doc: expand on representation of collinearity

doc: expand on representations of lines
doc: another unicode pseudo-space
2024-10-20 20:02:51 -07:00 · 2024-10-20 19:53:36 -07:00 · 2024-10-20 19:41:05 -07:00 · 2024-10-20 19:33:33 -07:00 · 2024-10-20 19:27:05 -07:00 · 2024-10-20 16:17:18 -07:00
18 changed files with 13 additions and 2394 deletions
--- a/app-proto/.gitignore
+++ b/app-proto/.gitignore
@ -1,4 +0,0 @@
 target
 dist
 profiling
 Cargo.lock
--- a/app-proto/Cargo.toml
+++ b/app-proto/Cargo.toml
@ -1,44 +0,0 @@
 [package]
 name = "dyna3"
 version = "0.1.0"
 authors = ["Aaron Fenyes", "Glen Whitney"]
 edition = "2021"
 [features]
 default = ["console_error_panic_hook"]
 [dependencies]
 itertools = "0.13.0"
 js-sys = "0.3.70"
 lazy_static = "1.5.0"
 nalgebra = "0.33.0"
 rustc-hash = "2.0.0"
 slab = "0.4.9"
 sycamore = "0.9.0-beta.3"
 # The `console_error_panic_hook` crate provides better debugging of panics by
 # logging them with `console.error`. This is great for development, but requires
 # all the `std::fmt` and `std::panicking` infrastructure, so isn't great for
 # code size when deploying.
 console_error_panic_hook = { version = "0.1.7", optional = true }
 [dependencies.web-sys]
 version = "0.3.69"
 features = [
  'HtmlCanvasElement',
  'HtmlInputElement',
  'Performance',
  'WebGl2RenderingContext',
  'WebGlBuffer',
  'WebGlProgram',
  'WebGlShader',
  'WebGlUniformLocation',
  'WebGlVertexArrayObject'
 ]
 [dev-dependencies]
 wasm-bindgen-test = "0.3.34"
 [profile.release]
 opt-level = "s" # optimize for small code size
 debug = true # include debug symbols
--- a/app-proto/index.html
+++ b/app-proto/index.html
@ -1,11 +0,0 @@
 <!DOCTYPE html>
 <html>
  <head>
    <meta charset="utf-8"/>
    <title>dyna3</title>
    <link data-trunk rel="css" href="main.css"/>
    <link href="https://fonts.bunny.net/css?family=fira-sans:ital,wght@0,400;1,400&display=swap" rel="stylesheet">
    <link href="https://fonts.bunny.net/css?family=noto-emoji:wght@400&text=%f0%9f%94%97%e2%9a%a0&display=swap" rel="stylesheet">
  </head>
  <body></body>
 </html>
--- a/app-proto/main.css
+++ b/app-proto/main.css
@ -1,175 +0,0 @@
 :root {
  --text: #fcfcfc; /* almost white */
  --text-bright: white;
  --text-invalid: #f58fc2; /* bright pink */
  --border: #555; /* light gray */
  --border-focus: #aaa; /* bright gray */
  --border-invalid: #70495c; /* dusky pink */
  --selection-highlight: #444; /* medium gray */
  --page-background: #222; /* dark gray */
  --display-background: #020202; /* almost black */
 }
 body {
  margin: 0px;
  color: var(--text);
  background-color: var(--page-background);
  font-family: 'Fira Sans', sans-serif;
 }
 /* sidebar */
 #sidebar {
  display: flex;
  flex-direction: column;
  float: left;
  width: 450px;
  height: 100vh;
  margin: 0px;
  padding: 0px;
  border-width: 0px 1px 0px 0px;
  border-style: solid;
  border-color: var(--border);
 }
 /* add-remove */
 #add-remove {
  display: flex;
  gap: 8px;
  margin: 8px;
 }
 #add-remove > button {
  width: 32px;
  height: 32px;
  font-size: large;
 }
 /* KLUDGE */
 /*
  for convenience, we're using emoji as temporary icons for some buttons. these
  buttons need to be displayed in an emoji font
 */
 #add-remove > button.emoji {
  font-family: 'Noto Emoji', sans-serif;
 }
 /* outline */
 #outline {
  flex-grow: 1;
  margin: 0px;
  padding: 0px;
  overflow-y: scroll;
 }
 li {
  user-select: none;
 }
 summary {
  display: flex;
 }
 summary.selected {
  color: var(--text-bright);
  background-color: var(--selection-highlight);
 }
 summary > div, .constraint {
  padding-top: 4px;
  padding-bottom: 4px;
 }
 .element, .constraint {
  display: flex;
  flex-grow: 1;
  padding-left: 8px;
  padding-right: 8px;
 }
 .element-switch {
  width: 18px;
  padding-left: 2px;
  text-align: center;
 }
 details:has(li) .element-switch::after {
  content: '▸';
 }
 details[open]:has(li) .element-switch::after {
  content: '▾';
 }
 .element-label {
  flex-grow: 1;
 }
 .constraint-label {
  flex-grow: 1;
 }
 .element-representation {
  display: flex;
 }
 .element-representation > div {
  padding: 2px 0px 0px 0px;
  font-size: 10pt;
  font-variant-numeric: tabular-nums;
  text-align: right;
  width: 56px;
 }
 .constraint {
  font-style: italic;
 }
 .constraint.invalid {
  color: var(--text-invalid);
 }
 .constraint > input[type=checkbox] {
  margin: 0px 8px 0px 0px;
 }
 .constraint > input[type=text] {
  color: inherit;
  background-color: inherit;
  border: 1px solid var(--border);
  border-radius: 2px;
 }
 .constraint.invalid > input[type=text] {
  border-color: var(--border-invalid);
 }
 .status {
  width: 20px;
  padding-left: 4px;
  text-align: center;
  font-family: 'Noto Emoji';
  font-style: normal;
 }
 .invalid > .status::after, details:has(.invalid):not([open]) .status::after {
  content: '⚠';
  color: var(--text-invalid);
 }
 /* display */
 canvas {
  float: left;
  margin-left: 20px;
  margin-top: 20px;
  background-color: var(--display-background);
  border: 1px solid var(--border);
  border-radius: 16px;
 }
 canvas:focus {
  border-color: var(--border-focus);
 }
--- a/app-proto/run-examples
+++ b/app-proto/run-examples
@ -1,8 +0,0 @@
 # based on "Enabling print statements in Cargo tests", by Jon Almeida
 #
 #   https://jonalmeida.com/posts/2015/01/23/print-cargo/
 #
 cargo test -- --nocapture engine::tests::irisawa_hexlet_test
 cargo test -- --nocapture engine::tests::three_spheres_example
 cargo test -- --nocapture engine::tests::point_on_sphere_example
--- a/app-proto/src/add_remove.rs
+++ b/app-proto/src/add_remove.rs
@ -1,240 +0,0 @@
 use sycamore::prelude::*;
 use web_sys::{console, wasm_bindgen::JsValue};
 use crate::{engine, AppState, assembly::{Assembly, Constraint, Element}};
 /* DEBUG */
 // load an example assembly for testing. this code will be removed once we've
 // built a more formal test assembly system
 fn load_gen_assemb(assembly: &Assembly) {
    let _ = assembly.try_insert_element(
        Element::new(
            String::from("gemini_a"),
            String::from("Castor"),
            [1.00_f32, 0.25_f32, 0.00_f32],
            engine::sphere(0.5, 0.5, 0.0, 1.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            String::from("gemini_b"),
            String::from("Pollux"),
            [0.00_f32, 0.25_f32, 1.00_f32],
            engine::sphere(-0.5, -0.5, 0.0, 1.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            String::from("ursa_major"),
            String::from("Ursa major"),
            [0.25_f32, 0.00_f32, 1.00_f32],
            engine::sphere(-0.5, 0.5, 0.0, 0.75)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            String::from("ursa_minor"),
            String::from("Ursa minor"),
            [0.25_f32, 1.00_f32, 0.00_f32],
            engine::sphere(0.5, -0.5, 0.0, 0.5)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            String::from("moon_deimos"),
            String::from("Deimos"),
            [0.75_f32, 0.75_f32, 0.00_f32],
            engine::sphere(0.0, 0.15, 1.0, 0.25)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            String::from("moon_phobos"),
            String::from("Phobos"),
            [0.00_f32, 0.75_f32, 0.50_f32],
            engine::sphere(0.0, -0.15, -1.0, 0.25)
        )
    );
 }
 /* DEBUG */
 // load an example assembly for testing. this code will be removed once we've
 // built a more formal test assembly system
 fn load_low_curv_assemb(assembly: &Assembly) {
    let a = 0.75_f64.sqrt();
    let _ = assembly.try_insert_element(
        Element::new(
            "central".to_string(),
            "Central".to_string(),
            [0.75_f32, 0.75_f32, 0.75_f32],
            engine::sphere(0.0, 0.0, 0.0, 1.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            "assemb_plane".to_string(),
            "Assembly plane".to_string(),
            [0.75_f32, 0.75_f32, 0.75_f32],
            engine::sphere_with_offset(0.0, 0.0, 1.0, 0.0, 0.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            "side1".to_string(),
            "Side 1".to_string(),
            [1.00_f32, 0.00_f32, 0.25_f32],
            engine::sphere_with_offset(1.0, 0.0, 0.0, 1.0, 0.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            "side2".to_string(),
            "Side 2".to_string(),
            [0.25_f32, 1.00_f32, 0.00_f32],
            engine::sphere_with_offset(-0.5, a, 0.0, 1.0, 0.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            "side3".to_string(),
            "Side 3".to_string(),
            [0.00_f32, 0.25_f32, 1.00_f32],
            engine::sphere_with_offset(-0.5, -a, 0.0, 1.0, 0.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            "corner1".to_string(),
            "Corner 1".to_string(),
            [0.75_f32, 0.75_f32, 0.75_f32],
            engine::sphere(-4.0/3.0, 0.0, 0.0, 1.0/3.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            "corner2".to_string(),
            "Corner 2".to_string(),
            [0.75_f32, 0.75_f32, 0.75_f32],
            engine::sphere(2.0/3.0, -4.0/3.0 * a, 0.0, 1.0/3.0)
        )
    );
    let _ = assembly.try_insert_element(
        Element::new(
            String::from("corner3"),
            String::from("Corner 3"),
            [0.75_f32, 0.75_f32, 0.75_f32],
            engine::sphere(2.0/3.0, 4.0/3.0 * a, 0.0, 1.0/3.0)
        )
    );
 }
 #[component]
 pub fn AddRemove() -> View {
    /* DEBUG */
    let assembly_name = create_signal("general".to_string());
    create_effect(move || {
        // get name of chosen assembly
        let name = assembly_name.get_clone();
        console::log_1(
            &JsValue::from(format!("Showing assembly \"{}\"", name.clone()))
        );
        batch(|| {
            let state = use_context::<AppState>();
            let assembly = &state.assembly;
            // clear state
            assembly.elements.update(|elts| elts.clear());
            assembly.elements_by_id.update(|elts_by_id| elts_by_id.clear());
            state.selection.update(|sel| sel.clear());
            // load assembly
            match name.as_str() {
                "general" => load_gen_assemb(assembly),
                "low-curv" => load_low_curv_assemb(assembly),
                _ => ()
            };
        });
    });
    view! {
        div(id="add-remove") {
            button(
                on:click=|_| {
                    let state = use_context::<AppState>();
                    state.assembly.insert_new_element();
                    /* DEBUG */
                    // print updated list of elements by identifier
                    console::log_1(&JsValue::from("elements by identifier:"));
                    for (id, key) in state.assembly.elements_by_id.get_clone().iter() {
                        console::log_3(
                            &JsValue::from(" "),
                            &JsValue::from(id),
                            &JsValue::from(*key)
                        );
                    }
                }
            ) { "+" }
            button(
                class="emoji", /* KLUDGE */ // for convenience, we're using an emoji as a temporary icon for this button
                disabled={
                    let state = use_context::<AppState>();
                    state.selection.with(|sel| sel.len() != 2)
                },
                on:click=|_| {
                    let state = use_context::<AppState>();
                    let subjects = state.selection.with(
                        |sel| {
                            let subject_vec: Vec<_> = sel.into_iter().collect();
                            (subject_vec[0].clone(), subject_vec[1].clone())
                        }
                    );
                    let lorentz_prod = create_signal(0.0);
                    let lorentz_prod_valid = create_signal(false);
                    let active = create_signal(true);
                    state.assembly.insert_constraint(Constraint {
                        subjects: subjects,
                        lorentz_prod: lorentz_prod,
                        lorentz_prod_text: create_signal(String::new()),
                        lorentz_prod_valid: lorentz_prod_valid,
                        active: active,
                    });
                    state.selection.update(|sel| sel.clear());
                    /* DEBUG */
                    // print updated constraint list
                    console::log_1(&JsValue::from("Constraints:"));
                    state.assembly.constraints.with(|csts| {
                        for (_, cst) in csts.into_iter() {
                            console::log_5(
                                &JsValue::from(" "),
                                &JsValue::from(cst.subjects.0),
                                &JsValue::from(cst.subjects.1),
                                &JsValue::from(":"),
                                &JsValue::from(cst.lorentz_prod.get_untracked())
                            );
                        }
                    });
                    // update the realization when the constraint becomes active
                    // and valid, or is edited while active and valid
                    create_effect(move || {
                        console::log_1(&JsValue::from(
                            format!("Constraint ({}, {}) updated", subjects.0, subjects.1)
                        ));
                        lorentz_prod.track();
                        if active.get() && lorentz_prod_valid.get() {
                            state.assembly.realize();
                        }
                    });
                }
            ) { "🔗" }
            select(bind:value=assembly_name) { /* DEBUG */ // example assembly chooser
                option(value="general") { "General" }
                option(value="low-curv") { "Low-curvature" }
                option(value="empty") { "Empty" }
            }
        }
    }
 }
--- a/app-proto/src/assembly.rs
+++ b/app-proto/src/assembly.rs
@ -1,233 +0,0 @@
 use nalgebra::{DMatrix, DVector};
 use rustc_hash::FxHashMap;
 use slab::Slab;
 use std::{collections::BTreeSet, sync::atomic::{AtomicU64, Ordering}};
 use sycamore::prelude::*;
 use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
 use crate::engine::{realize_gram, PartialMatrix};
 // the types of the keys we use to access an assembly's elements and constraints
 pub type ElementKey = usize;
 pub type ConstraintKey = usize;
 pub type ElementColor = [f32; 3];
 /* KLUDGE */
 // we should reconsider this design when we build a system for switching between
 // assemblies. at that point, we might want to switch to hierarchical keys,
 // where each each element has a key that identifies it within its assembly and
 // each assembly has a key that identifies it within the sesssion
 static NEXT_ELEMENT_SERIAL: AtomicU64 = AtomicU64::new(0);
 #[derive(Clone, PartialEq)]
 pub struct Element {
    pub id: String,
    pub label: String,
    pub color: ElementColor,
    pub representation: Signal<DVector<f64>>,
    pub constraints: Signal<BTreeSet<ConstraintKey>>,
    // a serial number, assigned by `Element::new`, that uniquely identifies
    // each element
    pub serial: u64,
    // the configuration matrix column index that was assigned to this element
    // last time the assembly was realized
    column_index: usize
 }
 impl Element {
    pub fn new(
        id: String,
        label: String,
        color: ElementColor,
        representation: DVector<f64>
    ) -> Element {
        // take the next serial number, panicking if that was the last number we
        // had left. the technique we use to panic on overflow is taken from
        // _Rust Atomics and Locks_, by Mara Bos
        //
        //   https://marabos.nl/atomics/atomics.html#example-handle-overflow
        //
        let serial = NEXT_ELEMENT_SERIAL.fetch_update(
            Ordering::SeqCst, Ordering::SeqCst,
            |serial| serial.checked_add(1)
        ).expect("Out of serial numbers for elements");
        Element {
            id: id,
            label: label,
            color: color,
            representation: create_signal(representation),
            constraints: create_signal(BTreeSet::default()),
            serial: serial,
            column_index: 0
        }
    }
 }
 #[derive(Clone)]
 pub struct Constraint {
    pub subjects: (ElementKey, ElementKey),
    pub lorentz_prod: Signal<f64>,
    pub lorentz_prod_text: Signal<String>,
    pub lorentz_prod_valid: Signal<bool>,
    pub active: Signal<bool>
 }
 // a complete, view-independent description of an assembly
 #[derive(Clone)]
 pub struct Assembly {
    // elements and constraints
    pub elements: Signal<Slab<Element>>,
    pub constraints: Signal<Slab<Constraint>>,
    // indexing
    pub elements_by_id: Signal<FxHashMap<String, ElementKey>>
 }
 impl Assembly {
    pub fn new() -> Assembly {
        Assembly {
            elements: create_signal(Slab::new()),
            constraints: create_signal(Slab::new()),
            elements_by_id: create_signal(FxHashMap::default())
        }
    }
    // --- inserting elements and constraints ---
    // insert an element into the assembly without checking whether we already
    // have an element with the same identifier. any element that does have the
    // same identifier will get kicked out of the `elements_by_id` index
    fn insert_element_unchecked(&self, elt: Element) {
        let id = elt.id.clone();
        let key = self.elements.update(|elts| elts.insert(elt));
        self.elements_by_id.update(|elts_by_id| elts_by_id.insert(id, key));
    }
    pub fn try_insert_element(&self, elt: Element) -> bool {
        let can_insert = self.elements_by_id.with_untracked(
            |elts_by_id| !elts_by_id.contains_key(&elt.id)
        );
        if can_insert {
            self.insert_element_unchecked(elt);
        }
        can_insert
    }
    pub fn insert_new_element(&self) {
        // find the next unused identifier in the default sequence
        let mut id_num = 1;
        let mut id = format!("sphere{}", id_num);
        while self.elements_by_id.with_untracked(
            |elts_by_id| elts_by_id.contains_key(&id)
        ) {
            id_num += 1;
            id = format!("sphere{}", id_num);
        }
        // create and insert a new element
        self.insert_element_unchecked(
            Element::new(
                id,
                format!("Sphere {}", id_num),
                [0.75_f32, 0.75_f32, 0.75_f32],
                DVector::<f64>::from_column_slice(&[0.0, 0.0, 0.0, 0.5, -0.5])
            )
        );
    }
    pub fn insert_constraint(&self, constraint: Constraint) {
        let subjects = constraint.subjects;
        let key = self.constraints.update(|csts| csts.insert(constraint));
        let subject_constraints = self.elements.with(
            |elts| (elts[subjects.0].constraints, elts[subjects.1].constraints)
        );
        subject_constraints.0.update(|csts| csts.insert(key));
        subject_constraints.1.update(|csts| csts.insert(key));
    }
    // --- realization ---
    pub fn realize(&self) {
        // index the elements
        self.elements.update_silent(|elts| {
            for (index, (_, elt)) in elts.into_iter().enumerate() {
                elt.column_index = index;
            }
        });
        // set up the Gram matrix and the initial configuration matrix
        let (gram, guess) = self.elements.with_untracked(|elts| {
            // set up the off-diagonal part of the Gram matrix
            let mut gram_to_be = PartialMatrix::new();
            self.constraints.with_untracked(|csts| {
                for (_, cst) in csts {
                    if cst.active.get_untracked() && cst.lorentz_prod_valid.get_untracked() {
                        let subjects = cst.subjects;
                        let row = elts[subjects.0].column_index;
                        let col = elts[subjects.1].column_index;
                        gram_to_be.push_sym(row, col, cst.lorentz_prod.get_untracked());
                    }
                }
            });
            // set up the initial configuration matrix and the diagonal of the
            // Gram matrix
            let mut guess_to_be = DMatrix::<f64>::zeros(5, elts.len());
            for (_, elt) in elts {
                let index = elt.column_index;
                gram_to_be.push_sym(index, index, 1.0);
                guess_to_be.set_column(index, &elt.representation.get_clone_untracked());
            }
            (gram_to_be, guess_to_be)
        });
        /* DEBUG */
        // log the Gram matrix
        console::log_1(&JsValue::from("Gram matrix:"));
        gram.log_to_console();
        /* DEBUG */
        // log the initial configuration matrix
        console::log_1(&JsValue::from("Old configuration:"));
        for j in 0..guess.nrows() {
            let mut row_str = String::new();
            for k in 0..guess.ncols() {
                row_str.push_str(format!(" {:>8.3}", guess[(j, k)]).as_str());
            }
            console::log_1(&JsValue::from(row_str));
        }
        // look for a configuration with the given Gram matrix
        let (config, success, history) = realize_gram(
            &gram, guess, &[],
            1.0e-12, 0.5, 0.9, 1.1, 200, 110
        );
        /* DEBUG */
        // report the outcome of the search
        console::log_1(&JsValue::from(
            if success {
                "Target accuracy achieved!"
            } else {
                "Failed to reach target accuracy"
            }
        ));
        console::log_2(&JsValue::from("Steps:"), &JsValue::from(history.scaled_loss.len() - 1));
        console::log_2(&JsValue::from("Loss:"), &JsValue::from(*history.scaled_loss.last().unwrap()));
        if success {
            // read out the solution
            for (_, elt) in self.elements.get_clone_untracked() {
                elt.representation.update(
                    |rep| rep.set_column(0, &config.column(elt.column_index))
                );
            }
        }
    }
 }
--- a/app-proto/src/display.rs
+++ b/app-proto/src/display.rs
@ -1,464 +0,0 @@
 use core::array;
 use nalgebra::{DMatrix, Rotation3, Vector3};
 use sycamore::{prelude::*, motion::create_raf};
 use web_sys::{
    console,
    window,
    KeyboardEvent,
    WebGl2RenderingContext,
    WebGlProgram,
    WebGlShader,
    WebGlUniformLocation,
    wasm_bindgen::{JsCast, JsValue}
 };
 use crate::AppState;
 fn compile_shader(
    context: &WebGl2RenderingContext,
    shader_type: u32,
    source: &str,
 ) -> WebGlShader {
    let shader = context.create_shader(shader_type).unwrap();
    context.shader_source(&shader, source);
    context.compile_shader(&shader);
    shader
 }
 fn get_uniform_array_locations<const N: usize>(
    context: &WebGl2RenderingContext,
    program: &WebGlProgram,
    var_name: &str,
    member_name_opt: Option<&str>
 ) -> [Option<WebGlUniformLocation>; N] {
    array::from_fn(|n| {
        let name = match member_name_opt {
            Some(member_name) => format!("{var_name}[{n}].{member_name}"),
            None => format!("{var_name}[{n}]")
        };
        context.get_uniform_location(&program, name.as_str())
    })
 }
 // load the given data into the vertex input of the given name
 fn bind_vertex_attrib(
    context: &WebGl2RenderingContext,
    index: u32,
    size: i32,
    data: &[f32]
 ) {
    // create a data buffer and bind it to ARRAY_BUFFER
    let buffer = context.create_buffer().unwrap();
    context.bind_buffer(WebGl2RenderingContext::ARRAY_BUFFER, Some(&buffer));
    // load the given data into the buffer. the function `Float32Array::view`
    // creates a raw view into our module's `WebAssembly.Memory` buffer.
    // allocating more memory will change the buffer, invalidating the view.
    // that means we have to make sure we don't allocate any memory until the
    // view is dropped
    unsafe {
        context.buffer_data_with_array_buffer_view(
            WebGl2RenderingContext::ARRAY_BUFFER,
            &js_sys::Float32Array::view(&data),
            WebGl2RenderingContext::STATIC_DRAW,
        );
    }
    // allow the target attribute to be used
    context.enable_vertex_attrib_array(index);
    // take whatever's bound to ARRAY_BUFFER---here, the data buffer created
    // above---and bind it to the target attribute
    //
    //   https://developer.mozilla.org/en-US/docs/Web/API/WebGLRenderingContext/vertexAttribPointer
    //
    context.vertex_attrib_pointer_with_i32(
        index,
        size,
        WebGl2RenderingContext::FLOAT,
        false, // don't normalize
        0, // zero stride
        0, // zero offset
    );
 }
 #[component]
 pub fn Display() -> View {
    let state = use_context::<AppState>();
    // canvas
    let display = create_node_ref();
    // navigation
    let pitch_up = create_signal(0.0);
    let pitch_down = create_signal(0.0);
    let yaw_right = create_signal(0.0);
    let yaw_left = create_signal(0.0);
    let roll_ccw = create_signal(0.0);
    let roll_cw = create_signal(0.0);
    let zoom_in = create_signal(0.0);
    let zoom_out = create_signal(0.0);
    let turntable = create_signal(false); /* BENCHMARKING */
    // change listener
    let scene_changed = create_signal(true);
    create_effect(move || {
        state.assembly.elements.with(|elts| {
            for (_, elt) in elts {
                elt.representation.track();
            }
        });
        state.selection.track();
        scene_changed.set(true);
    });
    /* INSTRUMENTS */
    const SAMPLE_PERIOD: i32 = 60;
    let mut last_sample_time = 0.0;
    let mut frames_since_last_sample = 0;
    let mean_frame_interval = create_signal(0.0);
    on_mount(move || {
        // timing
        let mut last_time = 0.0;
        // viewpoint
        const ROT_SPEED: f64 = 0.4; // in radians per second
        const ZOOM_SPEED: f64 = 0.15; // multiplicative rate per second
        const TURNTABLE_SPEED: f64 = 0.1; /* BENCHMARKING */
        let mut orientation = DMatrix::<f64>::identity(5, 5);
        let mut rotation = DMatrix::<f64>::identity(5, 5);
        let mut location_z: f64 = 5.0;
        // display parameters
        const OPACITY: f32 = 0.5; /* SCAFFOLDING */
        const HIGHLIGHT: f32 = 0.2; /* SCAFFOLDING */
        const LAYER_THRESHOLD: i32 = 0; /* DEBUG */
        const DEBUG_MODE: i32 = 0; /* DEBUG */
        /* INSTRUMENTS */
        let performance = window().unwrap().performance().unwrap();
        // get the display canvas
        let canvas = display.get().unchecked_into::<web_sys::HtmlCanvasElement>();
        let ctx = canvas
            .get_context("webgl2")
            .unwrap()
            .unwrap()
            .dyn_into::<WebGl2RenderingContext>()
            .unwrap();
        // compile and attach the vertex and fragment shaders
        let vertex_shader = compile_shader(
            &ctx,
            WebGl2RenderingContext::VERTEX_SHADER,
            include_str!("identity.vert"),
        );
        let fragment_shader = compile_shader(
            &ctx,
            WebGl2RenderingContext::FRAGMENT_SHADER,
            include_str!("inversive.frag"),
        );
        let program = ctx.create_program().unwrap();
        ctx.attach_shader(&program, &vertex_shader);
        ctx.attach_shader(&program, &fragment_shader);
        ctx.link_program(&program);
        let link_status = ctx
            .get_program_parameter(&program, WebGl2RenderingContext::LINK_STATUS)
            .as_bool()
            .unwrap();
        let link_msg = if link_status {
            "Linked successfully"
        } else {
            "Linking failed"
        };
        console::log_1(&JsValue::from(link_msg));
        ctx.use_program(Some(&program));
        /* DEBUG */
        // print the maximum number of vectors that can be passed as
        // uniforms to a fragment shader. the OpenGL ES 3.0 standard
        // requires this maximum to be at least 224, as discussed in the
        // documentation of the GL_MAX_FRAGMENT_UNIFORM_VECTORS parameter
        // here:
        //
        //   https://registry.khronos.org/OpenGL-Refpages/es3.0/html/glGet.xhtml
        //
        // there are also other size limits. for example, on Aaron's
        // machine, the the length of a float or genType array seems to be
        // capped at 1024 elements
        console::log_2(
            &ctx.get_parameter(WebGl2RenderingContext::MAX_FRAGMENT_UNIFORM_VECTORS).unwrap(),
            &JsValue::from("uniform vectors available")
        );
        // find indices of vertex attributes and uniforms
        const SPHERE_MAX: usize = 200;
        let position_index = ctx.get_attrib_location(&program, "position") as u32;
        let sphere_cnt_loc = ctx.get_uniform_location(&program, "sphere_cnt");
        let sphere_sp_locs = get_uniform_array_locations::<SPHERE_MAX>(
            &ctx, &program, "sphere_list", Some("sp")
        );
        let sphere_lt_locs = get_uniform_array_locations::<SPHERE_MAX>(
            &ctx, &program, "sphere_list", Some("lt")
        );
        let color_locs = get_uniform_array_locations::<SPHERE_MAX>(
            &ctx, &program, "color_list", None
        );
        let highlight_locs = get_uniform_array_locations::<SPHERE_MAX>(
            &ctx, &program, "highlight_list", None
        );
        let resolution_loc = ctx.get_uniform_location(&program, "resolution");
        let shortdim_loc = ctx.get_uniform_location(&program, "shortdim");
        let opacity_loc = ctx.get_uniform_location(&program, "opacity");
        let layer_threshold_loc = ctx.get_uniform_location(&program, "layer_threshold");
        let debug_mode_loc = ctx.get_uniform_location(&program, "debug_mode");
        // create a vertex array and bind it to the graphics context
        let vertex_array = ctx.create_vertex_array().unwrap();
        ctx.bind_vertex_array(Some(&vertex_array));
        // set the vertex positions
        const VERTEX_CNT: usize = 6;
        let positions: [f32; 3*VERTEX_CNT] = [
            // northwest triangle
            -1.0, -1.0, 0.0,
            -1.0,  1.0, 0.0,
             1.0,  1.0, 0.0,
            // southeast triangle
            -1.0, -1.0, 0.0,
             1.0,  1.0, 0.0,
             1.0, -1.0, 0.0
        ];
        bind_vertex_attrib(&ctx, position_index, 3, &positions);
        // set up a repainting routine
        let (_, start_animation_loop, _) = create_raf(move || {
            // get the time step
            let time = performance.now();
            let time_step = 0.001*(time - last_time);
            last_time = time;
            // get the navigation state
            let pitch_up_val = pitch_up.get();
            let pitch_down_val = pitch_down.get();
            let yaw_right_val = yaw_right.get();
            let yaw_left_val = yaw_left.get();
            let roll_ccw_val = roll_ccw.get();
            let roll_cw_val = roll_cw.get();
            let zoom_in_val = zoom_in.get();
            let zoom_out_val = zoom_out.get();
            let turntable_val = turntable.get(); /* BENCHMARKING */
            // update the assembly's orientation
            let ang_vel = {
                let pitch = pitch_up_val - pitch_down_val;
                let yaw = yaw_right_val - yaw_left_val;
                let roll = roll_ccw_val - roll_cw_val;
                if pitch != 0.0 || yaw != 0.0 || roll != 0.0 {
                    ROT_SPEED * Vector3::new(-pitch, yaw, roll).normalize()
                } else {
                    Vector3::zeros()
                }
            } /* BENCHMARKING */ + if turntable_val {
                Vector3::new(0.0, TURNTABLE_SPEED, 0.0)
            } else {
                Vector3::zeros()
            };
            let mut rotation_sp = rotation.fixed_view_mut::<3, 3>(0, 0);
            rotation_sp.copy_from(
                Rotation3::from_scaled_axis(time_step * ang_vel).matrix()
            );
            orientation = &rotation * &orientation;
            // update the assembly's location
            let zoom = zoom_out_val - zoom_in_val;
            location_z *= (time_step * ZOOM_SPEED * zoom).exp();
            if scene_changed.get() {
                /* INSTRUMENTS */
                // measure mean frame interval
                frames_since_last_sample += 1;
                if frames_since_last_sample >= SAMPLE_PERIOD {
                    mean_frame_interval.set((time - last_sample_time) / (SAMPLE_PERIOD as f64));
                    last_sample_time = time;
                    frames_since_last_sample = 0;
                }
                // find the map from assembly space to world space
                let location = {
                    let u = -location_z;
                    DMatrix::from_column_slice(5, 5, &[
                        1.0, 0.0,   0.0, 0.0, 0.0,
                        0.0, 1.0,   0.0, 0.0, 0.0,
                        0.0, 0.0,   1.0, 0.0,   u,
                        0.0, 0.0, 2.0*u, 1.0, u*u,
                        0.0, 0.0,   0.0, 0.0, 1.0
                    ])
                };
                let assembly_to_world = &location * &orientation;
                // get the assembly
                let (
                    elt_cnt,
                    reps_world,
                    colors,
                    highlights
                ) = state.assembly.elements.with(|elts| {
                    (
                        // number of elements
                        elts.len() as i32,
                        // representation vectors in world coordinates
                        elts.iter().map(
                            |(_, elt)| elt.representation.with(|rep| &assembly_to_world * rep)
                        ).collect::<Vec<_>>(),
                        // colors
                        elts.iter().map(|(key, elt)| {
                            if state.selection.with(|sel| sel.contains(&key)) {
                                elt.color.map(|ch| 0.2 + 0.8*ch)
                            } else {
                                elt.color
                            }
                        }).collect::<Vec<_>>(),
                        // highlight levels
                        elts.iter().map(|(key, _)| {
                            if state.selection.with(|sel| sel.contains(&key)) {
                                1.0_f32
                            } else {
                                HIGHLIGHT
                            }
                        }).collect::<Vec<_>>()
                    )
                });
                // set the resolution
                let width = canvas.width() as f32;
                let height = canvas.height() as f32;
                ctx.uniform2f(resolution_loc.as_ref(), width, height);
                ctx.uniform1f(shortdim_loc.as_ref(), width.min(height));
                // pass the assembly
                ctx.uniform1i(sphere_cnt_loc.as_ref(), elt_cnt);
                for n in 0..reps_world.len() {
                    let v = &reps_world[n];
                    ctx.uniform3f(
                        sphere_sp_locs[n].as_ref(),
                        v[0] as f32, v[1] as f32, v[2] as f32
                    );
                    ctx.uniform2f(
                        sphere_lt_locs[n].as_ref(),
                        v[3] as f32, v[4] as f32
                    );
                    ctx.uniform3fv_with_f32_array(
                        color_locs[n].as_ref(),
                        &colors[n]
                    );
                    ctx.uniform1f(
                        highlight_locs[n].as_ref(),
                        highlights[n]
                    );
                }
                // pass the display parameters
                ctx.uniform1f(opacity_loc.as_ref(), OPACITY);
                ctx.uniform1i(layer_threshold_loc.as_ref(), LAYER_THRESHOLD);
                ctx.uniform1i(debug_mode_loc.as_ref(), DEBUG_MODE);
                // draw the scene
                ctx.draw_arrays(WebGl2RenderingContext::TRIANGLES, 0, VERTEX_CNT as i32);
                // clear the scene change flag
                scene_changed.set(
                    pitch_up_val != 0.0
                    || pitch_down_val != 0.0
                    || yaw_left_val != 0.0
                    || yaw_right_val != 0.0
                    || roll_cw_val != 0.0
                    || roll_ccw_val != 0.0
                    || zoom_in_val != 0.0
                    || zoom_out_val != 0.0
                    || turntable_val /* BENCHMARKING */
                );
            } else {
                frames_since_last_sample = 0;
                mean_frame_interval.set(-1.0);
            }
        });
        start_animation_loop();
    });
    let set_nav_signal = move |event: KeyboardEvent, value: f64| {
        let mut navigating = true;
        let shift = event.shift_key();
        match event.key().as_str() {
            "ArrowUp" if shift => zoom_in.set(value),
            "ArrowDown" if shift => zoom_out.set(value),
            "ArrowUp" => pitch_up.set(value),
            "ArrowDown" => pitch_down.set(value),
            "ArrowRight" if shift => roll_cw.set(value),
            "ArrowLeft" if shift => roll_ccw.set(value),
            "ArrowRight" => yaw_right.set(value),
            "ArrowLeft" => yaw_left.set(value),
            _ => navigating = false
        };
        if navigating {
            scene_changed.set(true);
            event.prevent_default();
        }
    };
    view! {
        /* TO DO */
        // switch back to integer-valued parameters when that becomes possible
        // again
        canvas(
            ref=display,
            width="600",
            height="600",
            tabindex="0",
            on:keydown=move |event: KeyboardEvent| {
                if event.key() == "Shift" {
                    roll_cw.set(yaw_right.get());
                    roll_ccw.set(yaw_left.get());
                    zoom_in.set(pitch_up.get());
                    zoom_out.set(pitch_down.get());
                    yaw_right.set(0.0);
                    yaw_left.set(0.0);
                    pitch_up.set(0.0);
                    pitch_down.set(0.0);
                } else {
                    if event.key() == "Enter" { /* BENCHMARKING */
                        turntable.set_fn(|turn| !turn);
                        scene_changed.set(true);
                    }
                    set_nav_signal(event, 1.0);
                }
            },
            on:keyup=move |event: KeyboardEvent| {
                if event.key() == "Shift" {
                    yaw_right.set(roll_cw.get());
                    yaw_left.set(roll_ccw.get());
                    pitch_up.set(zoom_in.get());
                    pitch_down.set(zoom_out.get());
                    roll_cw.set(0.0);
                    roll_ccw.set(0.0);
                    zoom_in.set(0.0);
                    zoom_out.set(0.0);
                } else {
                    set_nav_signal(event, 0.0);
                }
            },
            on:blur=move |_| {
                pitch_up.set(0.0);
                pitch_down.set(0.0);
                yaw_right.set(0.0);
                yaw_left.set(0.0);
                roll_ccw.set(0.0);
                roll_cw.set(0.0);
            }
        )
    }
 }
--- a/app-proto/src/engine.rs
+++ b/app-proto/src/engine.rs
@ -1,540 +0,0 @@
 use lazy_static::lazy_static;
 use nalgebra::{Const, DMatrix, DVector, Dyn};
 use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
 // --- elements ---
 #[cfg(test)]
 pub fn point(x: f64, y: f64, z: f64) -> DVector<f64> {
    DVector::from_column_slice(&[x, y, z, 0.5, 0.5*(x*x + y*y + z*z)])
 }
 // the sphere with the given center and radius, with inward-pointing normals
 pub fn sphere(center_x: f64, center_y: f64, center_z: f64, radius: f64) -> DVector<f64> {
    let center_norm_sq = center_x * center_x + center_y * center_y + center_z * center_z;
    DVector::from_column_slice(&[
        center_x / radius,
        center_y / radius,
        center_z / radius,
        0.5 / radius,
        0.5 * (center_norm_sq / radius - radius)
    ])
 }
 // the sphere of curvature `curv` whose closest point to the origin has position
 // `off * dir` and normal `dir`, where `dir` is a unit vector. setting the
 // curvature to zero gives a plane
 pub fn sphere_with_offset(dir_x: f64, dir_y: f64, dir_z: f64, off: f64, curv: f64) -> DVector<f64> {
    let norm_sp = 1.0 + off * curv;
    DVector::from_column_slice(&[
        norm_sp * dir_x,
        norm_sp * dir_y,
        norm_sp * dir_z,
        0.5 * curv,
        off * (1.0 + 0.5 * off * curv)
    ])
 }
 // --- partial matrices ---
 struct MatrixEntry {
    index: (usize, usize),
    value: f64
 }
 pub struct PartialMatrix(Vec<MatrixEntry>);
 impl PartialMatrix {
    pub fn new() -> PartialMatrix {
        PartialMatrix(Vec::<MatrixEntry>::new())
    }
    pub fn push_sym(&mut self, row: usize, col: usize, value: f64) {
        let PartialMatrix(entries) = self;
        entries.push(MatrixEntry { index: (row, col), value: value });
        if row != col {
            entries.push(MatrixEntry { index: (col, row), value: value });
        }
    }
    /* DEBUG */
    pub fn log_to_console(&self) {
        let PartialMatrix(entries) = self;
        for ent in entries {
            let ent_str = format!("  {} {} {}", ent.index.0, ent.index.1, ent.value);
            console::log_1(&JsValue::from(ent_str.as_str()));
        }
    }
    fn proj(&self, a: &DMatrix<f64>) -> DMatrix<f64> {
        let mut result = DMatrix::<f64>::zeros(a.nrows(), a.ncols());
        let PartialMatrix(entries) = self;
        for ent in entries {
            result[ent.index] = a[ent.index];
        }
        result
    }
    fn sub_proj(&self, rhs: &DMatrix<f64>) -> DMatrix<f64> {
        let mut result = DMatrix::<f64>::zeros(rhs.nrows(), rhs.ncols());
        let PartialMatrix(entries) = self;
        for ent in entries {
            result[ent.index] = ent.value - rhs[ent.index];
        }
        result
    }
 }
 // --- descent history ---
 pub struct DescentHistory {
    pub config: Vec<DMatrix<f64>>,
    pub scaled_loss: Vec<f64>,
    pub neg_grad: Vec<DMatrix<f64>>,
    pub min_eigval: Vec<f64>,
    pub base_step: Vec<DMatrix<f64>>,
    pub backoff_steps: Vec<i32>
 }
 impl DescentHistory {
    fn new() -> DescentHistory {
        DescentHistory {
            config: Vec::<DMatrix<f64>>::new(),
            scaled_loss: Vec::<f64>::new(),
            neg_grad: Vec::<DMatrix<f64>>::new(),
            min_eigval: Vec::<f64>::new(),
            base_step: Vec::<DMatrix<f64>>::new(),
            backoff_steps: Vec::<i32>::new(),
        }
    }
 }
 // --- gram matrix realization ---
 // the Lorentz form
 lazy_static! {
    static ref Q: DMatrix<f64> = DMatrix::from_row_slice(5, 5, &[
        1.0, 0.0, 0.0,  0.0,  0.0,
        0.0, 1.0, 0.0,  0.0,  0.0,
        0.0, 0.0, 1.0,  0.0,  0.0,
        0.0, 0.0, 0.0,  0.0, -2.0,
        0.0, 0.0, 0.0, -2.0,  0.0
    ]);
 }
 struct SearchState {
    config: DMatrix<f64>,
    err_proj: DMatrix<f64>,
    loss: f64
 }
 impl SearchState {
    fn from_config(gram: &PartialMatrix, config: DMatrix<f64>) -> SearchState {
        let err_proj = gram.sub_proj(&(config.tr_mul(&*Q) * &config));
        let loss = err_proj.norm_squared();
        SearchState {
            config: config,
            err_proj: err_proj,
            loss: loss
        }
    }
 }
 fn basis_matrix(index: (usize, usize), nrows: usize, ncols: usize) -> DMatrix<f64> {
    let mut result = DMatrix::<f64>::zeros(nrows, ncols);
    result[index] = 1.0;
    result
 }
 // use backtracking line search to find a better configuration
 fn seek_better_config(
    gram: &PartialMatrix,
    state: &SearchState,
    base_step: &DMatrix<f64>,
    base_target_improvement: f64,
    min_efficiency: f64,
    backoff: f64,
    max_backoff_steps: i32
 ) -> Option<(SearchState, i32)> {
    let mut rate = 1.0;
    for backoff_steps in 0..max_backoff_steps {
        let trial_config = &state.config + rate * base_step;
        let trial_state = SearchState::from_config(gram, trial_config);
        let improvement = state.loss - trial_state.loss;
        if improvement >= min_efficiency * rate * base_target_improvement {
            return Some((trial_state, backoff_steps));
        }
        rate *= backoff;
    }
    None
 }
 // seek a matrix `config` for which `config' * Q * config` matches the partial
 // matrix `gram`. use gradient descent starting from `guess`
 pub fn realize_gram(
    gram: &PartialMatrix,
    guess: DMatrix<f64>,
    frozen: &[(usize, usize)],
    scaled_tol: f64,
    min_efficiency: f64,
    backoff: f64,
    reg_scale: f64,
    max_descent_steps: i32,
    max_backoff_steps: i32
 ) -> (DMatrix<f64>, bool, DescentHistory) {
    // start the descent history
    let mut history = DescentHistory::new();
    // find the dimension of the search space
    let element_dim = guess.nrows();
    let assembly_dim = guess.ncols();
    let total_dim = element_dim * assembly_dim;
    // scale the tolerance
    let scale_adjustment = (gram.0.len() as f64).sqrt();
    let tol = scale_adjustment * scaled_tol;
    // convert the frozen indices to stacked format
    let frozen_stacked: Vec<usize> = frozen.into_iter().map(
        |index| index.1*element_dim + index.0
    ).collect();
    // use Newton's method with backtracking and gradient descent backup
    let mut state = SearchState::from_config(gram, guess);
    for _ in 0..max_descent_steps {
        // stop if the loss is tolerably low
        history.config.push(state.config.clone());
        history.scaled_loss.push(state.loss / scale_adjustment);
        if state.loss < tol { break; }
        // find the negative gradient of the loss function
        let neg_grad = 4.0 * &*Q * &state.config * &state.err_proj;
        let mut neg_grad_stacked = neg_grad.clone().reshape_generic(Dyn(total_dim), Const::<1>);
        history.neg_grad.push(neg_grad.clone());
        // find the negative Hessian of the loss function
        let mut hess_cols = Vec::<DVector<f64>>::with_capacity(total_dim);
        for col in 0..assembly_dim {
            for row in 0..element_dim {
                let index = (row, col);
                let basis_mat = basis_matrix(index, element_dim, assembly_dim);
                let neg_d_err =
                    basis_mat.tr_mul(&*Q) * &state.config
                    + state.config.tr_mul(&*Q) * &basis_mat;
                let neg_d_err_proj = gram.proj(&neg_d_err);
                let deriv_grad = 4.0 * &*Q * (
                    -&basis_mat * &state.err_proj
                    + &state.config * &neg_d_err_proj
                );
                hess_cols.push(deriv_grad.reshape_generic(Dyn(total_dim), Const::<1>));
            }
        }
        let mut hess = DMatrix::from_columns(hess_cols.as_slice());
        // regularize the Hessian
        let min_eigval = hess.symmetric_eigenvalues().min();
        if min_eigval <= 0.0 {
            hess -= reg_scale * min_eigval * DMatrix::identity(total_dim, total_dim);
        }
        history.min_eigval.push(min_eigval);
        // project the negative gradient and negative Hessian onto the
        // orthogonal complement of the frozen subspace
        let zero_col = DVector::zeros(total_dim);
        let zero_row = zero_col.transpose();
        for &k in &frozen_stacked {
            neg_grad_stacked[k] = 0.0;
            hess.set_row(k, &zero_row);
            hess.set_column(k, &zero_col);
            hess[(k, k)] = 1.0;
        }
        // compute the Newton step
        /*
          we need to either handle or eliminate the case where the minimum
          eigenvalue of the Hessian is zero, so the regularized Hessian is
          singular. right now, this causes the Cholesky decomposition to return
          `None`, leading to a panic when we unrap
        */
        let base_step_stacked = hess.cholesky().unwrap().solve(&neg_grad_stacked);
        let base_step = base_step_stacked.reshape_generic(Dyn(element_dim), Dyn(assembly_dim));
        history.base_step.push(base_step.clone());
        // use backtracking line search to find a better configuration
        match seek_better_config(
            gram, &state, &base_step, neg_grad.dot(&base_step),
            min_efficiency, backoff, max_backoff_steps
        ) {
            Some((better_state, backoff_steps)) => {
                state = better_state;
                history.backoff_steps.push(backoff_steps);
            },
            None => return (state.config, false, history)
        };
    }
    (state.config, state.loss < tol, history)
 }
 // --- tests ---
 #[cfg(test)]
 mod tests {
    use std::{array, f64::consts::PI};
    use super::*;
    #[test]
    fn sub_proj_test() {
        let target = PartialMatrix(vec![
            MatrixEntry { index: (0, 0), value: 19.0 },
            MatrixEntry { index: (0, 2), value: 39.0 },
            MatrixEntry { index: (1, 1), value: 59.0 },
            MatrixEntry { index: (1, 2), value: 69.0 }
        ]);
        let attempt = DMatrix::<f64>::from_row_slice(2, 3, &[
            1.0, 2.0, 3.0,
            4.0, 5.0, 6.0
        ]);
        let expected_result = DMatrix::<f64>::from_row_slice(2, 3, &[
            18.0, 0.0, 36.0,
            0.0, 54.0, 63.0
        ]);
        assert_eq!(target.sub_proj(&attempt), expected_result);
    }
    #[test]
    fn zero_loss_test() {
        let gram = PartialMatrix({
            let mut entries = Vec::<MatrixEntry>::new();
            for j in 0..3 {
                for k in 0..3 {
                    entries.push(MatrixEntry {
                        index: (j, k),
                        value: if j == k { 1.0 } else { -1.0 }
                    });
                }
            }
            entries
        });
        let config = {
            let a: f64 = 0.75_f64.sqrt();
            DMatrix::from_columns(&[
                sphere(1.0, 0.0, 0.0, a),
                sphere(-0.5, a, 0.0, a),
                sphere(-0.5, -a, 0.0, a)
            ])
        };
        let state = SearchState::from_config(&gram, config);
        assert!(state.loss.abs() < f64::EPSILON);
    }
    // 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/
    //
    #[test]
    fn irisawa_hexlet_test() {
        let gram = PartialMatrix({
            let mut entries = Vec::<MatrixEntry>::new();
            for s in 0..9 {
                // each sphere is represented by a spacelike vector
                entries.push(MatrixEntry { index: (s, s), value: 1.0 });
                // the circumscribing sphere is tangent to all of the other
                // spheres, with matching orientation
                if s > 0 {
                    entries.push(MatrixEntry { index: (0, s), value: 1.0 });
                    entries.push(MatrixEntry { index: (s, 0), value: 1.0 });
                }
                if s > 2 {
                    // each chain sphere is tangent to the "sun" and "moon"
                    // spheres, with opposing orientation
                    for n in 1..3 {
                        entries.push(MatrixEntry { index: (s, n), value: -1.0 });
                        entries.push(MatrixEntry { index: (n, s), value: -1.0 });
                    }
                    // each chain sphere is tangent to the next chain sphere,
                    // with opposing orientation
                    let s_next = 3 + (s-2) % 6;
                    entries.push(MatrixEntry { index: (s, s_next), value: -1.0 });
                    entries.push(MatrixEntry { index: (s_next, s), value: -1.0 });
                }
            }
            entries
        });
        let guess = DMatrix::from_columns(
            [
                sphere(0.0, 0.0, 0.0, 15.0),
                sphere(0.0, 0.0, -9.0, 5.0),
                sphere(0.0, 0.0, 11.0, 3.0)
            ].into_iter().chain(
                (1..=6).map(
                    |k| {
                        let ang = (k as f64) * PI/3.0;
                        sphere(9.0 * ang.cos(), 9.0 * ang.sin(), 0.0, 2.5)
                    }
                )
            ).collect::<Vec<_>>().as_slice()
        );
        let frozen: [(usize, usize); 4] = array::from_fn(|k| (3, k));
        const SCALED_TOL: f64 = 1.0e-12;
        let (config, success, history) = realize_gram(
            &gram, guess, &frozen,
            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
        );
        let entry_tol = SCALED_TOL.sqrt();
        let solution_diams = [30.0, 10.0, 6.0, 5.0, 15.0, 10.0, 3.75, 2.5, 2.0 + 8.0/11.0];
        for (k, diam) in solution_diams.into_iter().enumerate() {
            assert!((config[(3, k)] - 1.0 / diam).abs() < entry_tol);
        }
        print!("\nCompleted Gram matrix:{}", config.tr_mul(&*Q) * &config);
        if success {
            println!("Target accuracy achieved!");
        } else {
            println!("Failed to reach target accuracy");
        }
        println!("Steps: {}", history.scaled_loss.len() - 1);
        println!("Loss: {}", history.scaled_loss.last().unwrap());
        if success {
            println!("\nChain diameters:");
            println!("  {} sun (given)", 1.0 / config[(3, 3)]);
            for k in 4..9 {
                println!("  {} sun", 1.0 / config[(3, k)]);
            }
        }
        println!("\nStep │ Loss\n─────┼────────────────────────────────");
        for (step, scaled_loss) in history.scaled_loss.into_iter().enumerate() {
            println!("{:<4} │ {}", step, scaled_loss);
        }
    }
    // --- process inspection examples ---
    // these tests are meant for human inspection, not automated use. run them
    // one at a time in `--nocapture` mode and read through the results and
    // optimization histories that they print out. the `run-examples` script
    // will run all of them
    #[test]
    fn three_spheres_example() {
        let gram = PartialMatrix({
            let mut entries = Vec::<MatrixEntry>::new();
            for j in 0..3 {
                for k in 0..3 {
                    entries.push(MatrixEntry {
                        index: (j, k),
                        value: if j == k { 1.0 } else { -1.0 }
                    });
                }
            }
            entries
        });
        let guess = {
            let a: f64 = 0.75_f64.sqrt();
            DMatrix::from_columns(&[
                sphere(1.0, 0.0, 0.0, 1.0),
                sphere(-0.5, a, 0.0, 1.0),
                sphere(-0.5, -a, 0.0, 1.0)
            ])
        };
        println!();
        let (config, success, history) = realize_gram(
            &gram, guess, &[],
            1.0e-12, 0.5, 0.9, 1.1, 200, 110
        );
        print!("\nCompleted Gram matrix:{}", config.tr_mul(&*Q) * &config);
        if success {
            println!("Target accuracy achieved!");
        } else {
            println!("Failed to reach target accuracy");
        }
        println!("Steps: {}", history.scaled_loss.len() - 1);
        println!("Loss: {}", history.scaled_loss.last().unwrap());
        println!("\nStep │ Loss\n─────┼────────────────────────────────");
        for (step, scaled_loss) in history.scaled_loss.into_iter().enumerate() {
            println!("{:<4} │ {}", step, scaled_loss);
        }
    }
    #[test]
    fn point_on_sphere_example() {
        let gram = PartialMatrix({
            let mut entries = Vec::<MatrixEntry>::new();
            for j in 0..2 {
                for k in 0..2 {
                    entries.push(MatrixEntry {
                        index: (j, k),
                        value: if (j, k) == (1, 1) { 1.0 } else { 0.0 }
                    });
                }
            }
            entries
        });
        let guess = DMatrix::from_columns(&[
            point(0.0, 0.0, 2.0),
            sphere(0.0, 0.0, 0.0, 1.0)
        ]);
        let frozen = [(3, 0)];
        println!();
        let (config, success, history) = realize_gram(
            &gram, guess, &frozen,
            1.0e-12, 0.5, 0.9, 1.1, 200, 110
        );
        print!("\nCompleted Gram matrix:{}", config.tr_mul(&*Q) * &config);
        print!("Configuration:{}", config);
        if success {
            println!("Target accuracy achieved!");
        } else {
            println!("Failed to reach target accuracy");
        }
        println!("Steps: {}", history.scaled_loss.len() - 1);
        println!("Loss: {}", history.scaled_loss.last().unwrap());
        println!("\nStep │ Loss\n─────┼────────────────────────────────");
        for (step, scaled_loss) in history.scaled_loss.into_iter().enumerate() {
            println!("{:<4} │ {}", step, scaled_loss);
        }
    }
    /* TO DO */
    // --- new test placed here to avoid merge conflict ---
    // at the frozen indices, the optimization steps should have exact zeros,
    // and the realized configuration should match the initial guess
    #[test]
    fn frozen_entry_test() {
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            for j in 0..2 {
                for k in j..2 {
                    gram_to_be.push_sym(j, k, if (j, k) == (1, 1) { 1.0 } else { 0.0 });
                }
            }
            gram_to_be
        };
        let guess = DMatrix::from_columns(&[
            point(0.0, 0.0, 2.0),
            sphere(0.0, 0.0, 0.0, 1.0)
        ]);
        let frozen = [(3, 0), (3, 1)];
        println!();
        let (config, success, history) = realize_gram(
            &gram, guess.clone(), &frozen,
            1.0e-12, 0.5, 0.9, 1.1, 200, 110
        );
        assert_eq!(success, true);
        for base_step in history.base_step.into_iter() {
            for index in frozen {
                assert_eq!(base_step[index], 0.0);
            }
        }
        for index in frozen {
            assert_eq!(config[index], guess[index]);
        }
    }
 }
--- a/app-proto/src/identity.vert
+++ b/app-proto/src/identity.vert
@ -1,7 +0,0 @@
 #version 300 es
 in vec4 position;
 void main() {
    gl_Position = position;
 }
--- a/app-proto/src/inversive.frag
+++ b/app-proto/src/inversive.frag
@ -1,234 +0,0 @@
 #version 300 es
 precision highp float;
 out vec4 outColor;
 // --- inversive geometry ---
 struct vecInv {
    vec3 sp;
    vec2 lt;
 };
 // --- uniforms ---
 // assembly
 const int SPHERE_MAX = 200;
 uniform int sphere_cnt;
 uniform vecInv sphere_list[SPHERE_MAX];
 uniform vec3 color_list[SPHERE_MAX];
 uniform float highlight_list[SPHERE_MAX];
 // view
 uniform vec2 resolution;
 uniform float shortdim;
 // controls
 uniform float opacity;
 uniform int layer_threshold;
 uniform bool debug_mode;
 // light and camera
 const float focal_slope = 0.3;
 const vec3 light_dir = normalize(vec3(2., 2., 1.));
 const float ixn_threshold = 0.005;
 const float INTERIOR_DIMMING = 0.7;
 // --- sRGB ---
 // map colors from RGB space to sRGB space, as specified in the sRGB standard
 // (IEC 61966-2-1:1999)
 //
 //   https://www.color.org/sRGB.pdf
 //   https://www.color.org/chardata/rgb/srgb.xalter
 //
 // in RGB space, color value is proportional to light intensity, so linear
 // color-vector interpolation corresponds to physical light mixing. in sRGB
 // space, the color encoding used by many monitors, we use more of the value
 // interval to represent low intensities, and less of the interval to represent
 // high intensities. this improves color quantization
 float sRGB(float t) {
    if (t <= 0.0031308) {
        return 12.92*t;
    } else {
        return 1.055*pow(t, 5./12.) - 0.055;
    }
 }
 vec3 sRGB(vec3 color) {
    return vec3(sRGB(color.r), sRGB(color.g), sRGB(color.b));
 }
 // --- shading ---
 struct Fragment {
    vec3 pt;
    vec3 normal;
    vec4 color;
 };
 Fragment sphere_shading(vecInv v, vec3 pt, vec3 base_color) {
    // the expression for normal needs to be checked. it's supposed to give the
    // negative gradient of the lorentz product between the impact point vector
    // and the sphere vector with respect to the coordinates of the impact
    // point. i calculated it in my head and decided that the result looked good
    // enough for now
    vec3 normal = normalize(-v.sp + 2.*v.lt.s*pt);
    float incidence = dot(normal, light_dir);
    float illum = mix(0.4, 1.0, max(incidence, 0.0));
    return Fragment(pt, normal, vec4(illum * base_color, opacity));
 }
 float intersection_dist(Fragment a, Fragment b) {
    float intersection_sin = length(cross(a.normal, b.normal));
    vec3 disp = a.pt - b.pt;
    return max(
        abs(dot(a.normal, disp)),
        abs(dot(b.normal, disp))
    ) / intersection_sin;
 }
 // --- ray-casting ---
 struct TaggedDepth {
    float depth;
    float dimming;
    int id;
 };
 // if `a/b` is less than this threshold, we approximate `a*u^2 + b*u + c` by
 // the linear function `b*u + c`
 const float DEG_THRESHOLD = 1e-9;
 // the depths, represented as multiples of `dir`, where the line generated by
 // `dir` hits the sphere represented by `v`. if both depths are positive, the
 // smaller one is returned in the first component. if only one depth is
 // positive, it could be returned in either component
 vec2 sphere_cast(vecInv v, vec3 dir) {
    float a = -v.lt.s * dot(dir, dir);
    float b = dot(v.sp, dir);
    float c = -v.lt.t;
    float adjust = 4.*a*c/(b*b);
    if (adjust < 1.) {
        // as long as `b` is non-zero, the linear approximation of
        //
        //   a*u^2 + b*u + c
        //
        // at `u = 0` will reach zero at a finite depth `u_lin`. the root of the
        // quadratic adjacent to `u_lin` is stored in `lin_root`. if both roots
        // have the same sign, `lin_root` will be the one closer to `u = 0`
        float square_rect_ratio = 1. + sqrt(1. - adjust);
        float lin_root = -(2.*c)/b / square_rect_ratio;
        if (abs(a) > DEG_THRESHOLD * abs(b)) {
            return vec2(lin_root, -b/(2.*a) * square_rect_ratio);
        } else {
            return vec2(lin_root, -1.);
        }
    } else {
        // the line through `dir` misses the sphere completely
        return vec2(-1., -1.);
    }
 }
 void main() {
    vec2 scr = (2.*gl_FragCoord.xy - resolution) / shortdim;
    vec3 dir = vec3(focal_slope * scr, -1.);
    // cast rays through the spheres
    const int LAYER_MAX = 12;
    TaggedDepth top_hits [LAYER_MAX];
    int layer_cnt = 0;
    for (int id = 0; id < sphere_cnt; ++id) {
        // find out where the ray hits the sphere
        vec2 hit_depths = sphere_cast(sphere_list[id], dir);
        // insertion-sort the points we hit into the hit list
        float dimming = 1.;
        for (int side = 0; side < 2; ++side) {
            float depth = hit_depths[side];
            if (depth > 0.) {
                for (int layer = layer_cnt; layer >= 0; --layer) {
                    if (layer < 1 || top_hits[layer-1].depth <= depth) {
                        // we're not as close to the screen as the hit before
                        // the empty slot, so insert here
                        if (layer < LAYER_MAX) {
                            top_hits[layer] = TaggedDepth(depth, dimming, id);
                        }
                        break;
                    } else {
                        // we're closer to the screen than the hit before the
                        // empty slot, so move that hit into the empty slot
                        top_hits[layer] = top_hits[layer-1];
                    }
                }
                layer_cnt = min(layer_cnt + 1, LAYER_MAX);
                dimming = INTERIOR_DIMMING;
            }
        }
    }
    /* DEBUG */
    // in debug mode, show the layer count instead of the shaded image
    if (debug_mode) {
        // at the bottom of the screen, show the color scale instead of the
        // layer count
        if (gl_FragCoord.y < 10.) layer_cnt = int(16. * gl_FragCoord.x / resolution.x);
        // convert number to color
        ivec3 bits = layer_cnt / ivec3(1, 2, 4);
        vec3 color = mod(vec3(bits), 2.);
        if (layer_cnt % 16 >= 8) {
            color = mix(color, vec3(0.5), 0.5);
        }
        outColor = vec4(color, 1.);
        return;
    }
    // composite the sphere fragments
    vec3 color = vec3(0.);
    int layer = layer_cnt - 1;
    TaggedDepth hit = top_hits[layer];
    Fragment frag_next = sphere_shading(
        sphere_list[hit.id],
        hit.depth * dir,
        hit.dimming * color_list[hit.id]
    );
    float highlight_next = highlight_list[hit.id];
    --layer;
    for (; layer >= layer_threshold; --layer) {
        // load the current fragment
        Fragment frag = frag_next;
        float highlight = highlight_next;
        // shade the next fragment
        hit = top_hits[layer];
        frag_next = sphere_shading(
            sphere_list[hit.id],
            hit.depth * dir,
            hit.dimming * color_list[hit.id]
        );
        highlight_next = highlight_list[hit.id];
        // highlight intersections
        float ixn_dist = intersection_dist(frag, frag_next);
        float max_highlight = max(highlight, highlight_next);
        float ixn_highlight = 0.5 * max_highlight * (1. - smoothstep(2./3.*ixn_threshold, 1.5*ixn_threshold, ixn_dist));
        frag.color = mix(frag.color, vec4(1.), ixn_highlight);
        frag_next.color = mix(frag_next.color, vec4(1.), ixn_highlight);
        // highlight cusps
        float cusp_cos = abs(dot(dir, frag.normal));
        float cusp_threshold = 2.*sqrt(ixn_threshold * sphere_list[hit.id].lt.s);
        float cusp_highlight = highlight * (1. - smoothstep(2./3.*cusp_threshold, 1.5*cusp_threshold, cusp_cos));
        frag.color = mix(frag.color, vec4(1.), cusp_highlight);
        // composite the current fragment
        color = mix(color, frag.color.rgb, frag.color.a);
    }
    color = mix(color, frag_next.color.rgb, frag_next.color.a);
    outColor = vec4(sRGB(color), 1.);
 }
--- a/app-proto/src/main.rs
+++ b/app-proto/src/main.rs
@ -1,42 +0,0 @@
 mod add_remove;
 mod assembly;
 mod display;
 mod engine;
 mod outline;
 use rustc_hash::FxHashSet;
 use sycamore::prelude::*;
 use add_remove::AddRemove;
 use assembly::{Assembly, ElementKey};
 use display::Display;
 use outline::Outline;
 #[derive(Clone)]
 struct AppState {
    assembly: Assembly,
    selection: Signal<FxHashSet<ElementKey>>
 }
 impl AppState {
    fn new() -> AppState {
        AppState {
            assembly: Assembly::new(),
            selection: create_signal(FxHashSet::default())
        }
    }
 }
 fn main() {
    sycamore::render(|| {
        provide_context(AppState::new());
        view! {
            div(id="sidebar") {
                AddRemove {}
                Outline {}
            }
            Display {}
        }
    });
 }
--- a/app-proto/src/outline.rs
+++ b/app-proto/src/outline.rs
@ -1,207 +0,0 @@
 use itertools::Itertools;
 use sycamore::prelude::*;
 use web_sys::{
    Event,
    HtmlInputElement,
    KeyboardEvent,
    MouseEvent,
    wasm_bindgen::JsCast
 };
 use crate::{AppState, assembly, assembly::{Constraint, ConstraintKey, ElementKey}};
 // an editable view of the Lorentz product representing a constraint
 #[component(inline_props)]
 fn LorentzProductInput(constraint: Constraint) -> View {
    view! {
        input(
            r#type="text",
            bind:value=constraint.lorentz_prod_text,
            on:change=move |event: Event| {
                let target: HtmlInputElement = event.target().unwrap().unchecked_into();
                match target.value().parse::<f64>() {
                    Ok(lorentz_prod) => batch(|| {
                        constraint.lorentz_prod.set(lorentz_prod);
                        constraint.lorentz_prod_valid.set(true);
                    }),
                    Err(_) => constraint.lorentz_prod_valid.set(false)
                };
            }
        )
    }
 }
 // a list item that shows a constraint in an outline view of an element
 #[component(inline_props)]
 fn ConstraintOutlineItem(constraint_key: ConstraintKey, element_key: ElementKey) -> View {
    let state = use_context::<AppState>();
    let assembly = &state.assembly;
    let constraint = assembly.constraints.with(|csts| csts[constraint_key].clone());
    let other_subject = if constraint.subjects.0 == element_key {
        constraint.subjects.1
    } else {
        constraint.subjects.0
    };
    let other_subject_label = assembly.elements.with(|elts| elts[other_subject].label.clone());
    let class = constraint.lorentz_prod_valid.map(
        |&lorentz_prod_valid| if lorentz_prod_valid { "constraint" } else { "constraint invalid" }
    );
    view! {
        li(class=class.get()) {
            input(r#type="checkbox", bind:checked=constraint.active)
            div(class="constraint-label") { (other_subject_label) }
            LorentzProductInput(constraint=constraint)
            div(class="status")
        }
    }
 }
 // a list item that shows an element in an outline view of an assembly
 #[component(inline_props)]
 fn ElementOutlineItem(key: ElementKey, element: assembly::Element) -> View {
    let state = use_context::<AppState>();
    let class = state.selection.map(
        move |sel| if sel.contains(&key) { "selected" } else { "" }
    );
    let label = element.label.clone();
    let rep_components = element.representation.map(
        |rep| rep.iter().map(
            |u| format!("{:.3}", u).replace("-", "\u{2212}")
        ).collect()
    );
    let constrained = element.constraints.map(|csts| csts.len() > 0);
    let constraint_list = element.constraints.map(
        |csts| csts.clone().into_iter().collect()
    );
    let details_node = create_node_ref();
    view! {
        li {
            details(ref=details_node) {
                summary(
                    class=class.get(),
                    on:keydown={
                        move |event: KeyboardEvent| {
                            match event.key().as_str() {
                                "Enter" => {
                                    if event.shift_key() {
                                        state.selection.update(|sel| {
                                            if !sel.remove(&key) {
                                                sel.insert(key);
                                            }
                                        });
                                    } else {
                                        state.selection.update(|sel| {
                                            sel.clear();
                                            sel.insert(key);
                                        });
                                    }
                                    event.prevent_default();
                                },
                                "ArrowRight" if constrained.get() => {
                                    let _ = details_node
                                        .get()
                                        .unchecked_into::<web_sys::Element>()
                                        .set_attribute("open", "");
                                },
                                "ArrowLeft" => {
                                    let _ = details_node
                                        .get()
                                        .unchecked_into::<web_sys::Element>()
                                        .remove_attribute("open");
                                },
                                _ => ()
                            }
                        }
                    }
                ) {
                    div(
                        class="element-switch",
                        on:click=|event: MouseEvent| event.stop_propagation()
                    )
                    div(
                        class="element",
                        on:click={
                            move |event: MouseEvent| {
                                if event.shift_key() {
                                    state.selection.update(|sel| {
                                        if !sel.remove(&key) {
                                            sel.insert(key);
                                        }
                                    });
                                } else {
                                    state.selection.update(|sel| {
                                        sel.clear();
                                        sel.insert(key);
                                    });
                                }
                                event.stop_propagation();
                                event.prevent_default();
                            }
                        }
                    ) {
                        div(class="element-label") { (label) }
                        div(class="element-representation") {
                            Indexed(
                                list=rep_components,
                                view=|coord_str| view! {
                                    div { (coord_str) }
                                }
                            )
                        }
                        div(class="status")
                    }
                }
                ul(class="constraints") {
                    Keyed(
                        list=constraint_list,
                        view=move |cst_key| view! {
                            ConstraintOutlineItem(
                                constraint_key=cst_key,
                                element_key=key
                            )
                        },
                        key=|cst_key| cst_key.clone()
                    )
                }
            }
        }
    }
 }
 // a component that lists the elements of the current assembly, showing the
 // constraints on each element as a collapsible sub-list. its implementation
 // is based on Kate Morley's HTML + CSS tree views:
 //
 //   https://iamkate.com/code/tree-views/
 //
 #[component]
 pub fn Outline() -> View {
    let state = use_context::<AppState>();
    // list the elements alphabetically by ID
    let element_list = state.assembly.elements.map(
        |elts| elts
            .clone()
            .into_iter()
            .sorted_by_key(|(_, elt)| elt.id.clone())
            .collect()
    );
    view! {
        ul(
            id="outline",
            on:click={
                let state = use_context::<AppState>();
                move |_| state.selection.update(|sel| sel.clear())
            }
        ) {
            Keyed(
                list=element_list,
                view=|(key, elt)| view! {
                    ElementOutlineItem(key=key, element=elt)
                },
                key=|(_, elt)| elt.serial
            )
        }
    }
 }
--- a/engine-proto/gram-test/Engine.jl
+++ b/engine-proto/gram-test/Engine.jl
@ -8,8 +8,7 @@ using Optim
 export
  rand_on_shell, Q, DescentHistory,
-  realize_gram_gradient, realize_gram_newton, realize_gram_optim,
+  realize_gram_gradient, realize_gram_newton, realize_gram_optim, realize_gram
  realize_gram_alt_proj, realize_gram
 # === guessing ===
@ -144,7 +143,7 @@ function realize_gram_gradient(
      break
    end
-    # find the negative gradient of the loss function
+    # find negative gradient of loss function
    neg_grad = 4*Q*L*Δ_proj
    slope = norm(neg_grad)
    dir = neg_grad / slope
@ -233,7 +232,7 @@ function realize_gram_newton(
      break
    end
-    # find the negative gradient of the loss function
+    # find the negative gradient of loss function
    neg_grad = 4*Q*L*Δ_proj
    # find the negative Hessian of the loss function
@ -314,129 +313,6 @@ function realize_gram_optim(
  )
 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`, with an
 # alternate technique for finding the projected base step from the unprojected
 # Hessian
 function realize_gram_alt_proj(
  gram::SparseMatrixCSC{T, <:Any},
  guess::Matrix{T},
  frozen = CartesianIndex[];
  scaled_tol = 1e-30,
  min_efficiency = 0.5,
  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
  # convert the frozen indices to stacked format
  frozen_stacked = [(index[2]-1)*element_dim + index[1] for index in frozen]
  # initialize search state
  L = copy(guess)
  Δ_proj = proj_diff(gram, L'*Q*L)
  loss = dot(Δ_proj, Δ_proj)
  # use Newton's method with backtracking and gradient descent backup
  for step in 1:max_descent_steps
    # stop if the loss is tolerably low
    if loss < tol
      break
    end
    # find the negative gradient of the 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_sym = Hermitian(hess)
    push!(history.hess, hess_sym)
    # regularize the Hessian
    min_eigval = minimum(eigvals(hess_sym))
    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)
    for k in frozen_stacked
      neg_grad_stacked[k] = 0
      hess[k, :] .= 0
      hess[:, k] .= 0
      hess[k, k] = 1
    end
    base_step_stacked = Hermitian(hess) \ neg_grad_stacked
    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
    base_target_improvement = dot(neg_grad, base_step)
    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 * base_target_improvement
        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
 # 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(
@ -445,6 +321,7 @@ function realize_gram(
  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,
@ -475,19 +352,20 @@ function realize_gram(
    unfrozen_stacked = reshape(is_unfrozen, total_dim)
  end
-  # initialize search state
+  # initialize variables
  grad_rate = init_rate
  L = copy(guess)
  Δ_proj = proj_diff(gram, L'*Q*L)
  loss = dot(Δ_proj, Δ_proj)
  # 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 the loss function
+    # find the negative gradient of loss function
    neg_grad = 4*Q*L*Δ_proj
    # find the negative Hessian of the loss function
@ -542,7 +420,6 @@ function realize_gram(
    empty!(history.last_line_loss)
    rate = one(T)
    step_success = false
    base_target_improvement = dot(neg_grad, base_step)
    for backoff_steps in 0:max_backoff_steps
      history.stepsize[end] = rate
      L = L_last + rate * base_step
@ -551,7 +428,7 @@ function realize_gram(
      improvement = loss_last - loss
      push!(history.last_line_L, L)
      push!(history.last_line_loss, loss / scale_adjustment)
-      if improvement >= min_efficiency * rate * base_target_improvement
+      if improvement >= min_efficiency * rate * dot(neg_grad, base_step)
        history.backoff_steps[end] = backoff_steps
        step_success = true
        break
--- a/engine-proto/gram-test/irisawa-hexlet.jl
+++ b/engine-proto/gram-test/irisawa-hexlet.jl
@ -75,12 +75,3 @@ if success
    println("  ", 1 / L[4,k], " sun")
  end
 end
 # test an alternate technique for finding the projected base step from the
 # unprojected Hessian
 L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
 completed_gram_alt = L_alt'*Engine.Q*L_alt
 println("\nDifference in result using alternate projection:\n")
 display(completed_gram_alt - completed_gram)
 println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
 println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
--- a/engine-proto/gram-test/sphere-in-tetrahedron.jl
+++ b/engine-proto/gram-test/sphere-in-tetrahedron.jl
@ -65,12 +65,3 @@ else
 end
 println("Steps: ", size(history.scaled_loss, 1))
 println("Loss: ", history.scaled_loss[end], "\n")
 # test an alternate technique for finding the projected base step from the
 # unprojected Hessian
 L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
 completed_gram_alt = L_alt'*Engine.Q*L_alt
 println("\nDifference in result using alternate projection:\n")
 display(completed_gram_alt - completed_gram)
 println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
 println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
--- a/engine-proto/gram-test/tetrahedron-radius-ratio.jl
+++ b/engine-proto/gram-test/tetrahedron-radius-ratio.jl
@ -94,12 +94,3 @@ if success
  radius_ratio = dot(infty, Engine.Q * L[:,5]) / dot(infty, Engine.Q * L[:,6])
  println("\nCircumradius / inradius: ", radius_ratio)
 end
 # test an alternate technique for finding the projected base step from the
 # unprojected Hessian
 L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
 completed_gram_alt = L_alt'*Engine.Q*L_alt
 println("\nDifference in result using alternate projection:\n")
 display(completed_gram_alt - completed_gram)
 println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
 println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
--- a/notes/inversive.md
+++ b/notes/inversive.md
@ -41,25 +41,3 @@ I will have to work out formulas for the Euclidean distance between two entities
 In this vein, it seems as though if J1 and J2 are the reps of two points, then Q(J1,J2) = d^2/2. So then the sphere centered at J1 through J2 is (J1-(2Q(J1,J2),0,0,0,0))/sqrt(2Q(J1,J2)). Ugh has a sqrt in it. Similarly for sphere centered at J3 through J2, (J3-(2Q(J3,J2),0000))/sqrt(2Q(J3,J2)). J1,J2,J3 are collinear if these spheres are tangent, i.e. if those vectors have Q-inner-product 1, which is to say Q(J1,J3) - Q(J1,J2) - Q(J3,J2) = 2sqrt(Q(J1,J2)Q(J2,J3)). But maybe that's not the simplest way of putting it. After all, we can just say that the cross-product of the two differences is 0; that has no square roots in it.
 One conceivable way to canonicalize lines is to use the *perpendicular* plane that goes through the origin, that's uniquely defined, and anyway just amounts to I = (0,0,d) where d is the ordinary direction vector of the line; and a point J in that plane that the line goes through, which just amounts to J=(r^2,1,E) with Q(I,J) = 0, i.e. E\dot d = 0. It's also the point on the line closest to the origin. The reason that we don't usually use that point as the companion to the direction vector is that the resulting set of six coordinates is not homogeneous. But here that's not an issue, since we have our standard point coordinates and plane coordinates; and for a plane through the origin, only two of the direction coordinates are really free, and then we have the one dot-product relation, so only two of the point coordinates are really free, giving us the correct dimensionality of 4 for the set of lines. So in some sense this says that we could take naively as coordinates for a line the projection of the unit direction vector to the xy plane and the projection of the line's closest point to the origin to the xy plane. That doesn't seem to have any weird gimbal locks or discontinuities or anything. And with these coordinates, you can test if the point E=x,y,z is on the line (dx,dy,cx,cy) by extending (dx,dy) to d via dz = sqrt(1-dx^2 - dy^2), extending (cx,cy) to c by determining cz via d\dot c = 0, and then checking if d\cross(E-c) = 0. And you can see if two lines are parallel just by checking if they have the same direction vector, and if not, you can see if they are coplanar by projecting both of their closest points perpendicularly onto the line in the direction of the cross product of their directions, and if the projections match they are coplanar.
 #### Engine Conventions
 The coordinate conventions used in the engine are different from the ones used in these notes. Marking the engine vectors and coordinates with $'$, we have
 $$I' = (x', y', z', b', c'),$$
 where
 $$
 \begin{align*}
 x' & = x & b' & = b/2 \\
 y' & = y & c' & = c/2. \\
 z' & = z
 \end{align*}
 $$
 The engine uses the quadratic form $Q' = -Q$, which is expressed in engine coordinates as
 $$Q'(I'_1, I'_2) = x'_1 x'_2 + y'_1 y'_2 + z'_1 z'_2 - 2(b'_1c'_2 + c'_1 b'_2).$$
 In the `engine` module, the matrix of $Q'$ is encoded in the lazy static variable `Q`.
 In the engine's coordinate conventions, a sphere with radius $r > 0$ centered on $P = (P_x, P_y, P_z)$ is represented by the vector
 $$I'_s = \left(\frac{P_x}{r}, \frac{P_y}{r}, \frac{P_z}{r}, \frac1{2r}, \frac{\|P\|^2 - r^2}{2r}\right),$$
 which has the normalization $Q'(I'_s, I'_s) = 1$. The point $P$ is represented by the vector
 $$I'_P = \left(P_x, P_y, P_z, \frac{1}{2}, \frac{\|P\|^2}{2}\right).$$
 In the `engine` module, these formulas are encoded in the `sphere` and `point` functions.
Author	SHA1	Message	Date
Glen Whitney	cbab33fb39	doc: expand on representation of collinearity	2024-10-20 20:02:51 -07:00
Glen Whitney	5918b30bb2	doc: expand on representations of lines	2024-10-20 19:53:36 -07:00
Glen Whitney	7fb4e3adee	doc: another unicode pseudo-space	2024-10-20 19:41:05 -07:00
Glen Whitney	cbad99d356	doc: punctuation	2024-10-20 19:33:33 -07:00
Glen Whitney	47c2217bcf	doc: fix more weird Unicode characters, probably from marktest editor	2024-10-20 19:27:05 -07:00
Glen Whitney	bb881f806c	doc: Won't tex formula immediately followed by dash	2024-10-20 16:17:18 -07:00
Glen Whitney	d3e5a0bc37	doc: Weird unicode minus	2024-10-20 16:15:27 -07:00
Glen Whitney	ce62a94ddb	doc: Two more small typos	2024-10-20 16:10:01 -07:00
Glen Whitney	160cb47f55	doc: Fix typos and expand some comments a bit in notes/inversive.md	2024-10-20 16:06:11 -07:00
Aaron Fenyes	48732093f7	doc: Clean up engine comments	2024-10-16 16:00:36 -07:00
Aaron Fenyes	db1b315df2	Merge branch 'engine-proto' of code.studioinfinity.org:glen/dyna3 into engine-proto	2024-10-15 14:41:20 -07:00
Aaron Fenyes	609cd2f814	doc: Correct and clarify various relations	2024-10-15 14:24:36 -07:00
Glen Whitney	ed88c7b9f9	doc: fix latex conflict with markdown table format	2024-10-03 21:20:32 -07:00
Glen Whitney	0e929123d4	doc: correct condition that two sphere/planes intersect	2024-10-03 20:39:15 -07:00
Aaron Fenyes	3f0cedfaab	doc: Clarify characterization of center of sphere In the process, clarify the signed distance from a point to a sphere and add inversion across a sphere.	2024-09-26 01:07:22 -07:00
Aaron Fenyes	2c1a42e251	doc: Clarify orientation convention	2024-09-25 23:07:52 -07:00
Glen Whitney	23ecca3963	doc: Another note about inversive coordinates	2024-09-18 20:27:04 -07:00
Glen Whitney	a182b66301	doc: More elaboration of plane coordinates in inversive notes	2024-09-18 20:11:20 -07:00
Glen Whitney	bd3e3506e5	doc: typo in inversive notes	2024-09-18 19:52:03 -07:00
Glen Whitney	8084fdeab0	doc: Slight mods to the inversive notes	2024-09-18 19:50:23 -07:00
Aaron Fenyes	d7dbee4c05	Stow algebraic engine prototype We're using the Gram matrix engine for the next stage of development, so the algebraic engine shouldn't be at the top level anymore.	2024-07-28 20:50:04 -07:00
Aaron Fenyes	9d69a900e2	Irisawa hexlet: use Abe's terminology in comments Abe uses the names "sun" and "moon" for what Wikipedia calls the nucleus spheres.	2024-07-18 03:39:41 -07:00
Aaron Fenyes	8a77cd7484	Irisawa hexlet: drop unviable approach The approach in the deleted file can't work, because the "sun" and "moon" spheres can't be placed arbitrarily.	2024-07-18 03:21:46 -07:00
Aaron Fenyes	a26f1e3927	Add Irisawa hexlet example Hat tip Romy, who sent me the article on sangaku that led me to this problem.	2024-07-18 03:16:57 -07:00
Aaron Fenyes	19a4d49497	Clean up example formatting	2024-07-18 01:48:05 -07:00
Aaron Fenyes	71c10adbdd	Overlapping pyramids: drop outdated comment	2024-07-18 01:12:49 -07:00
Aaron Fenyes	33c09917d0	Correct scope of guess constants	2024-07-18 01:05:13 -07:00
Aaron Fenyes	b24dcc9af8	Report success correctly when step limit is reached	2024-07-18 01:04:40 -07:00
Aaron Fenyes	b040bbb7fe	Drop old code from examples	2024-07-18 00:50:48 -07:00
Aaron Fenyes	9007c8bc7c	Circles in triangle: jiggle the guess	2024-07-18 00:49:09 -07:00
Aaron Fenyes	a7f9545a37	Circles in triangle: correct frozen variables Since the self-product of the point at infinity is left unspecified, the first three components can vary without violating any constraints. To keep the point at infinity where it's supposed to be, we freeze all of its components.	2024-07-18 00:43:00 -07:00
Aaron Fenyes	3764fde2f6	Clean up formatting of notes	2024-07-18 00:27:10 -07:00
Aaron Fenyes	24dae6807b	Clarify notes on tangency	2024-07-18 00:16:23 -07:00
Aaron Fenyes	74c7f64b0c	Correct sign of normal in plane utility Clarify the relevant notes too.	2024-07-18 00:03:12 -07:00
Aaron Fenyes	d0340c0b65	Correct point utility again The balance between the light cone basis vectors was wrong, throwing the point's coordinates off by a factor of two.	2024-07-17 23:37:28 -07:00
Aaron Fenyes	69a704d414	Use notes' sign convention for light cone basis	2024-07-17 23:07:34 -07:00
Aaron Fenyes	01f44324c1	Tetrahedron radius ratio: find radius ratio	2024-07-17 22:45:17 -07:00
Aaron Fenyes	96ffc59642	Tetrahedron radius ratio: tweak guess Jiggle the vertex guesses. Put the circumscribed sphere guess on-shell.	2024-07-17 19:01:34 -07:00
Aaron Fenyes	a02b76544a	Tetrahedron radius ratio: add circumscribed sphere	2024-07-17 18:55:36 -07:00
Aaron Fenyes	6e719f9943	Tetrahedron radius ratio: correct vertex guesses	2024-07-17 18:27:58 -07:00
Aaron Fenyes	d51d43f481	Correct point utility	2024-07-17 18:27:22 -07:00
Aaron Fenyes	6d233b5ee9	Tetrahedron radius ratio: correct signs	2024-07-17 18:08:36 -07:00
Aaron Fenyes	5abd4ca6e1	Revert "Give spheres positive radii in examples" This reverts commit `4728959ae0`, which actually gave the spheres negative radii! I got confused by the sign convention differences between the notes and the engine.	2024-07-17 17:49:43 -07:00
Aaron Fenyes	ea640f4861	Start tetrahedron radius ratio example Add the vertices of the tetrahedron to the `sphere-in-tetrahedron` example.	2024-07-17 17:33:32 -07:00
Aaron Fenyes	4728959ae0	Give spheres positive radii in examples This changes the meaning of `indep_val` in the overlapping pyramids example, so we adjust `indep_val` to get a nice-looking construction.	2024-07-17 17:22:33 -07:00
Aaron Fenyes	2038103d80	Write examples directly in light cone basis	2024-07-17 15:37:14 -07:00
Aaron Fenyes	bde42ebac0	Switch engine to light cone basis	2024-07-17 14:30:43 -07:00
Aaron Fenyes	e6cf08a9b3	Make tetrahedron faces planar	2024-07-15 23:54:59 -07:00
Aaron Fenyes	7c77481f5e	Don't constrain self-product of frozen vector	2024-07-15 23:39:05 -07:00
Aaron Fenyes	1ce609836b	Implement frozen variables	2024-07-15 22:11:54 -07:00
Aaron Fenyes	b185fd4b83	Switch to backtracking Newton's method in Optim This performs much better than the trust region Newton's method for the actual `circles-in-triangle` problem. (The trust region method performs better for the simplified problem produced by the conversion bug.)	2024-07-15 15:52:38 -07:00
Aaron Fenyes	94e0d321d5	Switch back to BigFloat precision in examples	2024-07-15 14:31:30 -07:00
Aaron Fenyes	53d8c38047	Preserve explicit zeros in Gram matrix conversion In previous commits, the `circles-in-triangle` example converged much more slowly in BigFloat precision than in Float64 precision. This turned out to be a sign of a bug in the Float64 computation: converting the Gram matrix using `Float64.()` dropped the explicit zeros, removing many constraints and making the problem much easier to solve. This commit corrects the Gram matrix conversion. The Float64 search now solves the same problem as the BigFloat search, with comparable performance.	2024-07-15 14:08:57 -07:00
Aaron Fenyes	7b3efbc385	Clean up backtracking gradient descent code Drop experimental singularity handling strategies. Reduce the default tolerance to within 64-bit floating point precision. Report success.	2024-07-15 13:15:15 -07:00
Aaron Fenyes	25b09ebf92	Sketch backtracking Newton's method This code is a mess, but I'm committing it to record a working state before I start trying to clean up.	2024-07-15 11:32:04 -07:00
Aaron Fenyes	3910b9f740	Use Newton's method for polishing	2024-07-11 13:43:52 -07:00
Aaron Fenyes	d538cbf716	Correct improvement threshold by using unit step Our formula for the improvement theshold works when the step size is an absolute distance. However, in commit `4d5ea06`, the step size was measured relative to the current gradient instead. This commit scales the base step to unit length, so now the step size really is an absolute distance.	2024-07-10 23:31:44 -07:00
Aaron Fenyes	4d5ea062a3	Record gradient and last line search in history	2024-07-09 15:00:13 -07:00
Aaron Fenyes	5652719642	Require triangle sides to be planar	2024-07-09 14:10:23 -07:00
Aaron Fenyes	f84d475580	Visualize neighborhoods of global minima	2024-07-09 14:01:30 -07:00
Aaron Fenyes	77bc124170	Change loss function to match gradient	2024-07-09 14:00:24 -07:00
Aaron Fenyes	023759a267	Start "circles in triangle" from a very close guess	2024-07-08 14:21:10 -07:00
Aaron Fenyes	610fc451f0	Track slope in gradient descent history	2024-07-08 14:19:25 -07:00
Aaron Fenyes	93dd05c317	Add required package to "sphere in tetrahedron" example	2024-07-08 14:19:05 -07:00
Aaron Fenyes	9efa99e8be	Test gradient descent for circles in triangle	2024-07-08 12:56:28 -07:00
Aaron Fenyes	828498b3de	Add sphere and plane utilities to engine	2024-07-08 12:56:14 -07:00
Aaron Fenyes	736ac50b07	Test gradient descent for sphere in tetrahedron	2024-07-07 17:58:55 -07:00
Aaron Fenyes	ea354b6c2b	Randomize guess in gradient descent test Randomly perturb the pre-solved part of the guess, and randomly choose the unsolved part.	2024-07-07 17:56:12 -07:00
Aaron Fenyes	d39244d308	Host Ganja.js locally	2024-07-06 21:35:09 -07:00
Aaron Fenyes	7e94fef19e	Improve random vector generator	2024-07-06 21:32:43 -07:00
Aaron Fenyes	abc53b4705	Sketch random vector generator This needs to be rewritten: it can fail at generating spacelike vectors.	2024-07-02 17:16:31 -07:00
Aaron Fenyes	17fefff61e	Name gradient descent test more specifically	2024-07-02 17:16:19 -07:00
Aaron Fenyes	133519cacb	Encapsulate gradient descent code The completed gram matrix from this commit matches the one from commit `e7dde58` to six decimal places.	2024-07-02 15:02:59 -07:00
Aaron Fenyes	e7dde5800c	Do gradient descent entirely in BigFloat The previos version accidentally returned steps in Float64.	2024-07-02 12:35:12 -07:00
Aaron Fenyes	242d630cc6	Get Ganja.js to display planes	2024-06-27 21:49:53 -07:00
Aaron Fenyes	8eb1ebb8d2	Merge branch 'ganja' into gram	2024-06-26 15:57:07 -07:00
Aaron Fenyes	05a824834d	Let visibility controls scroll	2024-06-26 15:56:51 -07:00
Aaron Fenyes	a113f33635	Merge branch 'ganja' into gram Get visibility controls.	2024-06-26 15:52:20 -07:00
Aaron Fenyes	5ea32ac53c	Streamline visibility controls	2024-06-26 15:51:57 -07:00
Aaron Fenyes	3eb4fc6c91	Add element visibility controls	2024-06-26 15:24:31 -07:00
Aaron Fenyes	7aaf134a36	Size the viewer window automatically	2024-06-26 13:15:54 -07:00
Aaron Fenyes	c933e07312	Switch to Ganja.js basis ordering	2024-06-26 11:39:34 -07:00
Aaron Fenyes	2b6c4f4720	Avoid naming conflict with identity transformation	2024-06-26 11:28:47 -07:00
Aaron Fenyes	5aadfecf6c	Merge branch 'ganja' into gram Visualize low-rank factorization results.	2024-06-26 11:12:24 -07:00
Aaron Fenyes	4a28a47520	Update namespace of AbstractAlgebra.Rationals	2024-06-26 01:06:27 -07:00
Aaron Fenyes	a3b1f4920c	Build construction viewer module	2024-06-26 00:41:21 -07:00
Aaron Fenyes	665cb30ce0	Correct indentation of CSS	2024-06-25 23:31:00 -07:00
Aaron Fenyes	182b5bb9f6	Generate palette automatically	2024-06-25 17:57:16 -07:00
Aaron Fenyes	b7b5b9386b	Load elements from Julia into Ganja.js	2024-06-25 16:30:19 -07:00
Aaron Fenyes	06a9dda5bb	Play with reflections Try configuration of five tangent spheres.	2024-06-25 13:40:40 -07:00
Aaron Fenyes	69a9baa8ee	Add live updates to Ganja.js visualization	2024-06-25 03:11:50 -07:00
Aaron Fenyes	3b10c95d5f	Clean up examples Declare JavaScript variables. Revise Julia comments to match new code.	2024-06-25 02:58:39 -07:00
Aaron Fenyes	3c34481519	Get familiar with Ganja.js inline syntax	2024-06-25 01:54:01 -07:00
Aaron Fenyes	d1ce91d2aa	Get a Ganja.js visualization running in Blink	2024-06-24 19:37:57 -07:00
Aaron Fenyes	58a5c38e62	Try numerical low-rank factorization The best technique I've found so far is the homemade gradient descent routine in `descent-test.jl`.	2024-05-30 00:36:03 -07:00
Aaron Fenyes	ef33b8ee10	Correct signature	2024-03-01 13:26:20 -05:00
Aaron Fenyes	717e5a6200	Extend Gram matrix automatically The signature of the Minkowski form on the subspace spanned by the Gram matrix should tell us what the big Gram matrix has to look like	2024-02-21 03:00:06 -05:00
Aaron Fenyes	16826cf07c	Try out the Gram matrix approach	2024-02-20 22:35:24 -05:00
Aaron Fenyes	3170a933e4	Clean up example of three mutually tangent spheres	2024-02-15 17:16:37 -08:00
Aaron Fenyes	f2000e5731	Test different sign patterns for cosines It seems like there are real solutions if and only if the product of the cosines is positive.	2024-02-15 16:25:09 -08:00
Aaron Fenyes	ba365174d3	Find real solutions for three mutually tangent spheres I'm not sure why the solver wasn't working before. It might've been just an unlucky random number draw.	2024-02-15 16:16:06 -08:00
Aaron Fenyes	ae5db0f9ea	Make results reproducible	2024-02-15 16:00:46 -08:00
Aaron Fenyes	8d8bc9162c	Store elements in arrays to keep order stable This seems to restore reproducibility.	2024-02-15 15:42:26 -08:00
Aaron Fenyes	291d5c8ff6	Study mutually tangent spheres with two fixed	2024-02-15 13:28:01 -08:00
Aaron Fenyes	e41bcc7e13	Explore the performance wall Three points on two spheres is too much.	2024-02-13 04:02:14 -05:00
Aaron Fenyes	31d5e7e864	Play with two points on two spheres Guess conditions that make the scaling constraint impossible to satisfy.	2024-02-12 22:48:16 -05:00
Aaron Fenyes	a450f701fb	Try displaying a chain of spheres For three mutually tangent spheres, I couldn't find real solutions.	2024-02-12 21:14:07 -05:00
Aaron Fenyes	6cf07dc6a1	Evaluate and display elements	2024-02-12 20:34:12 -05:00
Aaron Fenyes	1f173708eb	Move random cut routine into engine	2024-02-10 17:39:26 -05:00
Aaron Fenyes	6f18d4efcc	Test lots of uniformly distributed hyperplanes	2024-02-10 15:10:48 -05:00
Aaron Fenyes	621c4c5776	Try uniformly distributed hyperplane orientations Unit normals are uniformly distributed over the sphere.	2024-02-10 15:02:26 -05:00
Aaron Fenyes	b3b7c2026d	Separate the algebraic and numerical parts of the engine	2024-02-10 14:50:50 -05:00
Aaron Fenyes	af1d31f6e6	Test a scale constraint In all but a few cases (for example, a single point on a plane), we should be able to us the radius-coradius boost symmetry to make the average co-radius—representing the "overall scale"—roughly one.	2024-02-10 14:21:52 -05:00
Aaron Fenyes	8e33987f59	Systematically try out different cut planes	2024-02-10 13:46:01 -05:00
Aaron Fenyes	06872a04af	Say how many sample solutions we found	2024-02-10 01:06:06 -05:00
Aaron Fenyes	becefe0c47	Try switching to compiled system	2024-02-10 00:59:50 -05:00
Aaron Fenyes	34358a8728	Find witnesses on random rational hyperplanes Choose hyperplanes that go through the trivial solution.	2024-02-09 23:44:10 -05:00
Aaron Fenyes	95c0ff14b2	Show explicitly that all coefficients are 1 in first cut equation	2024-02-09 17:09:43 -05:00
Aaron Fenyes	f97090c997	Try a cut that goes through the trivial solution The previous cut was supposed to do this, but I was missing some parentheses.	2024-02-08 01:58:12 -05:00
Aaron Fenyes	45aaaafc8f	Seek sample solutions by cutting with a hyperplane The example hyperplane yields a single solution, with multiplicity six. You can find it analytically by hand, and homotopy continuation finds it numerically.	2024-02-08 01:53:55 -05:00
Aaron Fenyes	43cbf8a3a0	Add relations to center and orient the construction	2024-02-05 00:10:13 -05:00
Aaron Fenyes	21f09c4a4d	Switch element abbreviation from "elem" to "elt"	2024-02-04 16:08:13 -05:00
Aaron Fenyes	a3f3f6a31b	Order spheres before points within each coordinate block In the cases I've tried so far, this leads to substantially smaller Gröbner bases.	2024-02-01 16:13:22 -05:00
Aaron Fenyes	65d23fb667	Use module names as filenames You're right: this naming convention seems to be standard for Julia modules now.	2024-01-30 02:49:33 -05:00
Aaron Fenyes	4e02ee16fc	Find dimension of solution variety	2024-01-30 02:45:14 -05:00
Aaron Fenyes	6349f298ae	Extend AbstractAlgebra ideals to rational coefficients The extension should also let us work over finite fields of prime order, although we don't need to do that.	2024-01-29 19:11:21 -05:00
Aaron Fenyes	0731c7aac1	Correct relation equations	2024-01-29 12:41:07 -05:00
Aaron Fenyes	59a527af43	Correct Minkowski product; build chain of three spheres	2024-01-29 12:28:57 -05:00
Aaron Fenyes	c29000d912	Write a simple solver for the hitting set problem I think we need this to find the dimension of the solution variety.	2024-01-28 01:34:13 -05:00
Aaron Fenyes	86dbd9ea45	Order variables by coordinate and then element In other words, order coordinates like (rₛ₁, rₛ₂, sₛ₁, sₛ₂, xₛ₁, xₛ₂, xₚ₃, yₛ₁, yₛ₂, yₚ₃, zₛ₁, zₛ₂, zₚ₃) instead of like (rₛ₁, sₛ₁, xₛ₁, yₛ₁, zₛ₁, rₛ₂, sₛ₂, xₛ₂, yₛ₂, zₛ₂, xₚ₃, yₚ₃, zₚ₃). In the test cases, this really cuts down the size of the Gröbner basis.	2024-01-27 14:21:03 -05:00
Aaron Fenyes	463a3b21e1	Realize relations as equations	2024-01-27 12:28:29 -05:00
Aaron Fenyes	4d5aa3b327	Realize geometric elements as symbolic vectors	2024-01-26 11:14:32 -05:00
Aaron Fenyes	b864cf7866	Start drafting engine prototype	2024-01-24 11:16:24 -05:00