,0,0)).Dual.Dot(Element.Vector(0,0,0,1)).s, ff = front>0?1:-1; + var left = Element.sw(options.camera,Element.Trivector(0,0,1,0)).Dual.Dot(Element.Vector(0,0,0,1)).s, ll = left>0?1:-1; + var fa = Math.max(0,Math.min(1,5*Math.abs(left))), la = Math.max(0,Math.min(1,5*Math.abs(front))); + return [ + ...lines3d(20,[-1,-1,-1,1],[1,-1,1,1],2,options.labels?ff:0, 0, 0.05), + ...lines3d(20,[-1,-1,-1,1],[1,-1,1,1],0,options.labels?ll:0, 0, 0.05), + ...lines3d(20,[-1,-1,ll,1],[1,1,ll,1],0,0,0,0,fa), + ...lines3d(20,[-1,1,ll,1],[1,-1,ll,1],1,!options.labels?0:(ff!=-1)?1:2, ll*ff*-0.05, 0, fa), + ...lines3d(20,[ff,1,-1,1],[ff,-1,1,1],1,!options.labels?0:(ll!=-1)?1:2, ll*ff*0.05, 0, la), + ...lines3d(20,[ff,-1,-1,1],[ff,1,1,1],2,0,0,0,la), + ].join(''); + } + const s = options.scale, n = (10/s)|0, cx = options.camera.e02, cy = options.camera.e01, alpha = Math.min(1,(s-0.2)*10); if (options.scale<0.1) return; + const lines = (n,dir,space,width,color)=>[...Array(2*n+1)].map((x,xi)=>``) + return [``,...lines(n*2,0,0.2,0.005,'#DDD'),...lines(n*2,1,0.2,0.005,'#DDD'),...lines(n,0,1,0.005,'#AAA'),...lines(n,1,1,0.005,'#AAA'),...lines(n,0,5,0.005,'#444'),...lines(n,1,5,0.005,'#444')] + .concat(options.labels?[...Array(4*n+1)].map((x,xi)=>(xi-n*2==0)?``:`${((xi-n*2)*0.2).toFixed(1)}`):[]) + .concat(options.labels?[...Array(4*n+1)].map((x,xi)=>`${((xi-n*2)*-0.2).toFixed(1)}`):[]).join('')+''; + })():''} + // Handle conformal 2D elements. + ${options.conformal?f.map&&f.map((o,oidx)=>{ + // Optional animation handling. + if((o==Element.graph && or!==false)||(oidx==0&&options.animate&&or!==false)) { anim=true; requestAnimationFrame(()=>{var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; }); if (!options.animate) return; } + // Resolve expressions passed in. + while (o.call) o=o(); + if (options.ipns && o instanceof Element) o = o.Dual; + var sc = options.scale; + var lineWidth = options.lineWidth || 1; + var pointRadius = options.pointRadius || 1; + var dash_for_r2 = (r2, render_r, target_width) => { + // imaginary circles are dotted + if (r2 >= 0) return 'none'; + var half_circum = render_r*Math.PI; + var width = half_circum / Math.max(Math.round(half_circum / target_width), 1); + return `${width} ${width}`; + }; + // Arrays are rendered as segments or polygons. (2 or more elements) + if (o instanceof Array) { lx=ly=lr=0; o=o.map(o=>{ while(o.call)o=o(); return o.Scale(-1/o.Dot(ni).s); }); o.forEach((o)=>{lx+=sc*(o.e1);ly+=sc*(-o.e2)});lx/=o.length;ly/=o.length; return o.length>2?``:``; } + // Allow insertion of literal svg strings. + if (typeof o =='string' && o[0]=='<') { return o; } + // Strings are rendered at the current cursor position. + if (typeof o =='string') { var res2=(o[0]=='_')?'':` ${o} `; ly+=0.14; return res2; } + // Numbers change the current color. + if (typeof o =='number') { color='#'+(o+(1<<25)).toString(16).slice(-6); return ''; }; + // All other elements are rendered .. + var ni_part = o.Dot(no.Scale(-1)); // O_i + n_o O_oi + var no_part = ni.Scale(-1).Dot(o); // O_o + O_oi n_i + if (ni_part.VLength * 1e-6 > no_part.VLength) { + // direction or dual - nothing to render + return ""; + } + var no_ni_part = no_part.Dot(no.Scale(-1)); // O_oi + var no_only_part = ni.Wedge(no_part).Dot(no.Scale(-1)); // O_o + + /* Note: making 1e-6 smaller increases the maximum circle radius before they are drawn as lines */ + if (no_ni_part.VLength * 1e-6 > no_only_part.VLength) { + var is_flat = true; + var direction = no_ni_part; + } + else { + var is_flat = false; + var direction = no_only_part; + } + // normalize to make the direction unitary + var dl = direction.Length; + o = o.Scale(1/dl); + direction = direction.Scale(1/dl) + + var b0=direction.Grade(0).VLength>0.001,b1=direction.Grade(1).VLength>0.001,b2=direction.Grade(2).VLength>0.001; + if (!is_flat && b0 && !b1 && !b2) { + // Points + if (direction.s < 0) { o = Element.Sub(o); } + lx=sc*(o.e1); ly=sc*(-o.e2); lr=0; return res2=``; + } else if (is_flat && !b0 && b1 && !b2) { + // Lines. + var loc=minus_no.LDot(o).Div(o), att=ni.Dot(o); + lx=sc*(-loc.e1); ly=sc*(loc.e2); lr=Math.atan2(-o[14],o[13])/Math.PI*180; return ``; + } else if (!is_flat && !b0 && !b1 && b2) { + // Circles + var loc=o.Div(ni.LDot(o)); lx=sc*(-loc.e1); ly=sc*(loc.e2); + var r2=o.Mul(o.Conjugate).s; + var r = Math.sqrt(Math.abs(r2))*sc; + return ``; + } else if (!is_flat && !b0 && b1 && !b2) { + // Point Pairs. + lr=0; var ei=ni,eo=no, nix=o.Wedge(ei), sqr=o.LDot(o).s/nix.LDot(nix).s, r=Math.sqrt(Math.abs(sqr)), attitude=((ei.Wedge(eo)).LDot(nix)).Normalized.Mul(Element.Scalar(r)), pos=o.Div(nix); pos=pos.Div( pos.LDot(Element.Sub(ei))); + if (nix==0) { pos = o.Dot(Element.Coeff(4,-1)); sqr=-1; } + lx=sc*(pos.e1); ly=sc*(-pos.e2); + if (sqr==0) return ``; + // Draw imaginary pairs hollow + if (sqr > 0) var fill = color||'green', stroke = 'none', dash_array = 'none'; + else var fill = 'none', stroke = color||'green'; + lx=sc*(pos.e1+attitude.e1); ly=sc*(-pos.e2-attitude.e2); + var res2=``; + lx=sc*(pos.e1-attitude.e1); ly=sc*(-pos.e2+attitude.e2); + return res2+``; + } else { + /* Unrecognized */ + return ""; + } + // Handle projective 2D and 3D elements. + }):f.map&&f.map((o,oidx)=>{ if((o==Element.graph && or!==false)||(oidx==0&&options.animate&&or!==false)) { anim=true; requestAnimationFrame(()=>{var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; }); if (!options.animate) return; } while (o instanceof Function) o=o(); o=(o instanceof Array)?o.map(project):project(o); if (o===undefined) return; + // dual option dualizes before render + if (options.dual && o instanceof Element) o = o.Dual; + // line segments and polygons + if (o instanceof Array && o.length>1) { lx=ly=lr=0; o.forEach((o)=>{while (o.call) o=o(); lx+=options.scale*((drm[1]==6||drm[1]==14)?-1:1)*o[drm[2]]/o[drm[1]];ly+=options.scale*o[drm[3]]/o[drm[1]]});lx/=o.length;ly/=o.length; return o.length>2?``:``; } + // svg + if (typeof o =='string' && o[0]=='<') { return o; } + // Labels + if (typeof o =='string') { var res2=(o[0]=='_')?'':` ${o} `; ly-=0.14; return res2; } + // Colors + if (typeof o =='number') { color='#'+(o+(1<<25)).toString(16).slice(-6); return ''; }; + // Points + if (o[to2d[6]]**2 >0.0001) { lx=options.scale*o[drm[2]]/o[drm[1]]; if (drm[1]==6||drm[1]==14) lx*=-1; ly=options.scale*o[drm[3]]/o[drm[1]]; lr=0; var res2=``; ly+=0.05; lx-=0.1; return res2; } + // Lines + if (o[to2d[2]]**2+o[to2d[3]]**2>0.0001) { var l=Math.sqrt(o[to2d[2]]**2+o[to2d[3]]**2); o[to2d[2]]/=l; o[to2d[3]]/=l; o[to2d[1]]/=l; lx=0.5; ly=options.scale*((drm[1]==6)?-1:-1)*o[to2d[1]]; lr=-Math.atan2(o[to2d[2]],o[to2d[3]])/Math.PI*180; var res2=``; ly+=0.05; return res2; } + // Vectors + if (o[to2d[4]]**2+o[to2d[5]]**2>0.0001) { lr=0; ly+=0.05; lx+=0.1; var res2=``; ly=ly+o.e01/4*3-0.05; lx=lx-o.e02/4*3; return res2; } + }).join()}`,'text/html').body; + // return the inside of the created svg element. + return svg.removeChild(svg.firstChild); + }; + // Create the initial svg and install the mousehandlers. + res=build(f); res.value=f; res.options=options; res.setAttribute("stroke-width",options.lineWidth*0.005||0.005); + res.remake = (animate)=>{ options.animate = animate; if (animate) { var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; }; return res;}; + //onmousedown="if(evt.target==this)this.sel=undefined" + var mousex,mousey,cammove=false; + res.onwheel=(e)=>{ e.preventDefault(); options.scale = Math.min(5,Math.max(0.1,(options.scale||1)-e.deltaY*0.0001)); if (!anim) {var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; } } + res.onmousedown=(e)=>{ if (e.target == res) res.sel=undefined; mousex = e.clientX; mousey = e.clientY; cammove = true; } + res.onmousemove=(e)=>{ + if (cammove && tot==4 && !options.conformal) { + if (!e.buttons) { cammove=false; return; }; + var [dx,dy] = [(options.scale || 1)*(e.clientX - mousex)*3, 3*(options.scale || 1)*(e.clientY - mousey)]; + [mousex,mousey] = [e.clientX,e.clientY]; + if (res.sel !== undefined && f[res.sel].set) { + var [cw,ch] = [res.clientWidth, res.clientHeight]; + var ox = (1/(options.scale || 1)) * ((e.offsetX / cw) - 0.5) * (cw>ch?(cw/ch):1); + var oy = (1/(options.scale || 1)) * ((e.offsetY / ch) - 0.5) * (ch>cw?(ch/cw):1); + var tb = Element.sw(options.camera,f[res.sel]); + var z = -(tb.e012/tb.e123+5)/5*4; tb.e023 = ox*z*tb.e123; tb.e013 = oy*z*tb.e123; + f[res.sel].set(Element.sw(options.camera.Reverse, tb)); + //f[res.sel].set( Element.sw(Element.sw(options.camera.Reverse,Element.Bivector(-dx/res.clientWidth,dy/res.clientHeight,0,0,0,0).Exp()),f[res.sel]) ); + } else { + options.h = (options.h||0) + dx/300; + options.p = (options.p||0) - dy/600; + if (options.camera) options.camera.set( ( Element.Bivector(0,0,0,0,0,options.p).Exp() ).Mul( Element.Bivector(0,0,0,0,options.h,0).Exp() )/*.Mul(options.camera)*/ ) + } + if (!anim) {var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; } + return; + } + if (res.sel===undefined || f[res.sel] == undefined || f[res.sel].set == undefined || !e.buttons) return; + var resx=res.getBoundingClientRect().width,resy=res.getBoundingClientRect().height, + x=((e.clientX-res.getBoundingClientRect().left)/(resx/4||128)-2)*(resx>resy?resx/resy:1),y=((e.clientY-res.getBoundingClientRect().top)/(resy/4||128)-2)*(resy>resx?resy/resx:1); + x/=options.scale;y/=options.scale; + if (options.conformal) { f[res.sel].set(this.Coeff(1,x,2,-y).Add(no).Add(ni.Scale(0.5*(x*x+y*y))) ) } + else { f[res.sel][drm[2]]=((drm[1]==6)?-x:x)-((tot<4)?2*options.camera.e01:0); f[res.sel][drm[3]]=-y+((tot<4)?2*options.camera.e02:0); f[res.sel][drm[1]]=1; f[res.sel].set(f[res.sel].Normalized)} + if (!anim) {var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; } + res.dispatchEvent(new CustomEvent('input')) }; + return res; + } + // 1d and 2d functions are rendered on a canvas. + cvs=cvs||document.createElement('canvas'); if(ww)cvs.width=ww; if(hh)cvs.height=hh; var w=cvs.width,h=cvs.height,context=cvs.getContext('2d'), data=context.getImageData(0,0,w,h); + // Grid support for the canvas. + const [x_from,x_to,y_from,y_to]=options.range||[-1,1,1,-1]; + function drawGrid() { + const [X,Y]=[x=>(x-x_from)*w/(x_to-x_from),y=>(y-y_from)*h/(y_to-y_from)] + context.strokeStyle = "#008800"; context.lineWidth = 1; + // X and Y axis + context.beginPath(); + context.moveTo(X(x_from), Y(0)); context.lineTo(X(x_to ), Y(0)); context.stroke(); + context.moveTo(X(0), Y(y_from)); context.lineTo(X(0), Y(y_to )); context.stroke(); + // Draw ticks + context.strokeStyle = "#00FF00"; context.lineWidth = 2; context.font = "10px Arial"; context.fillStyle = "#448844"; + for (var i=x_from,j=y_from,ii=0; ii<=10; ++ii) { + context.beginPath(); j+= (y_to-y_from)/10; i+=(x_to-x_from)/10; + context.moveTo(X(i), Y(-(y_to - y_from)/200)); context.lineTo(X(i), Y((y_to - y_from)/200)); context.stroke(); + if(i.toFixed(1)!=0) context.fillText(i.toFixed(1), X(i-(x_to-x_from)/100), Y(-(y_to-y_from)/40)); + context.moveTo(X((x_to-x_from)/200), Y(j)); context.lineTo(X(-(x_to-x_from)/200), Y(j)); context.stroke(); + if(j.toFixed(1)!=0) context.fillText(j.toFixed(1), X((x_to-x_from)/100), Y(j)); + } + } + // two parameter functions .. evaluate for both and set resulting color. + if (f.length==2) for (var px=0; pxx*255).concat([255]),py*w*4+px*4); } + // one parameter function.. go over x range, use result as y. + else if (f.length==1) for (var px=0; px 0 && res < h-1) data.data.set([0,0,0,255],res*w*4+px*4); } + context.putImageData(data,0,0); + if (f.length == 1 || f.length == 2) if (options.grid) drawGrid(); + return cvs; + } + + // webGL2 Graphing function. (for OPNS/IPNS implicit 2D and 1D surfaces in 3D space). + static graphGL2(f,options) { + // Create canvas, get webGL2 context. + var canvas=document.createElement('canvas'); canvas.style.width=options.width||''; canvas.style.height=options.height||''; canvas.style.backgroundColor='#EEE'; + if (options.width && options.width.match && options.width.match(/px/i)) canvas.width = parseFloat(options.width)*(options.devicePixelRatio||devicePixelRatio||1); if (options.height && options.height.match && options.height.match(/px/i)) canvas.height = parseFloat(options.height)*(options.devicePixelRatio||devicePixelRatio||1); + var gl=canvas.getContext('webgl2',{alpha:options.alpha||false,preserveDrawingBuffer:true,antialias:true,powerPreference:'high-performance'}); + var gl2=!!gl; if (!gl) gl=canvas.getContext('webgl',{alpha:options.alpha||false,preserveDrawingBuffer:true,antialias:true,powerPreference:'high-performance'}); + gl.clearColor(240/255,240/255,240/255,1.0); gl.enable(gl.DEPTH_TEST); if (!gl2) { gl.getExtension("EXT_frag_depth"); gl.va = gl.getExtension('OES_vertex_array_object'); } + else gl.va = { createVertexArrayOES : gl.createVertexArray.bind(gl), bindVertexArrayOES : gl.bindVertexArray.bind(gl), deleteVertexArrayOES : gl.deleteVertexArray.bind(gl) } + // Compile vertex and fragment shader, return program. + var compile=(vs,fs)=>{ + var s=[gl.VERTEX_SHADER,gl.FRAGMENT_SHADER].map((t,i)=>{ + var r=gl.createShader(t); gl.shaderSource(r,[vs,fs][i]); gl.compileShader(r); + return gl.getShaderParameter(r, gl.COMPILE_STATUS)&&r||console.error(gl.getShaderInfoLog(r)); + }); + var p = gl.createProgram(); gl.attachShader(p, s[0]); gl.attachShader(p, s[1]); gl.linkProgram(p); + gl.getProgramParameter(p, gl.LINK_STATUS)||console.error(gl.getProgramInfoLog(p)); + return p; + }; + // Create vertex array and buffers, upload vertices and optionally texture coordinates. + var createVA=function(vtx) { + var r = gl.va.createVertexArrayOES(); gl.va.bindVertexArrayOES(r); + var b = gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b); + gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(vtx), gl.STATIC_DRAW); + gl.vertexAttribPointer(0, 3, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(0); + return {r,b} + }, + // Destroy Vertex array and delete buffers. + destroyVA=function(va) { + if (va.b) gl.deleteBuffer(va.b); if (va.r) gl.va.deleteVertexArrayOES(va.r); + } + // Drawing function + var M=[1,0,0,0,0,1,0,0,0,0,1,0,0,0,5,1]; + var draw=function(p, tp, vtx, color, color2, ratio, texc, va, b,color3,r,g){ + gl.useProgram(p); gl.uniformMatrix4fv(gl.getUniformLocation(p, "mv"),false,M); + gl.uniformMatrix4fv(gl.getUniformLocation(p, "p"),false, [5,0,0,0,0,5*(ratio||1),0,0,0,0,1,2,0,0,-1,0]) + gl.uniform3fv(gl.getUniformLocation(p, "color"),new Float32Array(color)); + gl.uniform3fv(gl.getUniformLocation(p, "color2"),new Float32Array(color2)); + if (color3) gl.uniform3fv(gl.getUniformLocation(p, "color3"),new Float32Array(color3)); + if (b) gl.uniform1fv(gl.getUniformLocation(p, "b"),(new Float32Array(counts[g])).map((x,i)=>b[g][i]||0)); + if (texc) gl.uniform1i(gl.getUniformLocation(p, "texc"),0); + if (r) gl.uniform1f(gl.getUniformLocation(p,"ratio"),r); + var v; if (!va) v = createVA(vtx); else gl.va.bindVertexArrayOESOES(va.r); + gl.drawArrays(tp, 0, (va&&va.tcount)||vtx.length/3); + if (v) destroyVA(v); + } + // Compile the OPNS renderer. (sphere tracing) + var programs = [], genprog = grade=>compile(`${gl2?"#version 300 es":""} + ${gl2?"in":"attribute"} vec4 position; ${gl2?"out":"varying"} vec4 Pos; uniform mat4 mv; uniform mat4 p; + void main() { Pos=mv*position; gl_Position = p*Pos; }`, + `${!gl2?"#extension GL_EXT_frag_depth : enable":"#version 300 es"} + precision highp float; + uniform vec3 color; uniform vec3 color2; + uniform vec3 color3; uniform float b[${counts[grade]}]; + uniform float ratio; ${gl2?"out vec4 col;":""} + ${gl2?"in":"varying"} vec4 Pos; + float product_len (in float z, in float y, in float x, in float[${counts[grade]}] b) { + ${this.nVector(options.up.length>tot?2:1,[])[options.IPNS?"IPNS_GLSL":"OPNS_GLSL"](this.nVector(grade,[]), options.up)} + return sqrt(abs(sum)); + } + vec3 find_root (in vec3 start, vec3 dir, in float thresh) { + vec3 orig=start; + float lastd = 1000.0; + const int count=${(options.maxSteps||80)}; + for (int i=0; i0.0) { + vec3 n = normalize(vec3( + product_len(d2[0]+h,d2[1],d2[2],b)-product_len(d2[0]-h,d2[1],d2[2],b), + product_len(d2[0],d2[1]+h,d2[2],b)-product_len(d2[0],d2[1]-h,d2[2],b), + product_len(d2[0],d2[1],d2[2]+h,b)-product_len(d2[0],d2[1],d2[2]-h,b) + )); + ${gl2?"gl_FragDepth":"gl_FragDepthEXT"} = dl2/50.0; + ${gl2?"col":"gl_FragColor"} = vec4(max(0.2,abs(dot(n,normalize(L-d2))))*color3 + pow(abs(dot(n,normalize(normalize(L-d2)+dir))),100.0),1.0); + } else discard; + }`),genprog2D = grade=>compile(`${gl2?"#version 300 es":""} + ${gl2?"in":"attribute"} vec4 position; ${gl2?"out":"varying"} vec4 Pos; uniform mat4 mv; uniform mat4 p; + void main() { Pos=mv*position; gl_Position = p*Pos; }`, + `${!gl2?"#extension GL_EXT_frag_depth : enable":"#version 300 es"} + precision highp float; + uniform vec3 color; uniform vec3 color2; + uniform vec3 color3; uniform float b[${counts[grade]}]; + uniform float ratio; ${gl2?"out vec4 col;":""} + ${gl2?"in":"varying"} vec4 Pos; + float product_len (in float z, in float y, in float x, in float[${counts[grade]}] b) { + ${this.nVector(1,[])[options.IPNS?"IPNS_GLSL":"OPNS_GLSL"](this.nVector(grade,[]), options.up)} + return sqrt(abs(sum)); + } + void main() { + vec3 p = -5.0*normalize(color2) -Pos[0]/5.0*color + color2 + vec3(0.0,Pos[1]/5.0*ratio,0.0); + float d2 = 1.0 - 150.0*pow(product_len( p[0]*5.0, p[1]*5.0, p[2]*5.0, b),2.0); + if (d2>0.0) { + ${gl2?"col":"gl_FragColor"} = vec4(color3,d2); + } else discard; + }`) + // canvas update will (re)render the content. + var armed=0; + canvas.update = (x)=>{ + // Start by updating canvas size if needed and viewport. + var s = getComputedStyle(canvas); if (s.width) { canvas.width = parseFloat(s.width)*(options.devicePixelRatio||devicePixelRatio||1); canvas.height = parseFloat(s.height)*(options.devicePixelRatio||devicePixelRatio||1); } + gl.viewport(0,0, canvas.width|0,canvas.height|0); var r=canvas.width/canvas.height; + // Defaults, resolve function input + var a,p=[],l=[],t=[],c=[.5,.5,.5],alpha=0,lastpos=[-2,2,0.2]; gl.clear(gl.COLOR_BUFFER_BIT+gl.DEPTH_BUFFER_BIT); while (x.call) x=x(); + // Loop over all items to render. + for (var i=0,ll=x.length;i>>24)&0xff)/255; c[0]=((e>>>16)&0xff)/255; c[1]=((e>>>8)&0xff)/255; c[2]=(e&0xff)/255; } + if (e instanceof Element){ + var tt = options.spin?-performance.now()*options.spin/1000:-options.h||0; tt+=Math.PI/2; var r = canvas.height/canvas.width; + var g=tot-1; while(!e[g]&&g>1) g--; + if (!programs[tot-1-g]) programs[tot-1-g] = (options.up.find(x=>x.match&&x.match("z")))?genprog(g):genprog2D(g); + gl.enable(gl.BLEND); gl.blendFunc(gl.ONE, gl.ONE_MINUS_SRC_ALPHA); + draw(programs[tot-1-g],gl.TRIANGLES,[-2,-2,0,-2,2,0,2,-2,0,-2,2,0,2,-2,0,2,2,0],[Math.cos(tt),0,-Math.sin(tt)],[Math.sin(tt),0,Math.cos(tt)],undefined,undefined,undefined,e,c,r,g); + gl.disable(gl.BLEND); + } + } + // if we're no longer in the page .. stop doing the work. + armed++; if (document.body.contains(canvas)) armed=0; if (armed==2) return; + canvas.value=x; if (options&&!options.animate) canvas.dispatchEvent(new CustomEvent('input')); + if (options&&options.animate) { requestAnimationFrame(canvas.update.bind(canvas,f,options)); } + if (options&&options.still) { canvas.value=x; canvas.dispatchEvent(new CustomEvent('input')); canvas.im.width=canvas.width; canvas.im.height=canvas.height; canvas.im.src = canvas.toDataURL(); } + } + // Basic mouse interactivity. needs more love. + var sel=-1; canvas.oncontextmenu = canvas.onmousedown = (e)=>{ e.preventDefault(); e.stopPropagation(); sel=-2; + var rc = canvas.getBoundingClientRect(), mx=(e.x-rc.left)/(rc.right-rc.left)*2-1, my=((e.y-rc.top)/(rc.bottom-rc.top)*-4+2)*canvas.height/canvas.width; + canvas.onwheel=e=>{e.preventDefault(); e.stopPropagation(); options.z = (options.z||5)+e.deltaY/100; if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options));} + canvas.onmouseup=e=>sel=-1; canvas.onmouseleave=e=>sel=-1; + canvas.onmousemove=(e)=>{ + var rc = canvas.getBoundingClientRect(); + var mx =(e.movementX)/(rc.right-rc.left)*2, my=((e.movementY)/(rc.bottom-rc.top)*-2)*canvas.height/canvas.width; + if (sel==-2) { options.h = (options.h||0)+mx; if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options)); return; }; if (sel < 0) return; + } + } + canvas.value = f.call?f():f; canvas.options = options; + if (options&&options.still) { + var i=new Image(); canvas.im = i; return requestAnimationFrame(canvas.update.bind(canvas,f,options)),i; + } else return requestAnimationFrame(canvas.update.bind(canvas,f,options)),canvas; + + } + + + // webGL Graphing function. (for parametric defined objects) + static graphGL(f,options) { + // Create a canvas, webgl2 context and set some default GL options. + var canvas=document.createElement('canvas'); canvas.style.width=options.width||''; canvas.style.height=options.height||''; canvas.style.backgroundColor='#EEE'; + if (options.width && options.width.match && options.width.match(/px/i)) canvas.width = parseFloat(options.width); if (options.height && options.height.match && options.height.match(/px/i)) canvas.height = parseFloat(options.height); + var gl=canvas.getContext('webgl',{alpha:options.alpha||false,antialias:true,preserveDrawingBuffer:options.still||true,powerPreference:'high-performance'}); + gl.enable(gl.DEPTH_TEST); gl.depthFunc(gl.LEQUAL); if (!options.alpha) gl.clearColor(240/255,240/255,240/255,1.0); gl.getExtension("OES_standard_derivatives"); gl.va=gl.getExtension("OES_vertex_array_object"); + // Compile vertex and fragment shader, return program. + var compile=(vs,fs)=>{ + var s=[gl.VERTEX_SHADER,gl.FRAGMENT_SHADER].map((t,i)=>{ + var r=gl.createShader(t); gl.shaderSource(r,[vs,fs][i]); gl.compileShader(r); + return gl.getShaderParameter(r, gl.COMPILE_STATUS)&&r||console.error(gl.getShaderInfoLog(r)); + }); + var p = gl.createProgram(); gl.attachShader(p, s[0]); gl.attachShader(p, s[1]); gl.linkProgram(p); + gl.getProgramParameter(p, gl.LINK_STATUS)||console.error(gl.getProgramInfoLog(p)); + return p; + }; + // Create vertex array and buffers, upload vertices and optionally texture coordinates. + var createVA=function(vtx, texc, idx, clr) { + var r = gl.va.createVertexArrayOES(); gl.va.bindVertexArrayOES(r); + var b = gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b); + gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(vtx), gl.STATIC_DRAW); + gl.vertexAttribPointer(0, 3, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(0); + if (texc){ + var b2=gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b2); + gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(texc), gl.STATIC_DRAW); + gl.vertexAttribPointer(1, 2, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(1); + } + if (clr){ + var b3=gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b3); + gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(clr), gl.STATIC_DRAW); + gl.vertexAttribPointer(texc?2:1, 3, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(texc?2:1); + } + if (idx) { + var b4=gl.createBuffer(); gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, b4); + gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, new Uint16Array(idx), gl.STATIC_DRAW); + } + return {r,b,b2,b4,b3} + }, + // Destroy Vertex array and delete buffers. + destroyVA=function(va) { + [va.b,va.b2,va.b4,va.b3].forEach(x=>{if(x) gl.deleteBuffer(x)}); if (va.r) gl.va.deleteVertexArrayOES(va.r); + } + // Default modelview matrix, convert camera to matrix (biquaternion->matrix) + var M=[1,0,0,0,0,1,0,0,0,0,1,0,0,0,5,1], mtx = (x,iscam=true)=>{ var t=options.spin?performance.now()*options.spin/1000:-options.h||0, t2=options.p||0; + var ct = Math.cos(t), st= Math.sin(t), ct2 = Math.cos(t2), st2 = Math.sin(t2), xx=options.posx||0, y=options.posy||0, z=options.posz||0, zoom=options.z||5; + if (tot==5) return [ct,st*-st2,st*ct2,0,0,ct2,st2,0,-st,ct*-st2,ct*ct2,0,xx*ct+z*-st,y*ct2+(xx*st+z*ct)*-st2,y*st2+xx*st+z*ct*ct2+zoom,1]; + x=x.Normalized; var y=x.Mul(x.Dual),X=x.e23,Y=-x.e13,Z=-x.e12,W=x.s; + var xx = X*X, xy = X*Y, xz = X*Z, xw = X*W, yy = Y*Y, yz = Y*Z, yw = Y*W, zz = Z*Z, zw = Z*W; + var mtx = [ 1-2*(yy+zz), 2*(xy+zw), 2*(xz-yw), 0, 2*(xy-zw), 1-2*(xx+zz), 2*(yz+xw), 0, 2*(xz+yw), 2*(yz-xw), 1-2*(xx+yy), 0, -2*y.e23, -2*y.e13, 2*y.e12+(iscam?5:0), 1]; + return mtx; + } + // Render the given vertices. (autocreates/destroys vertex array if not supplied). + var draw=function(p, tp, vtx, color, color2, ratio, texc, va, cbuf, allowcull=true){ + gl.useProgram(p); gl.uniformMatrix4fv(gl.getUniformLocation(p, "mv"),false,M); + gl.uniformMatrix4fv(gl.getUniformLocation(p, "p"),false, [5,0,0,0,0,5*(ratio||2),0,0,0,0,1,2,0,0,-1,0]) + gl.uniform3fv(gl.getUniformLocation(p, "color"),new Float32Array(color)); + gl.uniform3fv(gl.getUniformLocation(p, "color2"),new Float32Array(color2)); + //if (texc) gl.uniform1i(gl.getAttribLocation(p, "texc"),0); + var v; if (!va) v = createVA(vtx, texc, undefined, cbuf, p); else gl.va.bindVertexArrayOES(va.r); + if (options.cull && allowcull) gl.enable(gl.CULL_FACE); + if (va && va.b4) { + gl.drawElements(tp, va.tcount, gl.UNSIGNED_SHORT, 0); + } else { + gl.drawArrays(tp, 0, (va&&va.tcount)||vtx.length/3); + } + if (v) destroyVA(v); + if (options.cull) gl.disable(gl.CULL_FACE); + } + // Program for the geometry. Derivative based normals. Basic lambert shading. + var program = compile(`attribute vec4 position; varying vec4 Pos; uniform mat4 mv; uniform mat4 p; + void main() { gl_PointSize=12.0; Pos=mv*position; gl_Position = p*Pos; }`, + `#extension GL_OES_standard_derivatives : enable + precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos; + void main() { vec3 ldir = normalize(Pos.xyz - vec3(2.0,2.0,-4.0)); + vec3 normal = normalize(cross(dFdx(Pos.xyz), dFdy(Pos.xyz))); float l=dot(normal,ldir); + vec3 E = normalize(-Pos.xyz); vec3 R = normalize(reflect(ldir,normal)); + gl_FragColor = vec4(max(0.0,l)*color+vec3(0.5*pow(max(dot(R,E),0.0),20.0))+color2, 1.0); }`); + var programSphere = compile(`attribute vec4 position; varying vec4 Pos; varying vec3 N; uniform mat4 mv; uniform mat4 p; + void main() { gl_PointSize=12.0; Pos=mv*position; N = normalize(position.xzy); gl_Position = p*Pos; }`, + `#extension GL_OES_standard_derivatives : enable + precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos; varying vec3 N; + void main() { vec3 ldir = normalize(Pos.xyz - vec3(2.0,2.0,-4.0)); + vec3 normal = N; float l=dot(normal,ldir); + vec3 E = normalize(-Pos.xyz); vec3 R = normalize(reflect(ldir,normal)); + gl_FragColor = vec4(max(0.0,l)*color+vec3(0.5*pow(max(dot(R,E),0.0),20.0))+color2, 1.0); }`); + var programPoint = compile(`attribute vec4 position; varying vec4 Pos; uniform mat4 mv; uniform mat4 p; + void main() { gl_PointSize=${((options.pointRadius||1)*(options.devicePixelRatio||devicePixelRatio||1)*8.0).toFixed(2)}; Pos=mv*position; gl_Position = p*Pos; }`, + `precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos; + void main() { float distanceToCenter = length(gl_PointCoord - vec2(0.5)); if (distanceToCenter>0.5) discard; + gl_FragColor = vec4(color+color2, (distanceToCenter<0.5?1.0:0.0)); }`); + var programline = compile(` + attribute vec4 position; // current point. + attribute vec2 texc; // x = +w or -w, alternating. y = opacity. + attribute vec4 col; // next point. (extrapolated for end point) + uniform vec3 color; // r=aspect g=thickness + uniform mat4 mv,p; // modelview and projection matrix + varying vec2 tc; + void main() { + // Convert to clipspace. + vec4 cp = p*mv*vec4(position.xyz,1.0); + vec2 cs = cp.xy / abs(cp.w); + vec4 np = p*mv*vec4(col.xyz,1.0); + vec2 ns = np.xy / abs(np.w); + // compensate aspect + cs.x *= color.r; + ns.x *= color.r; + // clipspace line direction. + vec2 dir = normalize(cs-ns); + // Calculate screenspace normal. + vec2 normal = vec2( -dir.y, dir.x); + // Line scaling and aspect fix. + normal *= color.g * 5.0; + normal.x /= color.r; + // Pass through texture coordinates for edge softening + tc = vec2(texc.x / abs(texc.x), texc.y); + gl_Position = cp + vec4(normal*texc.x,0.0,0.0); + }`, + `precision highp float; + uniform vec3 color2; + varying vec2 tc; + void main() { +// gl_FragColor = vec4(abs(tc.x),abs(tc.x),abs(tc.x),1.0-abs(tc.x)); + gl_FragColor = vec4(color2,(1.0-pow(abs(tc.x),2.0))*tc.y); + }`); + var programcol = compile(`attribute vec4 position; attribute vec3 col; varying vec3 Col; varying vec4 Pos; uniform mat4 mv; uniform mat4 p; + void main() { gl_PointSize=6.0; Pos=mv*position; gl_Position = p*Pos; Col=col; }`, + `#extension GL_OES_standard_derivatives : enable + precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos; varying vec3 Col; + void main() { vec3 ldir = normalize(Pos.xyz - vec3(1.0,1.0,2.0)); + vec3 normal = normalize(cross(dFdx(Pos.xyz), dFdy(Pos.xyz))); float l=dot(normal,ldir); + vec3 E = normalize(-Pos.xyz); vec3 R = normalize(reflect(ldir,normal)); + gl_FragColor = vec4(max(0.3,l)*Col+vec3(pow(max(dot(R,E),0.0),20.0))+color2, 1.0); ${options.shader||''} }`); + var programmot = compile(`attribute vec4 position; attribute vec2 texc; attribute vec3 col; varying vec3 Col; varying vec4 Pos; uniform mat4 mv; uniform mat4 p; uniform vec3 color2; + void main() { gl_PointSize=2.0; float blend=fract(color2.x+texc.r)*0.5; Pos=mv*(position*(1.0-blend) + (blend)*vec4(col,1.0)); gl_Position = p*Pos; Col=vec3(length(col-position.xyz)*1.); gl_PointSize = 8.0 - Col.x; Col.y=sin(blend*2.*3.1415); }`, + `precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos; varying vec3 Col; + void main() { float distanceToCenter = length(gl_PointCoord - vec2(0.5));gl_FragColor = vec4(1.0-pow(Col.x,2.0),0.0,0.0,(.6-Col.x*0.05)*(distanceToCenter<0.5?1.0:0.0)*Col.y); }`); + gl.lineWidth(options.lineWidth||1); // doesn't work yet (nobody supports it) + // Create a font texture, lucida console or otherwise monospaced. + var fw=33, font = Object.assign(document.createElement('canvas'),{width:(19+94)*fw,height:48}), + ctx = Object.assign(font.getContext('2d'),{font:'bold 48px lucida console, monospace'}), + ftx = gl.createTexture(); gl.activeTexture(gl.TEXTURE0); gl.bindTexture(gl.TEXTURE_2D, ftx); + for (var i=33; i<127; i++) ctx.fillText(String.fromCharCode(i),(i-33)*fw,40); + var specialChars = "∞≅¹²³₀₁₂₃₄₅₆₇₈₉⋀⋁∆⋅"; specialChars.split('').forEach((x,i)=>ctx.fillText(x,(i-33+127)*fw,40)); + // 2.0 gl.texImage2D(gl.TEXTURE_2D,0,gl.RGBA,94*fw,32,0,gl.RGBA,gl.UNSIGNED_BYTE,font); + gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, gl.RGBA, gl.UNSIGNED_BYTE, font); + gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MAG_FILTER, gl.LINEAR); gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR); + gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_S, gl.CLAMP_TO_EDGE); gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_T, gl.CLAMP_TO_EDGE); + // Font rendering program. Renders billboarded fonts, transforms offset passed as color2. + var program2 = compile(`attribute vec4 position; attribute vec2 texc; varying vec2 tex; varying vec4 Pos; uniform mat4 mv; uniform mat4 p; uniform vec3 color2; + void main() { tex=texc; gl_PointSize=6.0; vec4 o=mv*vec4(color2,0.0); Pos=(-1.0/(o.z-mv[3][2]))*position+vec4(mv[3][0],mv[3][1],mv[3][2],0.0)+o; gl_Position = p*Pos; }`, + `precision highp float; uniform vec3 color; varying vec4 Pos; varying vec2 tex; + uniform sampler2D texm; void main() { vec4 c = texture2D(texm,tex); if (c.a<0.01) discard; gl_FragColor = vec4(color,c.a);}`); + // Helpers for line drawing. Convert line segments to triangles. + const line_to_tri = ([ax,ay,az,bx,by,bz]) => [ax,ay,az,ax,ay,az,bx,by,bz,bx,by,bz,ax,ay,az,bx,by,bz]; + const line_to_tri2 = ([ax,ay,az,bx,by,bz]) => [bx,by,bz,bx,by,bz,2*bx-ax,2*by-ay,2*bz-az,2*bx-ax,2*by-ay,2*bz-az,bx,by,bz,2*bx-ax,2*by-ay,2*bz-az]; + // Conformal space needs a bit extra magic to extract euclidean parametric representations. + if (tot==5 && options.conformal) var ni = Element.Coeff(4,1).Add(Element.Coeff(5,1)), no = Element.Coeff(4,0.5).Sub(Element.Coeff(5,0.5)); + var interprete = (x)=>{ + if (!(x instanceof Element)) return { tp:0 }; + if (options.ipns) x=x.Dual; + // tp = { 0:unknown 1:point 2:line, 3:plane, 4:circle, 5:sphere + var X2 = (x.Mul(x)).s, tp=0, weight2, opnix = ni.Wedge(x), ipnix = ni.LDot(x), + attitude, pos, normal, tg,btg,epsilon = 0.000001/(options.scale||1), I3=Element.Coeff(16,-1); + var x2zero = Math.abs(X2) < epsilon, ipnixzero = ipnix.VLength < epsilon, opnixzero = opnix.VLength < epsilon; + if (opnixzero && ipnixzero) { // free flat + } else if (opnixzero && !ipnixzero) { // bound flat (lines) + attitude = no.Wedge(ni).LDot(x); + weight2 = Math.abs(attitude.LDot(attitude).s)**.5; + pos = attitude.LDot(x.Reverse); //Inverse); + pos = [-pos.e15/pos.e45,-pos.e25/pos.e45,-pos.e34/pos.e45]; + if (x.Grade(3).VLength) { + normal = [attitude.e1/weight2,attitude.e2/weight2,attitude.e3/weight2]; tp=2; + } else if (x.Grade(2).VLength) { // point pair with ni + tp = 1; + } else { + normal = Element.LDot(Element.Mul(attitude,1/weight2),I3).Normalized; + var r=normal.Mul(Element.Coeff(3,1)); if (r[0]==-1) r[0]=1; else {r[0]+=1; r=r.Normalized;} + tg = [...r.Mul(Element.Coeff(1,1)).Mul(r.Conjugate)].slice(1,4); + btg = [...r.Mul(Element.Coeff(2,1)).Mul(r.Conjugate)].slice(1,4); + normal = [...normal.slice(1,4)]; tp=3; + } + } else if (!opnixzero && ipnixzero) { // dual bound flat + } else if (x2zero) { // bound vec,biv,tri (points) + if (options.ipns) x=x.Dual; + attitude = ni.Wedge(no).LDot(ni.Wedge(x)); + pos = [...(Element.LDot(1/(ni.LDot(x)).s,x)).slice(1,4)].map(x=>-x); + tp=1; + } else if (!x2zero) { // round (point pair,circle,sphere) + tp = x.Grade(3).VLength?4:x.Grade(2).VLength?6:5; + var nix = ni.Wedge(x), nix2 = (nix.Mul(nix)).s; + attitude = ni.Wedge(no).LDot(nix); + pos = [...(x.Mul(ni).Mul(x)).slice(1,4)].map(x=>-x/(2.0*nix2)); + weight2 = Math.abs((x.LDot(x)).s / nix2)**.5; + if (tp==4) { + if (x.LDot(x).s < 0) { weight2 = -weight2; } + normal = Element.LDot(Element.Mul(attitude,1/weight2),I3).Normalized; + var r=normal.Mul(Element.Coeff(3,1)); if (r[0]==-1) r[0]=1; else {r[0]+=1; r=r.Normalized;} + tg = [...r.Mul(Element.Coeff(1,1)).Mul(r.Conjugate)].slice(1,4); + btg = [...r.Mul(Element.Coeff(2,1)).Mul(r.Conjugate)].slice(1,4); + normal = [...normal.slice(1,4)]; + } else if (tp==6) { + weight2 = (x.LDot(x).s < 0)?-(weight2):weight2; + normal = Element.Mul(attitude.Normalized,weight2).slice(1,4); + } else { + normal = [...((Element.LDot(Element.Mul(attitude,1/weight2),I3)).Normalized).slice(1,4)]; + } + } + return {tp,pos:pos?pos.map(x=>x*(options.scale||1)):[0,0,0],normal,tg,btg,weight2:weight2*(options.scale||1)} + }; + // canvas update will (re)render the content. + var armed=0,sphere,e14 = Element.Coeff(14,1); + canvas.update = (x)=>{ + if (!canvas.parentElement) return; + // restore from still.. + if (options && !options.still && canvas.im && canvas.im.parentElement) { canvas.im.parentElement.insertBefore(canvas,canvas.im); canvas.im.parentElement.removeChild(canvas.im); } + // Start by updating canvas size if needed and viewport. + var s = getComputedStyle(canvas); if (s.width) { canvas.width = parseFloat(s.width)*(options.devicePixelRatio||devicePixelRatio||1); canvas.height = parseFloat(s.height)*(options.devicePixelRatio||devicePixelRatio||1); } + gl.viewport(0,0, canvas.width|0,canvas.height|0); var r=canvas.width/canvas.height; + // Defaults, resolve function input + var a,p=[],l=[],t=[],c=[.5,.5,.5],alpha=0,lastpos=[-1.95,1.5,0,1]; gl.clear(gl.COLOR_BUFFER_BIT+gl.DEPTH_BUFFER_BIT); while (x.call) x=x(); + // Create default camera matrix and initial lastposition (contra-compensated for camera) + M = mtx(options.camera); + var a = new this(); a.set([1,-2,1.90*canvas.height/canvas.width,0],1); a = options.camera.Conjugate.Mul(a.Dual).Mul(options.camera); + lastpos = a.slice(11,14).map((y,i)=>(i<=1?1:-1)*y/a[14]).reverse(); + var linediff = new this(); linediff.set([0,0,-0.12*2000/canvas.width*(options.fontSize||1),0],1); + linediff = options.camera.Conjugate.Mul(linediff.Dual).Mul(options.camera).slice(11,14).map((y,i)=>(i<=1?1:-1)*y/a[14]).reverse(); + // Grid. + if (options.grid) { + const gr = options.gridSize||1; + if (!options.gridLines) { options.gridLines=[[],[],[]]; for (var i=-gr; i<=gr; i+=gr/10) { + options.gridLines[0].push(i,-gr,gr, i,-gr,-gr, gr,-gr,i, -gr,-gr,i); + options.gridLines[1].push(i,gr,gr, i,-gr,gr, gr,i,gr, -gr,i,gr); + options.gridLines[2].push(-gr,i,gr, -gr,i,-gr, -gr,gr,i, -gr,-gr,i); + }} + var ltest = [], ltest2 = [], ttest = []; for (var j=0; j<3; ++j) for (var i=0; i{ while (x.call) x=x.call(); x=interprete(x);l.push.apply(l,x.pos); }); var d = {tp:-1}; } + else if (e instanceof Array && e.length==3) { e.forEach(x=>{ while (x.call) x=x.call(); x=interprete(x);t.push.apply(t,x.pos); }); var d = {tp:-1}; } + else var d = interprete(e); + if (d.tp) lastpos=d.pos; + if (d.tp==1) p.push.apply(p,d.pos); + if (d.tp==2) { l.push.apply(l,d.pos.map((x,i)=>x-d.normal[i]*3)); l.push.apply(l,d.pos.map((x,i)=>x+d.normal[i]*3)); } + if (d.tp==3) { t.push.apply(t,d.pos.map((x,i)=>x+d.tg[i]+d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x-d.tg[i]+d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x+d.tg[i]-d.btg[i])); + t.push.apply(t,d.pos.map((x,i)=>x-d.tg[i]+d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x+d.tg[i]-d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x-d.tg[i]-d.btg[i])); } + if (d.tp==4) { + var ne=0,la=0; + if (d.weight2<0) { c[0]=1;c[1]=0;c[2]=0; } + for (var j=0; j<65; j++) { + ne = d.pos.map((x,i)=>x+Math.cos(j/32*Math.PI)*d.weight2*d.tg[i]+Math.sin(j/32*Math.PI)*d.weight2*d.btg[i]); if (ne&&la&&(d.weight2>0||j%2==0)) { l.push.apply(l,la); l.push.apply(l,ne); }; la=ne; + } + } + if (d.tp==6) { + if (d.weight2<0) { c[0]=1;c[1]=0;c[2]=0; } + if (options.useUnnaturalLineDisplayForPointPairs) { + l.push.apply(l,d.pos.map((x,i)=>x-d.normal[i]*(options.scale||1))); + l.push.apply(l,d.pos.map((x,i)=>x+d.normal[i]*(options.scale||1))); + } + p.push.apply(p,d.pos.map((x,i)=>x-d.normal[i]*(options.scale||1))); + p.push.apply(p,d.pos.map((x,i)=>x+d.normal[i]*(options.scale||1))); + } + if (d.tp==5) { + if (!sphere) { + var pnts = [], tris=[], S=Math.sin, C=Math.cos, pi=Math.PI, W=96, H=48; + for (var j=0; jx.s); + gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,1-(alpha||0.1)); gl.enable(gl.CULL_FACE) + draw(programSphere,gl.TRIANGLES,undefined,c,[0,0,0],r,undefined,sphere.va); + gl.disable(gl.BLEND); gl.disable(gl.CULL_FACE); + M = oldM; + } + if (i==ll-1 || d.tp==0) { + // render triangles, lines, points. + if (alpha) { gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,1-alpha); } + if (t.length) { draw(program,gl.TRIANGLES,t,c,[0,0,0],r); t.forEach((x,i)=>{ if (i%9==0) lastpos=[0,0,0]; lastpos[i%3]+=x/3; }); t=[]; } + if (l.length) { + var ltest = [], ltest2 = [], ttest = []; for (var li=0; lix%(countx*county))); e.va3.tcount = (countx-1)*county*2*3; + } + if ( e.call && e.length==1 && !e.va2) { var countx=e.dx||256; + var temp=new Float32Array(3*countx),o=new Float32Array(3),et=[]; + for (var ii=0; ii{ + if (e instanceof Array && e.length==3) { tc++; e.forEach(x=>{ while (x.call) x=x.call(); x=interprete(x);et3.push.apply(et3,x.pos); }); var d = {tp:-1}; } + else { + var d = interprete(e); + if (d.tp==1) { pc++; et.push(...d.pos); } + if (d.tp==2) { lc++; et2.push(...d.pos.map((x,i)=>x-d.normal[i]*10),...d.pos.map((x,i)=>x+d.normal[i]*10)); } + } + }); + e.va = createVA(et,undefined); e.va.tcount = pc; + e.va2 = createVA(et2,undefined); e.va2.tcount = lc*2; + e.va3 = createVA(et3,undefined); e.va3.tcount = tc*3; + } + // render the vertex array. + if (e.va && e.va.tcount) { gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA); draw(programPoint,gl.POINTS,undefined,[0,0,0],c,r,undefined,e.va); gl.disable(gl.BLEND); }; + if (e.va2 && e.va2.tcount) draw(program,gl.LINES,undefined,[0,0,0],c,r,undefined,e.va2); + if (e.va3 && e.va3.tcount) draw(program,gl.TRIANGLES,undefined,c,[0,0,0],r,undefined,e.va3); + } + if (alpha) gl.disable(gl.BLEND); // no alpha for text printing. + // setup a new color + if (typeof e == "number") { alpha=((e>>>24)&0xff)/255; c[0]=((e>>>16)&0xff)/255; c[1]=((e>>>8)&0xff)/255; c[2]=(e&0xff)/255; } + if (typeof(e)=='string') { + if (options.htmlText) { + if (!x['_'+i]) { console.log('creating div'); Object.defineProperty(x,'_'+i, {value: document.body.appendChild(document.createElement('div')), enumerable:false }) }; + var rc = canvas.getBoundingClientRect(), div = x['_'+i]; + var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [...lastpos,1]).map(x=>x.s); + pos2 = Element.Mul( [[5,0,0,0],[0,5*(r||2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]); + Object.assign(div.style,{position:'fixed',pointerEvents:'none',left:rc.left + (rc.right-rc.left)*(pos2[0]/2+0.5),top: rc.top + (rc.bottom-rc.top)*(-pos2[1]/2+0.5) - 20}); + if (div.last != e) { div.innerHTML = e; div.last = e; if (self.renderMathInElement) self.renderMathInElement(div); } + } else { + gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA,gl.ONE_MINUS_SRC_ALPHA); + var fw = 113, mapChar = (x)=>{ var c = x.charCodeAt(0)-33; if (c>=94) c = 94+specialChars.indexOf(x); return c/fw; } + draw(program2,gl.TRIANGLES, + [...Array(e.length*6*3)].map((x,i)=>{ var x=0,z=-0.2, o=x+(i/18|0)*1.1; return (0.05*(options.z||5))*[o,-1,z,o+1.2,-1,z,o,1,z,o+1.2,-1,z,o+1.2,1,z,o,1,z][i%18]}),c,lastpos,r, + [...Array(e.length*6*2)].map((x,i)=>{ var o=mapChar(e[i/12|0]); return [o,1,o+1/fw,1,o,0,o+1/fw,1,o+1/fw,0,o,0][i%12]})); gl.disable(gl.BLEND); lastpos[1]+=linediff[1]; lastpos[0]+=linediff[0]; lastpos[2]+=linediff[2]; + } + } + } + continue; + } + // PGA + if (options.dual && e instanceof Element) e = e.Dual; + // Convert planes to polygons. + if (e instanceof Element && e.Grade(1).Length > 0.001) { + var m = Element.Add(1, Element.Mul(e.Normalized, Element.Coeff(3,1))).Normalized, e0 = 0; + e=Element.sw(m,[[-1,-1],[-1,1],[1,1],[-1,-1],[1,1],[1,-1]].map(([x,z])=>Element.Trivector(x*e.Length,e0,z*e.Length,1))); + } + // Convert lines to line segments. + if (e instanceof Element && e.Grade(2).Length) + e=[e.LDot(e14).Wedge(e).Add(e.Wedge(Element.Coeff(1,1)).Mul(Element.Coeff(0,-(options.clip||3)))),e.LDot(e14).Wedge(e).Add(e.Wedge(Element.Coeff(1,1)).Mul(Element.Coeff(0,options.clip||3)))] + .map(x=>x[14]<0?x.Scale(-1):x); + // If euclidean point, store as point, store line segments and triangles. + if (e.e123) p.push.apply(p,e.slice(11,14).map((y,i)=>(i<=1?1:-1)*y/e[14]).reverse()); + if (e instanceof Array && e.length==2) l=l.concat.apply(l,e.map(x=>[...x.slice(11,14).map((y,i)=>(i<=1?1:-1)*y/x[14]).reverse()])); + if (e instanceof Array && e.length%3==0) t=t.concat.apply(t,e.map(x=>[...x.slice(11,14).map((y,i)=>(i<=1?1:-1)*y/x[14]).reverse()])); + // Render orbits of parametrised motors, as well as lists of points.. + function sw_mot_orig(A,R){ + var a0=A[0],a1=A[5],a2=A[6],a3=A[7],a4=A[8],a5=A[9],a6=A[10],a7=A[15]; + R[2] = -2*(a0*a3+a4*a7-a6*a2-a5*a1); + R[1] = -2*(a4*a1-a0*a2-a6*a3+a5*a7); + R[0] = 2*(a0*a1+a4*a2+a5*a3+a6*a7); + return R + } + if ( e.call && e.length==1) { var count=e.dx||64; + for (var ismot,xx,o=new Float32Array(3),ii=0; ii1) l.push(xx[0],xx[1],xx[2]); + var m = e(ii/(count-1)); + if (ii==0) ismot = m[0]||m[5]||m[6]||m[7]||m[8]||m[9]||m[10]; + xx = ismot?sw_mot_orig(m,o):m.slice(11,14).map((y,i)=>(i<=1?1:-1)*y).reverse(); //Element.sw(e(ii/(count-1)),o); + l.push(xx[0],xx[1],xx[2]); + } + } + if ( e.call && e.length==2 && !e.va) { var countx=e.dx||64,county=e.dy||32; + var temp=new Float32Array(3*countx*county),o=new Float32Array(3),et=[]; + for (var pp=0,ii=0; iix%(countx*county))); e.va.tcount = (countx-1)*county*2*3; + } + // Experimental display of motors using particle systems. + if (e instanceof Object && e.motor) { + if (!e.va || e.recalc) { + var seed = 1; function random() { var x = Math.sin(seed++) * 10000; return x - Math.floor(x); } + e.xRange = e.xRange === undefined ? 1:e.xRange; e.yRange = e.yRange === undefined ? 1:e.yRange; e.zRange = e.zRange === undefined ? 1:e.zRange; + var vtx=[], tx=[], vtx2=[]; + for (var i=0; i<(e.zRange===0?5000:60000); i++) { + var p = Element.Trivector(random()*(2*e.xRange)-e.xRange,random()*2*e.yRange-e.yRange,random()*2*e.zRange-e.zRange,1); +// var p2 = Element.sw(e.motor,p); + var p2 = e.motor.Mul(p).Mul(e.motor.Inverse); + tx.push(random(), random()); + vtx.push(...p.slice(11,14).reverse()); vtx2.push(...p2.slice(11,14).reverse()); + } + e.va = createVA(vtx,tx,undefined,vtx2); e.va.tcount = vtx.length/3; + e.recalc = false; + } + var time = performance.now()/1000; + gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA); gl.disable(gl.DEPTH_TEST); + draw(programmot, gl.POINTS,t,c,[time%1,0,0],r,undefined,e.va); + gl.disable(gl.BLEND); gl.enable(gl.DEPTH_TEST); + } + // we could also be an object with cached vertex array of triangles .. + else if (e.va || (e instanceof Object && e.data)) { + // Create the vertex array and store it for re-use. + if (!e.va) { + if (e.idx) { + var et = e.data.map(x=>[...x.slice(11,14).map((y,i)=>(i<=1?1:-1)*y/x[14]).reverse()]).flat(); + } else { + var et=[]; e.data.forEach(e=>{if (e instanceof Array && e.length==3) et=et.concat.apply(et,e.map(x=>[...x.slice(11,14).map((y,i)=>(i<=1?1:-1)*y/x[14]).reverse()]));}); + } + e.va = createVA(et,undefined,e.idx,e.color?new Float32Array(e.color):undefined); e.va.tcount = (e.idx && e.idx.length)?e.idx.length:e.data.length*3; + } + // render the vertex array. + var M5 = Element.Scalar(1).Add(Element.Coeff(7,2.5)); + if (e.transform) { + var M1 = mtx(e.transform, false); + var M2 = mtx(M5.Mul(options.camera), false); + M = Array(16).fill(0); + for (var ii=0; ii<4; ++ii) for (var jj=0; jj<4; ++jj) for (var kk=0; kk<4; ++kk) M[ii*4+kk] += M1[ii*4+jj] * M2[jj*4+kk]; + } + if (alpha) { gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,1-alpha); } + draw(e.color?programcol:program,gl.TRIANGLES,t,c,[0,0,0],r,undefined,e.va); + if (alpha) gl.disable(gl.BLEND); + if (e.transform) { M=mtx(options.camera); } + } + // if we're a number (color), label or the last item, we output the collected items. + else if (typeof e=='number' || i==ll-1 || typeof e == 'string') { + // render triangles, lines, points. + if (alpha) { gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,1-alpha); } + if (t.length) { draw(program,gl.TRIANGLES,t,c,[0,0,0],r); t.forEach((x,i)=>{ if (i%9==0) lastpos=[0,0,0]; lastpos[i%3]+=x/3; }); t=[]; } + if (l.length) { + var ltest = [], ltest2 = [], ttest = [], w = (options.lineWidth||1); for (var li=0; li>>24)&0xff)/255; c[0]=((e>>>16)&0xff)/255; c[1]=((e>>>8)&0xff)/255; c[2]=(e&0xff)/255; } + // render a label + if (typeof(e)=='string') { + if (options.htmlText) { + if (!canvas['_'+i]) { console.log('creating div'); Object.defineProperty(canvas,'_'+i, {value: document.body.appendChild(document.createElement('div')), enumerable:false }) }; + var rc = canvas.getBoundingClientRect(), div = canvas['_'+i]; + var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [...lastpos,1]).map(x=>x.s); + pos2 = Element.Mul( [[5,0,0,0],[0,5*(r||2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]); + Object.assign(div.style,{position:'fixed',pointerEvents:'none',left:rc.left + (rc.right-rc.left)*(pos2[0]/2+0.5),top: rc.top + (rc.bottom-rc.top)*(-pos2[1]/2+0.5) - 20}); + if (div.last != e) { div.innerHTML = e; div.last = e; if (self.renderMathInElement) self.renderMathInElement(div,{output:'html'}); } + } else { + gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA,gl.ONE_MINUS_SRC_ALPHA); gl.disable(gl.DEPTH_TEST); + var fw = 113, mapChar = (x)=>{ var c = x.charCodeAt(0)-33; if (c>=94) c = 94+specialChars.indexOf(x); return c/fw; } + draw(program2,gl.TRIANGLES, + [...Array(e.length*6*3)].map((x,i)=>{ var x=0,z=0.2, o=x+(i/18|0)*1.1; return 0.2*(options.fontSize||1)*2000/canvas.width*[o,-1,z,o+1.2,-1,z,o,1,z,o+1.2,-1,z,o+1.2,1,z,o,1,z][i%18]}),c,lastpos,r, + [...Array(e.length*6*2)].map((x,i)=>{ var o=mapChar(e[i/12|0]); return [o,1,o+1/fw,1,o,0,o+1/fw,1,o+1/fw,0,o,0][i%12]})); gl.disable(gl.BLEND); lastpos[0] += linediff[0];lastpos[1] += linediff[1];lastpos[2] += linediff[2]; + if (!options.noZ) gl.enable(gl.DEPTH_TEST); + } + } + } + }; + // if we're no longer in the page .. stop doing the work. + armed++; if (document.body.contains(canvas)) armed=0; if (armed==2) return; + canvas.value=x; if (options&&!options.animate) canvas.dispatchEvent(new CustomEvent('input')); canvas.options=options; + if (options&&options.animate) { requestAnimationFrame(canvas.update.bind(canvas,f,options)); } + if (options&&options.still) { canvas.value=x; canvas.dispatchEvent(new CustomEvent('input')); canvas.im.style.width=canvas.style.width; canvas.im.style.height=canvas.style.height; canvas.im.src = canvas.toDataURL(); + var p=canvas.parentElement; if (p) { p.insertBefore(canvas.im,canvas); p.removeChild(canvas); } + } + } + // Basic mouse interactivity. needs more love. + var sel=-1; canvas.oncontextmenu = canvas.onmousedown = (e)=>{e.preventDefault(); e.stopPropagation(); if (e.detail===0) return; + var rc = canvas.getBoundingClientRect(), mx=(e.x-rc.left)/(rc.right-rc.left)*2-1, my=((e.y-rc.top)/(rc.bottom-rc.top)*4-2)*canvas.height/canvas.width; + sel = (e.button==2)?-3:-2; canvas.value.forEach((x,i)=>{ + if (tot != 5) { if (x[14]) { + var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [-x[13]/x[14],x[12]/x[14],x[11]/x[14],1]).map(x=>x.s); + pos2 = Element.Mul( [[5,0,0,0],[0,-5*(2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]); + if ((mx-pos2[0])**2 + ((my)-pos2[1])**2 < 0.001) sel=i; + }} else { + x = interprete(x); if (x.tp==1) { + var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [...x.pos,1]).map(x=>x.s); + pos2 = Element.Mul( [[5,0,0,0],[0,5*(r||2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]); + if ((mx-pos2[0])**2 + ((-my)-pos2[1])**2 < 0.01) sel=i; + } + } + }); + canvas.onwheel=e=>{e.preventDefault(); e.stopPropagation(); options.z = (options.z||5)+e.deltaY/100; if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options));} + canvas.onmouseup=e=>sel=-1; canvas.onmouseleave=e=>sel=-1; + var tx,ty; canvas.ontouchstart = (e)=>{e.preventDefault(); canvas.focus(); var x = e.changedTouches[0].pageX, y = e.changedTouches[0].pageY; tx=x; ty=y; } + canvas.ontouchmove = function (e) { e.preventDefault(); + var x = e.changedTouches[0].pageX, y = e.changedTouches[0].pageY, mx = (x-(tx||x))/1000, my = -(y-(ty||y))/1000; tx=x; ty=y; + options.h = (options.h||0)+mx; options.p = Math.max(-Math.PI/2,Math.min(Math.PI/2, (options.p||0)+my)); if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options)); return; + }; + canvas.onmousemove=(e)=>{ + var rc = canvas.getBoundingClientRect(),x; if (sel>=0) { if (tot==5) x=interprete(canvas.value[sel]); else { x=canvas.value[sel]; x={pos:[-x[13]/x[14],-x[12]/x[14],x[11]/x[14]]}; }} + var mx =(e.movementX)/(rc.right-rc.left)*2, my=((e.movementY)/(rc.bottom-rc.top)*2)*canvas.height/canvas.width; + if (sel==-2) { options.h = (options.h||0)+(options.conformal?-1:1)*mx/2; options.p = Math.max(-Math.PI/2,Math.min(Math.PI/2, (options.p||0)-my/2)); if (options.camera) options.camera.set( ( Element.Bivector(0,0,0,0,0,options.p).Exp() ).Mul( Element.Bivector(0,0,0,0,options.h,0).Exp() )); if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options)); return; }; + if (sel==-3) { var ct = Math.cos(options.h||0), st= Math.sin(options.h||0), ct2 = Math.cos(options.p||0), st2 = Math.sin(options.p||0); + if (e.shiftKey) { options.posy = (options.posy||0)+my; } else { options.posx = (options.posx||0)+mx*ct+my*st; options.posz = (options.posz||0)+mx*-st+my*ct*ct2; } if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options));return; }; if (sel < 0) return; + if (tot==5) { + x.pos[0] += (e.buttons!=2)?Math.cos((options.h||0))*mx:Math.sin(-(options.h||0))*-my; x.pos[1]+=(e.buttons!=2)?-my:0; x.pos[2]+=(e.buttons!=2)?Math.sin((options.h||0))*mx:Math.cos(-(options.h||0))*-my; + canvas.value[sel].set(Element.Mul(ni,(x.pos[0]**2+x.pos[1]**2+x.pos[2]**2)*0.5).Sub(no)); canvas.value[sel].set(x.pos,1); } + else if (x) { + var [cw,ch] = [rc.width, rc.height]; + var ox = (1/(options.scale || 1)) * ((e.offsetX / cw) - 0.5); + var oy = (1/(options.scale || 1)) * ((e.offsetY / ch) - 0.5) * (ch/cw); + var tb = Element.sw(options.camera,canvas.value[sel]); + var z = -(tb.e012/tb.e123+5)/5*4; tb.e023 = ox*z*tb.e123; tb.e013 = oy*z*tb.e123; + canvas.value[sel].set(Element.sw(options.camera.Reverse, tb)); + } + if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options)); + } + } + canvas.value = f.call?f():f; canvas.options=options; + if (options&&options.still) { + var i=new Image(); canvas.im = i; return requestAnimationFrame(canvas.update.bind(canvas,f,options)),canvas; + } else return requestAnimationFrame(canvas.update.bind(canvas,f,options)),canvas; + } + + // The inline function is a js to js translator that adds operator overloading and algebraic literals. + // It can be called with a function, a string, or used as a template function. + static inline(intxt) { + // If we are called as a template function. + if (arguments.length>1 || intxt instanceof Array) { + var args=[].slice.call(arguments,1); + return res.inline(new Function(args.map((x,i)=>'_template_'+i).join(),'return ('+intxt.map((x,i)=>(x||'')+(args[i]&&('_template_'+i)||'')).join('')+')')).apply(res,args); + } + // Get the source input text. + var txt = (intxt instanceof Function)?intxt.toString():`function(){return (${intxt})}`; + // Our tokenizer reads the text token by token and stores it in the tok array (as type/token tuples). + var tok = [], resi=[], t, possibleRegex=false, c, tokens = [/^[\s\uFFFF]|^[\u000A\u000D\u2028\u2029]|^\/\/[^\n]*\n|^\/\*[\s\S]*?\*\//g, // 0: whitespace/comments + /^\"\"|^\'\'|^\".*?[^\\]\"|^\'.*?[^\\]\'|^\`[\s\S]*?[^\\]\`/g, // 1: literal strings + /^\d+[.]{0,1}\d*[ei][\+\-_]{0,1}\d*|^\.\d+[ei][\+\-_]{0,1}\d*|^e_\d*/g, // 2: literal numbers in scientific notation (with small hack for i and e_ asciimath) + /^\d+[.]{0,1}\d*[E][+-]{0,1}\d*|^\.\d+[E][+-]{0,1}\d*|^0x\d+|^\d+[.]{0,1}\d*|^\.\d+/g, // 3: literal hex, nonsci numbers + /^\/.*?[^\\]\/[gmisuy]?/g, // 4: regex + /^(\.Normalized|\.Length|\.\.\.|>>>=|===|!==|>>>|<<=|>>=|=>|\|\||[<>\+\-\*%&|^\/!\=]=|\*\*|\+\+|\-\-|<<|>>|\&\&|\^\^|^[{}()\[\];.,<>\+\-\*%|&^!~?:=\/]{1})/g, // 5: punctuator + /^[$_\p{L}][$_\p{L}\p{Mn}\p{Mc}\p{Nd}\p{Pc}\u200C\u200D]*/gu] // 6: identifier + while (txt.length) for (t in tokens) { + if (t == 4 && !possibleRegex) continue; + if (resi = txt.match(tokens[t])) { + c = resi[0]; if (t!=0) {possibleRegex = c == '(' || c == '=' || c == '[' || c == ',' || c == ';';} tok.push([t | 0, c]); txt = txt.slice(c.length); break; + }} // tokenise + // Translate algebraic literals. (scientific e-notation to "this.Coeff" + tok=tok.map(t=>(t[0]==2)?[2,'Element.Coeff('+basis.indexOf((!options.Cayley?simplify:(x)=>x)('e'+t[1].split(/e_|e|i/)[1]||1).replace('-',''))+','+(simplify(t[1].split(/e_|e|i/)[1]||1).match('-')?"-1*":"")+parseFloat(t[1][0]=='e'?1:t[1].split(/e_|e|i/)[0])+')']:t); + // String templates (limited support - needs fundamental changes.). + tok=tok.map(t=>(t[0]==1 && t[1][0]=='`')?[1,t[1].replace(/\$\{(.*?)\}/g,a=>"${"+Element.inline(a.slice(2,-1)).toString().match(/return \((.*)\)/)[1]+"}")]:t); + // We support two syntaxes, standard js or if you pass in a text, asciimath. + var syntax = (intxt instanceof Function)?[[['.Normalized','Normalize',2],['.Length','Length',2]],[['~','Conjugate',1],['!','Dual',1]],[['**','Pow',0,1]],[['^','Wedge'],['&','Vee'],['<<','LDot']],[['*','Mul'],['/','Div']],[['|','Dot']],[['>>>','sw',0,1]],[['-','Sub'],['+','Add']],[['%','%']],[['==','eq'],['!=','neq'],['<','lt'],['>','gt'],['<=','lte'],['>=','gte']]] + :[[['pi','Math.PI'],['sin','Math.sin']],[['ddot','this.Reverse'],['tilde','this.Involute'],['hat','this.Conjugate'],['bar','this.Dual']],[['^','Pow',0,1]],[['^^','Wedge'],['*','LDot']],[['**','Mul'],['/','Div']],[['-','Sub'],['+','Add']],[['<','lt'],['>','gt'],['<=','lte'],['>=','gte']]]; + // For asciimath, some fixed translations apply (like pi->Math.PI) etc .. + tok=tok.map(t=>(t[0]!=6)?t:[].concat.apply([],syntax).filter(x=>x[0]==t[1]).length?[6,[].concat.apply([],syntax).filter(x=>x[0]==t[1])[0][1]]:t); + // Now the token-stream is translated recursively. + function translate(tokens) { + // helpers : first token to the left of x that is not of a type in the skip list. + var left = (x=ti-1,skip=[0])=>{ while(x>=0&&~skip.indexOf(tokens[x][0])) x--; return x; }, + // first token to the right of x that is not of a type in the skip list. + right= (x=ti+1,skip=[0])=>{ while(x{tokens.splice(x,y-x+1,[tp,...(sub||tokens.slice(x,y+1))])}, + // match O-C pairs. returns the 'matching bracket' position + match = (O="(",C=")")=>{var o=1,x=ti+1; while(o){if(tokens[x][1]==O)o++;if(tokens[x][1]==C)o--; x++;}; return x-1;}; + // grouping (resolving brackets). + for (var ti=0,t,si;t=tokens[ti];ti++) if (t[1]=="(") glue(ti,si=match(),7,[[5,"("],...translate(tokens.slice(ti+1,si)),[5,")"]]); + // [] dot call and new + for (var ti=0,t,si; t=tokens[ti];ti++) { + if (t[1]=="[") { glue(ti,si=match("[","]"),7,[[5,"["],...translate(tokens.slice(ti+1,si)),[5,"]"]]); if (ti)ti--;} // matching [] + else if (t[1]==".") { glue(left(),right()); ti--; } // dot operator + else if (t[0]==7 && ti && left()>=0 && tokens[left()][0]>=6 && tokens[left()][1]!="return") { glue(left(),ti--) } // collate ( and [ + else if (t[1]=='new') { glue(ti,right()) }; // collate new keyword + } + // ++ and -- + for (var ti=0,t; t=tokens[ti];ti++) if (t[1]=="++" || t[1]=="--") glue(left(),ti); + // unary - and + are handled separately from syntax .. + for (var ti=0,t,si; t=tokens[ti];ti++) + if (t[1]=="-" && (left()<0 || (tokens[left()]||[])[1]=='return'||(tokens[left()]||[5])[0]==5)) glue(ti,right(),6,["Element.Sub(",tokens[right()],")"]); // unary minus works on all types. + else if (t[1]=="+" && (left()<0 || (tokens[left()]||[])[1]=='return'|| (tokens[left()]||[0])[0]==5 && (tokens[left()]||[0])[1][0]!=".")) glue(ti,ti+1); // unary plus is glued, only on scalars. + // now process all operators in the syntax list .. + for (var si=0,s; s=syntax[si]; si++) for (var ti=s[0][3]?tokens.length-1:0,t; t=tokens[ti];s[0][3]?ti--:ti++) for (var opi=0,op; op=s[opi]; opi++) if (t[1]==op[0]) { + // exception case .. ".Normalized" and ".Length" properties are re-routed (so they work on scalars etc ..) + if (op[2]==2) { var arg=tokens[left()]; glue(ti-1,ti,6,["Element."+op[1],"(",arg,")"]); } + // unary operators (all are to the left) + else if (op[2]) { var arg=tokens[right()]; glue(ti, right(), 6, ["Element."+op[1],"(",arg,")"]); } + // binary operators + else { var l=left(),r=right(),a1=tokens[l],a2=tokens[r]; if (op[0]==op[1]) glue(l,r,6,[a1,op[1],a2]); else glue(l,r,6,["Element."+op[1],"(",a1,",",a2,")"]); ti-=2; } + } + return tokens; + } + // Glue all back together and return as bound function. + return eval( ('('+(function f(t){return t.map(t=>t instanceof Array?f(t):typeof t == "string"?t:"").join('');})(translate(tok))+')') ); + } + } + + if ((p==2 || p==3) && (r==1)) { + res.arrow = res.inline(( from_point, to_point, w=0.03, aspect=0.8, camera=1 )=>{ + from_point = from_point/(-from_point|!1e0); to_point = to_point/(-to_point|!1e0); + var line = ( from_point & to_point ), l = line.Length; + var shape = [[0,w],[l-5*w,w],[l-5*w,aspect*5*w],[l,0],[l-5*w,-aspect*5*w],[l-5*w,-w],[0,-w]].map(([x,y])=>!(1e0+x*1e1+y*1e2)); + var sqrt = R => R==-1?1e12:(1+R).Normalized; + var R = ((to_point - from_point).UnDual).Normalized * 1e1; + var R2 = sqrt(from_point/!1e0) * sqrt(R); + var p2 = R2 >>> 1e3; + if (p2 != 0) { var p1 = (((~(camera+0e1) >>> 1e3)|line)/line).Normalized; return sqrt(p1/p2) * R2 >>> shape; } + return R2 >>> shape; + }) + } + + if (options.dual) { + Object.defineProperty(res.prototype, 'Inverse', {configurable:true, get(){ var s = 1/this.s**2; return this.map((x,i)=>i?-x*s:1/x ); }}); + } else { + // Matrix-free inverses up to 5D. Should translate this to an inline call for readability. + // http://repository.essex.ac.uk/17282/1/TechReport_CES-534.pdf + Object.defineProperty(res.prototype, 'Inverse', {configurable: true, get(){ + // Shirokov inverse .. + if (tot > 5) { + for (var N=2**(((tot+1)/2)|0), Uk=this.Scale(1), k=1; kres.prototype[x] = options.over.inline(res.prototype[x])); + res.prototype.Coeff = function() { for (var i=0,l=arguments.length; ix==0?undefined:(i?'('+x+')'+basis[i]:x.toString())).filter(x=>x).join(' + '); } + } + + // Experimental differential operator. + var _D, _DT, _DA, totd = basis.length; + function makeD(transpose=false){ + _DA = _DA || Algebra({ p:p,q:q,r:r,basis:options.basis,even:options.even,over:Algebra({dual:totd})}); // same algebra, but over dual numbers. + return (func)=>{ + var dfunc = _DA.inline(func); // convert input function to dual algebra + return (val,...args)=>{ // return a new function (the derivative w.r.t 1st param) + if (!(val instanceof res)) val = res.Scalar(val); // allow to be called with scalars. + args = args.map(x=>{ var r = _DA.Scalar(0); for (var i=0; ival.slice()); // call the function in the dual algebra. + if (transpose) for (var i=0; i 0.5) +n_neg = count(eig.values .< -0.5) +if n_pos + n_neg == size(gram, 1) + printgood("Non-degenerate subspace") +else + printbad("Degenerate subspace") +end +sig_rem = Int64[ones(1-n_pos); -ones(4-n_neg)] +unk = hcat(a, b, c) +M = matrix_space(F, 5, 5) +big_gram = M(F.([ + diagm(sig_rem) unk; + transpose(unk) gram +])) + +r, p, L, U = lu(big_gram) +if isone(p) + printgood("Found a solution") +else + printbad("Didn't find a solution") +end +solution = transpose(L) +mform = U * inv(solution) + +vals = [0, 0, 0, 1, 0, -3//4] +solution_ex = [evaluate(entry, vals) for entry in solution] +mform_ex = [evaluate(entry, vals) for entry in mform] + +std_basis = [ + 0 0 0 1 1; + 0 0 0 1 -1; + 1 0 0 0 0; + 0 1 0 0 0; + 0 0 1 0 0 +] +std_solution = M(F.(std_basis)) * solution +std_solution_ex = std_basis * solution_ex + +println("Minkowski form:") +display(mform_ex) + +big_gram_recovered = transpose(solution_ex) * mform_ex * solution_ex +valid = all(iszero.( + [evaluate(entry, vals) for entry in big_gram] - big_gram_recovered +)) +if valid + printgood("Recovered Gram matrix:") +else + printbad("Didn't recover Gram matrix. Instead, got:") +end +display(big_gram_recovered) + +# this should be a solution +hand_solution = [0 0 1 0 0; 0 0 -1 2 2; 0 0 0 1 -1; 1 0 0 0 0; 0 1 0 0 0] +unmix = Rational{Int64}[[1//2 1//2; 1//2 -1//2] zeros(Int64, 2, 3); zeros(Int64, 3, 2) Matrix{Int64}(I, 3, 3)] +hand_solution_diag = unmix * hand_solution +big_gram_hand_recovered = transpose(hand_solution_diag) * diagm([1; -ones(Int64, 4)]) * hand_solution_diag +println("Gram matrix from hand-written solution:") +display(big_gram_hand_recovered) \ No newline at end of file diff --git a/engine-proto/gram-test/gram-test.sage b/engine-proto/gram-test/gram-test.sage new file mode 100644 index 0000000..a95ce97 --- /dev/null +++ b/engine-proto/gram-test/gram-test.sage @@ -0,0 +1,27 @@ +F = QQ['a', 'b', 'c'].fraction_field() +a, b, c = F.gens() + +# three mutually tangent spheres which are all perpendicular to the x, y plane +gram = matrix([ + [-1, 0, 0, 0, 0], + [0, -1, a, b, c], + [0, a, -1, 1, 1], + [0, b, 1, -1, 1], + [0, c, 1, 1, -1] +]) + +P, L, U = gram.LU() +solution = (P * L).transpose() +mform = U * L.transpose().inverse() + +concrete = solution.subs({a: 0, b: 1, c: -3/4}) + +std_basis = matrix([ + [0, 0, 0, 1, 1], + [0, 0, 0, 1, -1], + [1, 0, 0, 0, 0], + [0, 1, 0, 0, 0], + [0, 0, 1, 0, 0] +]) +std_solution = std_basis * solution +std_concrete = std_basis * concrete \ No newline at end of file diff --git a/engine-proto/gram-test/irisawa-hexlet.jl b/engine-proto/gram-test/irisawa-hexlet.jl new file mode 100644 index 0000000..67def8c --- /dev/null +++ b/engine-proto/gram-test/irisawa-hexlet.jl @@ -0,0 +1,77 @@ +include("Engine.jl") + +using SparseArrays + +# this problem is from a sangaku by Irisawa Shintarō Hiroatsu. the article below +# includes a nice translation of the problem statement, which was recorded in +# Uchida Itsumi's book _Kokon sankan_ (_Mathematics, Past and Present_) +# +# "Japan's 'Wasan' Mathematical Tradition", by Abe Haruki +# https://www.nippon.com/en/japan-topics/c12801/ +# + +# initialize the partial gram matrix +J = Int64[] +K = Int64[] +values = BigFloat[] +for s in 1:9 + # each sphere is represented by a spacelike vector + push!(J, s) + push!(K, s) + push!(values, 1) + + # the circumscribing sphere is internally tangent to all of the other spheres + if s > 1 + append!(J, [1, s]) + append!(K, [s, 1]) + append!(values, [1, 1]) + end + + if s > 3 + # each chain sphere is externally tangent to the "sun" and "moon" spheres + for n in 2:3 + append!(J, [s, n]) + append!(K, [n, s]) + append!(values, [-1, -1]) + end + + # each chain sphere is externally tangent to the next chain sphere + s_next = 4 + mod(s-3, 6) + append!(J, [s, s_next]) + append!(K, [s_next, s]) + append!(values, [-1, -1]) + end +end +gram = sparse(J, K, values) + +# make an initial guess +guess = hcat( + Engine.sphere(BigFloat[0, 0, 0], BigFloat(15)), + Engine.sphere(BigFloat[0, 0, -9], BigFloat(5)), + Engine.sphere(BigFloat[0, 0, 11], BigFloat(3)), + ( + Engine.sphere(9*BigFloat[cos(k*π/3), sin(k*π/3), 0], BigFloat(2.5)) + for k in 1:6 + )... +) +frozen = [CartesianIndex(4, k) for k in 1:4] + +# complete the gram matrix using Newton's method with backtracking +L, success, history = Engine.realize_gram(gram, guess, frozen) +completed_gram = L'*Engine.Q*L +println("Completed Gram matrix:\n") +display(completed_gram) +if success + println("\nTarget accuracy achieved!") +else + println("\nFailed to reach target accuracy") +end +println("Steps: ", size(history.scaled_loss, 1)) +println("Loss: ", history.scaled_loss[end], "\n") +if success + println("Chain diameters:") + println(" ", 1 / L[4,4], " sun (given)") + for k in 5:9 + println(" ", 1 / L[4,k], " sun") + end +end \ No newline at end of file diff --git a/engine-proto/gram-test/low-rank-test.jl b/engine-proto/gram-test/low-rank-test.jl new file mode 100644 index 0000000..d932a3d --- /dev/null +++ b/engine-proto/gram-test/low-rank-test.jl @@ -0,0 +1,49 @@ +using LowRankModels +using LinearAlgebra +using SparseArrays + +# testing Gram matrix recovery using the LowRankModels package + +# initialize the partial gram matrix for an arrangement of seven spheres in +# which spheres 1 through 5 are mutually tangent, and spheres 3 through 7 are +# also mutually tangent +I = Int64[] +J = Int64[] +values = Float64[] +for i in 1:7 + for j in 1:7 + if (i <= 5 && j <= 5) || (i >= 3 && j >= 3) + push!(I, i) + push!(J, j) + push!(values, i == j ? 1 : -1) + end + end +end +gram = sparse(I, J, values) + +# in this initial guess, the mutual tangency condition is satisfied for spheres +# 1 through 5 +X₀ = sqrt(0.5) * [ + 1 0 1 1 1; + 1 0 1 -1 -1; + 1 0 -1 1 -1; + 1 0 -1 -1 1; + 2 -sqrt(6) 0 0 0; + 0.2 0.3 -0.1 -0.2 0.1; + 0.1 -0.2 0.3 0.4 -0.1 +]' +Y₀ = diagm([-1, 1, 1, 1, 1]) * X₀ + +# search parameters +search_params = ProxGradParams( + 1.0; + max_iter = 100, + inner_iter = 1, + abs_tol = 1e-16, + rel_tol = 1e-9, + min_stepsize = 0.01 +) + +# complete gram matrix +model = GLRM(gram, QuadLoss(), ZeroReg(), ZeroReg(), 5, X = X₀, Y = Y₀) +X, Y, history = fit!(model, search_params) diff --git a/engine-proto/gram-test/overlap-test.jl b/engine-proto/gram-test/overlap-test.jl new file mode 100644 index 0000000..e75531a --- /dev/null +++ b/engine-proto/gram-test/overlap-test.jl @@ -0,0 +1,37 @@ +using LinearAlgebra +using AbstractAlgebra + +function printgood(msg) + printstyled("✓", color = :green) + println(" ", msg) +end + +function printbad(msg) + printstyled("✗", color = :red) + println(" ", msg) +end + +F, gens = rational_function_field(AbstractAlgebra.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"]) +x = gens[1] +t = gens[2:4] + +# three mutually tangent spheres which are all perpendicular to the x, y plane +M = matrix_space(F, 7, 7) +gram = M(F[ + 1 -1 -1 -1 -1 t[1] t[2]; + -1 1 -1 -1 -1 x t[3] + -1 -1 1 -1 -1 -1 -1; + -1 -1 -1 1 -1 -1 -1; + -1 -1 -1 -1 1 -1 -1; + t[1] x -1 -1 -1 1 -1; + t[2] t[3] -1 -1 -1 -1 1 +]) + +r, p, L, U = lu(gram) +if isone(p) + printgood("Found a solution") +else + printbad("Didn't find a solution") +end +solution = transpose(L) +mform = U * inv(solution) diff --git a/engine-proto/gram-test/overlapping-pyramids.jl b/engine-proto/gram-test/overlapping-pyramids.jl new file mode 100644 index 0000000..a4ae01a --- /dev/null +++ b/engine-proto/gram-test/overlapping-pyramids.jl @@ -0,0 +1,90 @@ +include("Engine.jl") + +using SparseArrays +using AbstractAlgebra +using PolynomialRoots +using Random + +# initialize the partial gram matrix for an arrangement of seven spheres in +# which spheres 1 through 5 are mutually tangent, and spheres 3 through 7 are +# also mutually tangent +J = Int64[] +K = Int64[] +values = BigFloat[] +for j in 1:7 + for k in 1:7 + if (j <= 5 && k <= 5) || (j >= 3 && k >= 3) + push!(J, j) + push!(K, k) + push!(values, j == k ? 1 : -1) + end + end +end +gram = sparse(J, K, values) + +# set the independent variable +indep_val = -9//5 +gram[6, 1] = BigFloat(indep_val) +gram[1, 6] = gram[6, 1] + +# in this initial guess, the mutual tangency condition is satisfied for spheres +# 1 through 5 +Random.seed!(50793) +guess = let + a = sqrt(BigFloat(3)/2) + hcat( + sqrt(1/BigFloat(2)) * BigFloat[ + 1 1 -1 -1 0 + 1 -1 1 -1 0 + 1 -1 -1 1 0 + 0.5 0.5 0.5 0.5 1+a + 0.5 0.5 0.5 0.5 1-a + ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)), + Engine.rand_on_shell(fill(BigFloat(-1), 2)) + ) +end + +# complete the gram matrix using Newton's method with backtracking +L, success, history = Engine.realize_gram(gram, guess) +completed_gram = L'*Engine.Q*L +println("Completed Gram matrix:\n") +display(completed_gram) +if success + println("\nTarget accuracy achieved!") +else + println("\nFailed to reach target accuracy") +end +println("Steps: ", size(history.scaled_loss, 1)) +println("Loss: ", history.scaled_loss[end], "\n") + +# === algebraic check === + +#= +R, gens = polynomial_ring(AbstractAlgebra.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"]) +x = gens[1] +t = gens[2:4] + +S, u = polynomial_ring(AbstractAlgebra.Rationals{BigInt}(), "u") + +M = matrix_space(R, 7, 7) +gram_symb = M(R[ + 1 -1 -1 -1 -1 t[1] t[2]; + -1 1 -1 -1 -1 x t[3] + -1 -1 1 -1 -1 -1 -1; + -1 -1 -1 1 -1 -1 -1; + -1 -1 -1 -1 1 -1 -1; + t[1] x -1 -1 -1 1 -1; + t[2] t[3] -1 -1 -1 -1 1 +]) +rank_constraints = det.([ + gram_symb[1:6, 1:6], + gram_symb[2:7, 2:7], + gram_symb[[1, 3, 4, 5, 6, 7], [1, 3, 4, 5, 6, 7]] +]) + +# solve for x and t +x_constraint = 25//16 * to_univariate(S, evaluate(rank_constraints[1], [2], [indep_val])) +t₂_constraint = 25//16 * to_univariate(S, evaluate(rank_constraints[3], [2], [indep_val])) +x_vals = PolynomialRoots.roots(x_constraint.coeffs) +t₂_vals = PolynomialRoots.roots(t₂_constraint.coeffs) +=# \ No newline at end of file diff --git a/engine-proto/gram-test/sphere-in-tetrahedron.jl b/engine-proto/gram-test/sphere-in-tetrahedron.jl new file mode 100644 index 0000000..97f0720 --- /dev/null +++ b/engine-proto/gram-test/sphere-in-tetrahedron.jl @@ -0,0 +1,67 @@ +include("Engine.jl") + +using SparseArrays +using Random + +# initialize the partial gram matrix for a sphere inscribed in a regular +# tetrahedron +J = Int64[] +K = Int64[] +values = BigFloat[] +for j in 1:6 + for k in 1:6 + filled = false + if j == 6 + if k <= 4 + push!(values, 0) + filled = true + end + elseif k == 6 + if j <= 4 + push!(values, 0) + filled = true + end + elseif j == k + push!(values, 1) + filled = true + elseif j <= 4 && k <= 4 + push!(values, -1/BigFloat(3)) + filled = true + else + push!(values, -1) + filled = true + end + if filled + push!(J, j) + push!(K, k) + end + end +end +gram = sparse(J, K, values) + +# set initial guess +Random.seed!(99230) +guess = hcat( + sqrt(1/BigFloat(3)) * BigFloat[ + 1 1 -1 -1 0 + 1 -1 1 -1 0 + 1 -1 -1 1 0 + 0 0 0 0 1.5 + 1 1 1 1 -0.5 + ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)), + BigFloat[0, 0, 0, 0, 1] +) +frozen = [CartesianIndex(j, 6) for j in 1:5] + +# complete the gram matrix using Newton's method with backtracking +L, success, history = Engine.realize_gram(gram, guess, frozen) +completed_gram = L'*Engine.Q*L +println("Completed Gram matrix:\n") +display(completed_gram) +if success + println("\nTarget accuracy achieved!") +else + println("\nFailed to reach target accuracy") +end +println("Steps: ", size(history.scaled_loss, 1)) +println("Loss: ", history.scaled_loss[end], "\n") \ No newline at end of file diff --git a/engine-proto/gram-test/tetrahedron-radius-ratio.jl b/engine-proto/gram-test/tetrahedron-radius-ratio.jl new file mode 100644 index 0000000..7ceb794 --- /dev/null +++ b/engine-proto/gram-test/tetrahedron-radius-ratio.jl @@ -0,0 +1,96 @@ +include("Engine.jl") + +using LinearAlgebra +using SparseArrays +using Random + +# initialize the partial gram matrix for a sphere inscribed in a regular +# tetrahedron +J = Int64[] +K = Int64[] +values = BigFloat[] +for j in 1:11 + for k in 1:11 + filled = false + if j == 11 + if k <= 4 + push!(values, 0) + filled = true + end + elseif k == 11 + if j <= 4 + push!(values, 0) + filled = true + end + elseif j == k + push!(values, j <= 6 ? 1 : 0) + filled = true + elseif j <= 4 + if k <= 4 + push!(values, -1/BigFloat(3)) + filled = true + elseif k == 5 + push!(values, -1) + filled = true + elseif 7 <= k <= 10 && k - j != 6 + push!(values, 0) + filled = true + end + elseif k <= 4 + if j == 5 + push!(values, -1) + filled = true + elseif 7 <= j <= 10 && j - k != 6 + push!(values, 0) + filled = true + end + elseif j == 6 && 7 <= k <= 10 || k == 6 && 7 <= j <= 10 + push!(values, 0) + filled = true + end + if filled + push!(J, j) + push!(K, k) + end + end +end +gram = sparse(J, K, values) + +# set initial guess +Random.seed!(99230) +guess = hcat( + sqrt(1/BigFloat(3)) * BigFloat[ + 1 1 -1 -1 0 0 + 1 -1 1 -1 0 0 + 1 -1 -1 1 0 0 + 0 0 0 0 1.5 0.5 + 1 1 1 1 -0.5 -1.5 + ] + 0.0*Engine.rand_on_shell(fill(BigFloat(-1), 6)), + Engine.point([-0.5, -0.5, -0.5] + 0.3*randn(3)), + Engine.point([-0.5, 0.5, 0.5] + 0.3*randn(3)), + Engine.point([ 0.5, -0.5, 0.5] + 0.3*randn(3)), + Engine.point([ 0.5, 0.5, -0.5] + 0.3*randn(3)), + BigFloat[0, 0, 0, 0, 1] +) +frozen = vcat( + [CartesianIndex(4, k) for k in 7:10], + [CartesianIndex(j, 11) for j in 1:5] +) + +# complete the gram matrix using Newton's method with backtracking +L, success, history = Engine.realize_gram(gram, guess, frozen) +completed_gram = L'*Engine.Q*L +println("Completed Gram matrix:\n") +display(completed_gram) +if success + println("\nTarget accuracy achieved!") +else + println("\nFailed to reach target accuracy") +end +println("Steps: ", size(history.scaled_loss, 1)) +println("Loss: ", history.scaled_loss[end]) +if success + infty = BigFloat[0, 0, 0, 0, 1] + radius_ratio = dot(infty, Engine.Q * L[:,5]) / dot(infty, Engine.Q * L[:,6]) + println("\nCircumradius / inradius: ", radius_ratio) +end \ No newline at end of file diff --git a/notes/inversive.md b/notes/inversive.md index 6de7ef2..5ee6329 100644 --- a/notes/inversive.md +++ b/notes/inversive.md @@ -2,28 +2,29 @@ (proposed by Alex Kontorovich as a practical system for doing 3D geometric calculations) -These coordinates are of form $I=(c, r, x, y, z)$ where we think of $c$ as the co-radius, $r$ as the radius, and $x, y, z$ as the "Euclidean" part, which we abbreviate $E_I$. There is an underlying basic quadratic form $Q(I_1,I_2) = (c_1r_2+c_2r_1)/2 - x_1x_2 -y_1y_2-z_1z_2$ which aids in calculation/verification of coordinates in this representation. We have: +These coordinates are of form $I=(c, b, x, y, z)$ where we think of $c$ as the co-radius, $b$ as the "bend" (reciprocal radius), and $x, y, z$ as the "Euclidean" part, which we abbreviate $E_I$. There is an underlying basic quadratic form $Q(I_1,I_2) = (c_1b_2+c_2b_1)/2 - x_1x_2 -y_1y_2-z_1z_2$ which aids in calculation/verification of coordinates in this representation. We have: -| Entity or Relationship | Representation | Comments/questions | -| ------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -| Sphere s with radius r>0 centered on P = (x,y,z) | $I_s = (1/c, 1/r, x/r, y/r, z/r)$ satisfying $Q(I_s,I_s) = -1$, i.e., $c = r/(\|P\|^2 - r^2)$. | Can also write $I_s = (\|P\|^2/r - r, 1/r, x/r. y/r, z/r)$ -- so there is no trouble if $\|E_{I_s}\| = r$, just get first coordinate to be 0. | -| Plane p with unit normal (x,y,z), a distance s from origin | $I_p = (2s, 0, x, y, z)$ | Note $Q(I_p, I_p)$ is still -1. Also, there are two representations for each plane through the origin, namely $(0,0,x,y,z)$ and $(0,0,-x,-y,-z)$ | -| Point P with Euclidean coordinates (x,y,z) | $I_P = (\|P\|^2, 1, x, y, z)$ | Note $Q(I_P,I_P) = 0$.  Because of this we might choose  some other scaling of the inversive coordinates, say $(\||P\||,1/\||P\||,x/\||P\||,y/\||P\||,z/\||P\||)$ instead, but that fails at the origin, and likely won't have some of the other nice properties listed below.  Note that scaling just the co-radius by $s$ and the radius by $1/s$ (which still preserves $Q=0$) dilates by a factor of $s$ about the origin, so that $(\|P\|, \|P\|, x, y, z)$, which might look symmetric, would actually have to represent the Euclidean point $(x/\||P\||, y/\||P\||, z/\||P\||)$ . | -| ∞, the "point at infinity" | $I_\infty = (1,0,0,0,0)$ | The only solution to $Q(I,I) = 0$ not covered by the above case. | -| P lies on sphere or plane given by I | $Q(I_P, I) = 0$ | | -| Sphere/planes represented by I and J are tangent | $Q(I,J) = 1$ (??, see note at right) | Seems as though this must be $Q(I,J) = \pm1$  ? For example, the $xy$ plane represented by (0,0,0,0,1)  is tangent to the unit circle centered at (0,0,1) rep'd by (0,1,0,0,1), but their Q-product is -1. And in general you can reflect any sphere tangent to any plane through the plane and it should flip the sign of $Q(I,J)$, if I am not mistaken. | -| Sphere/planes represented by I and J intersect (respectively, don't intersect) | $\|Q(I,J)\| < (\text{resp. }>)\; 1$ | Follows from the angle formula, at least conceptually. | -| P is center of sphere represented by I | Well, $Q(I_P, I)$ comes out to be $(\|P\|^2/r - r + \|P\|^2/r)/2 - \|P\|^2/r$ or just $-r/2$ . | Is it if and only if ?   No this probably doesn't work because center is not conformal quantity. | -| Distance between P and R is d | $Q(I_P, I_R) = d^2/2$ | | -| Distance between P and sphere/plane rep by I | | In the very simple case of a plane $I$ rep'd by $(2s, 0, x, y, z)$ and a point $P$ that lies on its perpendicular through the origin, rep'd by $(r^2, 1, rx, ry, rz)$ we get $Q(I, I_p) = s-r$, which is indeed the signed distance between $I$ and $P$. Not sure if this generalizes to other combinations? | -| Distance between sphere/planes rep by I and J | Note that for any two Euclidean-concentric spheres rep by $I$ and $J$ with radius $r$ and $s,$ $Q(I,J) = -\frac12\left(\frac rs  + \frac sr\right)$ depends only on the ratio of $r$ and $s$. So this can't give something that determines the Euclidean distance between the two spheres, which presumably grows as the two spheres are blown up proportionally. For another example, for any two parallel planes, $Q(I,J) = \pm1$. | Alex had said: Q(I,J)=cosh^2 (d/2) maybe where d is distance in usual hyperbolic metric. Or maybe cosh d. That may be right depending on what's meant by the hyperbolic metric there, but it seems like it won't determine a reasonable Euclidean distance between planes, which should differ between different pairs of parallel planes. | -| Sphere centered on P through R | | Probably just calculate distance etc. | -| Plane rep'd by I goes through center of sphere rep'd by J | I think this is equivalent to the plane being perpendicular to the sphere, i.e.$Q(I,J) = 0$. | | -| Dihedral angle between planes (or spheres?) rep by I and J | $\theta = \arccos(Q(I,J))$ | Aaron Fenyes points out: The angle between spheres in $S^3$ matches the angle between the planes they bound in $R^{(1,4)}$, which matches the angle between the spacelike vectors perpendicular to those planes. So we should have $Q(I,J) = \cos\theta$. Note that when the spheres do not intersect, we can interpret this as the "imaginary angle" between them, via $\cosh t = \cos it$. | -| R, P, S are collinear | Maybe just cross product of two differences is 0. Or, $R,P,S,\infty$ lie on a circle, or equivalently, $I_R,I_P,I_S,I_\infty$ span a plane (rather than a three-space). | Not a conformal property, but $R,P,S,\infty$ lying on a circle _is_. | -| Plane through noncollinear R, P, S | Should be, just solve Q(I, I_R) = 0 etc. | | -| circle | Maybe concentric sphere and the containing plane? Note it is easy to constrain the relationship between those two: they must be perpendicular. | Defn: circle is intersection of two spheres. That does cover lines. But you lose the canonicalness | -| line | Maybe two perpendicular containing planes? Maybe the plane perpendicular to the line and through origin, together with the point of the line on that plane? Or maybe just as a bag of collinear points? | The first is the limiting case of the possible circle rep, but it is not canonical. The second appears to be canonical, but I don't see a circle rep that corresponds to it. | +| Entity or Relationship | Representation | Comments/questions | +| ---------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | +| Sphere $s$ with radius $r>0$ centered on $P = (x,y,z)$ | $I_s = (\frac1{c}, \frac1{r}, \frac{x}{r}, \frac{y}{r}, \frac{z}{r})$ satisfying $Q(I_s,I_s) = -1,$ i.e., $c = r/(\|P\|^2 - r^2)$. | Note that $1/c = \|P\|^2/r - r$, so there is no trouble if $\|P\| = r$; we just get first coordinate to be 0. Using the point representation $I_P$ from below, let's orient the sphere so that its normals point into the "positive side," where $Q(I_P, I_s) > 0$. The vector $I_s$ then represents a sphere with outward normals, while $-I_s$ represents one with inward normals. | +| Plane $p$ with unit normal $(x,y,z)$ through the (Euclidean) point $(sx,sy,sz)$ | $I_p = (-2s, 0, -x, -y, -z)$ | Note that $Q(I_p, I_p)$ is still $-1$. We orient planes using the same convention we use for spheres. For example, $(-2, 0, -1/\sqrt3, -1/\sqrt3, -1/\sqrt3)$ and $(2, 0, 1/\sqrt3, 1/\sqrt3, 1/\sqrt3)$ represent planes that coincide in space, which have their normals pointing away from and toward the origin, respectively. Note that the ray from $(sx, sy, sz) \in p$ in direction $(-x, -y, -z)$ is the ray perpendicular to the plane through the origin; since $(-x, -y, -z)$ is a unit vector, $(sx, sy, sz)$ and hence $p$ is at distance $s$ from the origin. These coordinates are essentially the limit of a sphere's coordinates as its radius goes to infinity, or equivalently, as its bend goes to 0. | +| Point $P$ with Euclidean coordinates $(x,y,z)$ | $I_P = (\|P\|^2, 1, x, y, z)$ | Note $Q(I_P,I_P) = 0$. This gives us the freedom to choose a different normalization. For example, we could scale the representation shown here by $(\|P\|^2+1)^{-1}$, putting it on the sphere where the light cone intersects the plane where the first two coordinates sum to $1$. | +| ∞, the "point at infinity" | $I_\infty = (1,0,0,0,0)$ | The only solution to $Q(I,I) = 0$ not covered by (some normalization of) the above case. | +| Point $P$ lies on sphere or plane given by $I$ | $Q(I_P, I) = 0$ | Actually also works if $I$ is the coordinates of a point, in which case "lies on" simply means "coincides with". | +| Sphere/planes represented by $I$ and $J$ are tangent | If $I$ and $J$ have the same orientation where they touch, $Q(I,J) = -1$. If they have opposing orientations, $Q(I,J) = 1$. | For example, the $xy$ plane with normal $-e_z$, represented by $(0,0,0,0,1)$, is tangent with matching orientation to the unit sphere centered at $(0,0,1)$ with outward normals, represented by $(0,1,0,0,1).$ Accordingly, their $Q$ - product is $-1$. | +| Sphere/planes represented by $I$ and $J$ intersect (respectively, don't intersect) | $\lvert Q(I,J)\rvert \le (\text{resp. }>)\; 1$ | Follows from the angle formula and the tangency condition, at least conceptually. One subtlety: parallel planes have $Q$ - product $\pm 1$, because they intersect at infinity (and in fact, are "tangent" there)! | +| $P$ is center of sphere rep'd by $I$ | $Q(I, I_P) = -r/2$, where $1/r = 2Q(I_\infty, I)$ is the signed bend of the sphere, and $I_P$ is normalized in the standard way, which is to set $Q(I_\infty, I_P) = 1/2$ | This relationship is equivalent to both of the following. (1) The point $P$ has signed distance $-r$ from the sphere. (2) Inversion across the sphere maps $\infty$ to $P$. | +| Distance between points $P$ and $R$ is $d$ | $Q(I_P, I_R) = d^2/2$ | If $P$ and $R$ are represented by non-normalized vectors $V_P$ and $V_R$, the relation becomes $Q(V_P, V_R) = 2\,Q(V_P, I_\infty)\,Q(V_R, I_\infty)\,d^2$. This version of the relation makes it easier to see why $d$ goes to infinity as $P$ or $R$ approaches the point at infinity. | +| Signed distance between point rep'd by $V$ and sphere/plane rep'd by $I$ is $d$ | In general, $\frac{Q(I, V)}{2Q(I_\infty, V)} = Q(I_\infty, I)\,d^2 + d$. When $V$ is normalized in the usual way, this simplifies to $Q(I, V) = d^2/r + d$ for a sphere of radius $r$, and to $Q(I, V) = d$ for a plane. | We can use a Euclidean motion, represented linearly by a Lorentz transformation that fixes $I_\infty$, to put the point on the $z$ axis and put the nearest point on the sphere/plane at the origin with its normal pointing in the positive $z$ direction. Then the sphere/plane is represented by $I = (0, 1/r, 0, 0, -1)$, and the point can be represented by any multiple of $I_P = (d^2, 1, 0, 0, d)$, giving $Q(I, I_P) = d^2/2r + d.$ We turn this into a general expression by writing it in terms of Lorentz-invariant quantities and making it independent of the normalization of $I_P$. | +| Distance between sphere/planes rep by $I$ and $J$ | Note that for any two Euclidean-concentric spheres rep by $I$ and $J$ with radius $r$ and $s,$ $Q(I,J) = -\frac12\left(\frac rs + \frac sr\right)$ depends only on the ratio of $r$ and $s$. So this can't give something that determines the Euclidean distance between the two spheres, which presumably grows as the two spheres are blown up proportionally. For another example, for any two parallel planes, $Q(I,J) = \pm1$. | Alex had said: $Q(I,J)=\cosh(d/2)^2$ maybe where d is distance in usual hyperbolic metric. Or maybe $\cosh(d)$. That may be right depending on what's meant by the hyperbolic metric there, but it seems like it won't determine a reasonable Euclidean distance between planes, which should differ between different pairs of parallel planes. | +| Sphere centered on point $P$ through point $R$ | | Probably just calculate distance etc. | +| Plane rep'd by $I$ goes through center of sphere rep'd by $J$ | This is equivalent to the plane being perpendicular to the sphere: that is, $Q(I, J) = 0$. | | +| Dihedral angle between planes or spheres rep by $I$ and $J$ | $\theta = \arccos(Q(I,J))$ | Aaron Fenyes points out: The angle between spheres in $S^3$ matches the angle between the planes they bound in $R^{(1,4)}$, which matches the angle between the spacelike vectors perpendicular to those planes. So we should have $Q(I,J) = \cos(\theta)$. Note that when the spheres do not intersect, we can interpret this as the "imaginary angle" between them, via $\cosh(t) = \cos(it)$. | +| Points $R, P, S$ are collinear | Maybe just cross product of two differences is 0. Or, $R,P,S,\infty$ lie on a circle, or equivalently, $I_R,I_P,I_S,I_\infty$ span a plane (rather than a three-space). Or we can add two planes constrained to be perpendicular with one constrained to contain the origin, and all three points constrained to lie on both. But that's a lot of auxiliary entities and constraints... | $R,P,S$ lying on a line isn't a conformal property, but $R,P,S,\infty$ lying on a circle is. | +| Plane through noncollinear $R, P, S$ | Should be, just solve $Q(I, I_R) = 0$ etc. | | +| circle | Maybe concentric sphere and the containing plane? Note it is easy to constrain the relationship between those two: they must be perpendicular. | Defn: circle is intersection of two spheres. That does cover lines. But you lose the canonicalness | +| line | Maybe two perpendicular containing planes? Maybe the plane perpendicular to the line and through origin, together with the point of the line on that plane? Or maybe just as a bag of collinear points? | The first is the limiting case of the possible circle rep, but it is not canonical. However, there is a distinguished "standard" choice we could make: always choose one plane to contain the origin and the line, and the other to be the perpendicular plane containing the line. That choice or Plücker coordinates might be the best we can do. If we use the standardized perpendicular planes choice, then adding a line would be equivalent to adding two planes and the two constraints that one contains the origin and the other is perpendicular to it. That doesn't seem so bad. The second convention (perpendicular plane through the origin and a point on it) appears to be canonical, but there doesn't seem to be a circle representation that tends to it in the limit. | +| Inversion of entity represented by $v$ across sphere $s$, rep'd by $I_s$ | $v \mapsto v + 2Q(I_s, v)\,I_s$ | This is just an educated guess, but its behavior is consistent with inversion in at least two ways. (1) It fixes points on $s$ and spheres perpendicular to $s$. (2) It preserves dihedral angles with $s$. | The unification of spheres/planes is indeed attractive for a project like Dyna3. The relationship between this representation and Geometric Algebras is a bit murky; likely it somehow fits under the Geometric Algebra umbrella.