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Engine prototype (#13)

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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 <aaron.fenyes@fareycircles.ooo>
Co-authored-by: Glen Whitney <glen@studioinfinity.org>
Reviewed-on: #13
Co-authored-by: Vectornaut <vectornaut@nobody@nowhere.net>
Co-committed-by: Vectornaut <vectornaut@nobody@nowhere.net>
This commit is contained in:
Vectornaut 2024-10-21 03:18:47 +00:00 committed by Glen Whitney
parent c48d685ad6
commit b92be312e8
20 changed files with 3977 additions and 21 deletions
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223
engine-proto/alg-test/ConstructionViewer.jl Normal file
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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)

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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

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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

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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

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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

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<!DOCTYPE html>
<html>
<head>
<style>
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;
}
</style>
<script src="https://unpkg.com/ganja.js"></script>
</head>
<body>
<p><button onclick="flip()">Flip</button></p>
<script>
// in the default view, e4 + e5 is the point at infinity
let CGA3 = Algebra(4, 1);
let elements = [
CGA3.inline(() => Math.sqrt(0.5)*( 1e1 + 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*( 1e1 - 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*(-1e1 + 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*(-1e1 - 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => -Math.sqrt(3)*1e4 + Math.sqrt(2)*1e5)()
];
/*
these blocks of commented-out code can be used to confirm that a spacelike
vector and its Hodge dual represent the same generalized sphere
*/
/*let elements = [
CGA3.inline(() => Math.sqrt(0.5)*!( 1e1 + 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!( 1e1 - 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!(-1e1 + 1e2 - 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!(-1e1 - 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => !(-Math.sqrt(3)*1e4 + Math.sqrt(2)*1e5))()
];*/
/*let elements = [
CGA3.inline(() => 1e1 + 1e5)(),
CGA3.inline(() => 1e2 + 1e5)(),
CGA3.inline(() => 1e3 + 1e5)(),
CGA3.inline(() => -1e4 + 1e5)(),
CGA3.inline(() => Math.sqrt(0.5)*(1e1 + 1e2 + 1e3 + 1e5))(),
CGA3.inline(() => Math.sqrt(0.5)*!(1e1 + 1e2 + 1e3 - 0.01e4 + 1e5))()
];*/
// set up palette
var colorIndex;
var palette = [0xff00b0, 0x00ffb0, 0x00b0ff, 0x8040ff, 0xc0c0c0];
function nextColor() {
colorIndex = (colorIndex + 1) % palette.length;
return palette[colorIndex];
}
function resetColorCycle() {
colorIndex = palette.length - 1;
}
resetColorCycle();
// create scene function
function scene() {
commands = [];
resetColorCycle();
elements.forEach((elt) => commands.push(nextColor(), elt));
return commands;
}
// initialize graph
let graph = CGA3.graph(
scene,
{
conformal: true, gl: true, grid: true
}
)
document.body.appendChild(graph);
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) {
/*for (let k = 0; k < 26; ++k) {*/
elements[n][k] = 0;
}
}
requestAnimationFrame(graph.update.bind(graph, scene));
}
</script>
</body>
</html>

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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, """
<p><button id="flip-button" onclick="flip()">Flip</button></p>
""", 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

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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

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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

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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")

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engine-proto/gram-test/ganja-1.0.204.js Normal file
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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}(), ["a₁", "a₂", "b₁", "b₂", "c₁", "c₂"])
a = gens[1:2]
b = gens[3:4]
c = gens[5:6]
# three mutually tangent spheres which are all perpendicular to the x, y plane
gram = [
-1 1 1;
1 -1 1;
1 1 -1
]
eig = eigen(gram)
n_pos = count(eig.values .> 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)

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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

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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

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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)

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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)

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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)
=#

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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")

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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

43
notes/inversive.md
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@ -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.