Add kaleidocycle example

The engine successfully captures the kaleidocycle's twisting motion. However, nudging doesn't work quite the way we want. This is probably to be expected, since nudges don't commute with Euclidean motions in our current implementation.
2025-01-21 12:05:20 -08:00
16 changed files with 330 additions and 1594 deletions
--- a/.forgejo/setup-trunk/action.yaml
+++ b/.forgejo/setup-trunk/action.yaml
@ -1,22 +0,0 @@
 # set up the Trunk web build system
 #
 #   https://trunkrs.dev
 #
 # the `curl` call is based on David Tolnay's `rust-toolchain` action
 #
 #   https://github.com/dtolnay/rust-toolchain
 #
 runs:
  using: "composite"
  steps:
    - run: rustup target add wasm32-unknown-unknown
    # install the Trunk binary to `ci-bin` within the workspace directory, which
    # is determined by the `github.workspace` label and reflected in the
    # `GITHUB_WORKSPACE` environment variable. then, make the `trunk` command
    # available by placing the fully qualified path to `ci-bin` on the
    # workflow's search path
    - run: mkdir -p ci-bin
    - run: curl --output - --proto '=https' --tlsv1.2 --retry 10 --retry-connrefused --location --silent --show-error --fail 'https://github.com/trunk-rs/trunk/releases/download/v0.21.12/trunk-x86_64-unknown-linux-gnu.tar.gz' | tar --gunzip --extract --file -
      working-directory: ci-bin
    - run: echo "${{ github.workspace }}/ci-bin" >> $GITHUB_PATH
--- a/.forgejo/workflows/continuous-integration.yaml
+++ b/.forgejo/workflows/continuous-integration.yaml
@ -1,29 +0,0 @@
 on:
  pull_request:
  push:
    branches: [main]
 jobs:
  # run the automated tests, reporting success if the tests pass and were built
  # without warnings. the examples are run as tests, because we've configured
  # each example target with `test = true` and `harness = false` in Cargo.toml.
  # Trunk build failures caused by problems outside the Rust source code, like
  # missing assets, should be caught by `trunk_build_test`
  test:
    runs-on: docker
    container:
      image: cimg/rust:1.85-node
    defaults:
      run:
        # set the default working directory for each `run` step, relative to the
        # workspace directory. this default only affects `run` steps (and if we
        # tried to set the `working-directory` label for any other kind of step,
        # it wouldn't be recognized anyway)
        working-directory: app-proto
    steps:
      # Check out the repository so that its top-level directory is the
      # workspace directory (action variable `github.workspace`, environment
      # variable `$GITHUB_WORKSPACE`):
      - uses: https://code.forgejo.org/actions/checkout@v4
      - uses: ./.forgejo/setup-trunk
      - run: RUSTFLAGS='-D warnings' cargo test
--- a/.gitignore
+++ b/.gitignore
@ -1,2 +1,8 @@
-ci-bin
+node_modules
 site
 docbuild
 __tests__
 coverage
 dyna3.zip
 tmpproj
 *~
--- a/app-proto/Cargo.lock
+++ b/app-proto/Cargo.lock
@ -1,788 +0,0 @@
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 "bumpalo",
 "log",
 "once_cell",
 "proc-macro2",
 "quote",
 "syn",
 "wasm-bindgen-shared",
 ]
 [[package]]
 name = "wasm-bindgen-futures"
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 "cfg-if",
 "js-sys",
 "wasm-bindgen",
 "web-sys",
 ]
 [[package]]
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 "wasm-bindgen-macro-support",
 ]
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 "quote",
 "syn",
 "wasm-bindgen-backend",
 "wasm-bindgen-shared",
 ]
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 "js-sys",
 "minicov",
 "scoped-tls",
 "wasm-bindgen",
 "wasm-bindgen-futures",
 "wasm-bindgen-test-macro",
 ]
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 "safe_arch",
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 "windows-targets",
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 [[package]]
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 dependencies = [
 "windows_aarch64_gnullvm",
 "windows_aarch64_msvc",
 "windows_i686_gnu",
 "windows_i686_gnullvm",
 "windows_i686_msvc",
 "windows_x86_64_gnu",
 "windows_x86_64_gnullvm",
 "windows_x86_64_msvc",
 ]
 [[package]]
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 [[package]]
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 checksum = "1b9b4fd18abc82b8136838da5d50bae7bdea537c574d8dc1a34ed098d6c166f0"
 dependencies = [
 "byteorder",
 "zerocopy-derive",
 ]
 [[package]]
 name = "zerocopy-derive"
 version = "0.7.35"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "fa4f8080344d4671fb4e831a13ad1e68092748387dfc4f55e356242fae12ce3e"
 dependencies = [
 "proc-macro2",
 "quote",
 "syn",
 ]
--- a/app-proto/Cargo.toml
+++ b/app-proto/Cargo.toml
@ -13,7 +13,6 @@ itertools = "0.13.0"
 js-sys = "0.3.70"
 lazy_static = "1.5.0"
 nalgebra = "0.33.0"
 readonly = "0.2.12"
 rustc-hash = "2.0.0"
 slab = "0.4.9"
 sycamore = "0.9.0-beta.3"
@ -50,23 +49,3 @@ wasm-bindgen-test = "0.3.34"
 [profile.release]
 opt-level = "s" # optimize for small code size
 debug = true # include debug symbols
 [[example]]
 name = "irisawa-hexlet"
 test = true
 harness = false
 [[example]]
 name = "kaleidocycle"
 test = true
 harness = false
 [[example]]
 name = "point-on-sphere"
 test = true
 harness = false
 [[example]]
 name = "three-spheres"
 test = true
 harness = false
--- a/app-proto/examples/irisawa-hexlet.rs
+++ b/app-proto/examples/irisawa-hexlet.rs
@ -1,4 +1,4 @@
-use dyna3::engine::{Q, examples::realize_irisawa_hexlet};
+use dyna3::engine::{Q, irisawa::realize_irisawa_hexlet};
 fn main() {
    const SCALED_TOL: f64 = 1.0e-12;
--- a/app-proto/examples/kaleidocycle.rs
+++ b/app-proto/examples/kaleidocycle.rs
@ -1,10 +1,53 @@
-use nalgebra::{DMatrix, DVector};
+use nalgebra::DMatrix;
 use std::{array, f64::consts::PI};
-use dyna3::engine::{Q, examples::realize_kaleidocycle};
+use dyna3::engine::{Q, point, realize_gram, PartialMatrix};
 fn main() {
-    const SCALED_TOL: f64 = 1.0e-12;
+    // set up a kaleidocycle, made of points with fixed distances between them,
-    let (config, tangent, success, history) = realize_kaleidocycle(SCALED_TOL);
+    // and find its tangent space
    const N_POINTS: usize = 12;
    let gram = {
        let mut gram_to_be = PartialMatrix::new();
        for block in (0..N_POINTS).step_by(2) {
            let block_next = (block + 2) % N_POINTS;
            for j in 0..2 {
                // diagonal and hinge edges
                for k in j..2 {
                    gram_to_be.push_sym(block + j, block + k, if j == k { 0.0 } else { -0.5 });
                }
                // non-hinge edges
                for k in 0..2 {
                    gram_to_be.push_sym(block + j, block_next + k, -0.625);
                }
            }
        }
        gram_to_be
    };
    let guess = {
        const N_HINGES: usize = 6;
        let guess_elts = (0..N_HINGES).step_by(2).flat_map(
            |n| {
                let ang_hor = (n as f64) * PI/3.0;
                let ang_vert = ((n + 1) as f64) * PI/3.0;
                let x_vert = ang_vert.cos();
                let y_vert = ang_vert.sin();
                [
                    point(0.0, 0.0, 0.0),
                    point(ang_hor.cos(), ang_hor.sin(), 0.0),
                    point(x_vert, y_vert, -0.5),
                    point(x_vert, y_vert, 0.5)
                ]
            }
        ).collect::<Vec<_>>();
        DMatrix::from_columns(&guess_elts)
    };
    let frozen: [_; N_POINTS] = array::from_fn(|k| (3, k));
    let (config, tangent, success, history) = realize_gram(
        &gram, guess, &frozen,
        1.0e-12, 0.5, 0.9, 1.1, 200, 110
    );
    print!("Completed Gram matrix:{}", config.tr_mul(&*Q) * &config);
    print!("Configuration:{}", config);
    if success {
@ -15,15 +58,23 @@ fn main() {
    println!("Steps: {}", history.scaled_loss.len() - 1);
    println!("Loss: {}\n", history.scaled_loss.last().unwrap());
-    // find the kaleidocycle's twist motion by projecting onto the tangent space
+    // find the kaleidocycle's twist motion
    const N_POINTS: usize = 12;
    let up = DVector::from_column_slice(&[0.0, 0.0, 1.0, 0.0]);
    let down = -&up;
    let twist_motion: DMatrix<_> = (0..N_POINTS).step_by(4).flat_map(
-        |n| [
+        |n| {
-            tangent.proj(&up.as_view(), n),
+            let up_field = {
-            tangent.proj(&down.as_view(), n+1)
+                DMatrix::from_column_slice(5, 5, &[
-        ]
+                    0.0, 0.0, 0.0, 0.0, 0.0,
                    0.0, 0.0, 0.0, 0.0, 0.0,
                    0.0, 0.0, 0.0, 0.0, 1.0,
                    0.0, 0.0, 2.0, 0.0, 0.0,
                    0.0, 0.0, 0.0, 0.0, 0.0
                ])
            };
            [
                tangent.proj(&(&up_field * config.column(n)).as_view(), n),
                tangent.proj(&(-&up_field * config.column(n+1)).as_view(), n+1)
            ]
        }
    ).sum();
    let normalization = 5.0 / twist_motion[(2, 0)];
    print!("Twist motion:{}", normalization * twist_motion);
--- a/app-proto/main.css
+++ b/app-proto/main.css
@ -3,8 +3,7 @@
  --text-bright: white;
  --text-invalid: #f58fc2; /* bright pink */
  --border: #555; /* light gray */
-  --border-focus-dark: #aaa; /* bright gray */
+  --border-focus: #aaa; /* bright gray */
  --border-focus-light: white;
  --border-invalid: #70495c; /* dusky pink */
  --selection-highlight: #444; /* medium gray */
  --page-background: #222; /* dark gray */
@ -24,7 +23,7 @@ body {
  display: flex;
  flex-direction: column;
  float: left;
-  width: 500px;
+  width: 450px;
  height: 100vh;
  margin: 0px;
  padding: 0px;
@ -78,12 +77,12 @@ summary.selected {
  background-color: var(--selection-highlight);
 }
-summary > div, .regulator {
+summary > div, .constraint {
  padding-top: 4px;
  padding-bottom: 4px;
 }
-.element, .regulator {
+.element, .constraint {
  display: flex;
  flex-grow: 1;
  padding-left: 8px;
@ -108,7 +107,7 @@ details[open]:has(li) .element-switch::after {
  flex-grow: 1;
 }
-.regulator-label {
+.constraint-label {
  flex-grow: 1;
 }
@ -124,34 +123,26 @@ details[open]:has(li) .element-switch::after {
  width: 56px;
 }
-.regulator {
+.constraint {
  font-style: italic;
 }
-.regulator-type {
+.constraint.invalid {
-  padding: 2px 8px 0px 8px;
+  color: var(--text-invalid);
  font-size: 10pt;
 }
-.regulator-input {
+.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;
 }
-.regulator-input::placeholder {
+.constraint.invalid > input[type=text] {
  color: inherit;
  opacity: 54%;
  font-style: italic;
 }
 .regulator-input.constraint {
  background-color: var(--display-background);
 }
 .regulator-input.invalid {
  color: var(--text-invalid);
  border-color: var(--border-invalid);
 }
@ -163,7 +154,7 @@ details[open]:has(li) .element-switch::after {
  font-style: normal;
 }
-.regulator-input.invalid + .status::after, details:has(.invalid):not([open]) .status::after {
+.invalid > .status::after, details:has(.invalid):not([open]) .status::after {
  content: '⚠';
  color: var(--text-invalid);
 }
@ -180,11 +171,5 @@ canvas {
 }
 canvas:focus {
-  border-color: var(--border-focus-dark);
+  border-color: var(--border-focus);
  outline: none;
 }
 input:focus {
  border-color: var(--border-focus-light);
  outline: none;
 }
--- a/app-proto/src/add_remove.rs
+++ b/app-proto/src/add_remove.rs
@ -1,11 +1,7 @@
 use sycamore::prelude::*;
 use web_sys::{console, wasm_bindgen::JsValue};
-use crate::{
+use crate::{engine, AppState, assembly::{Assembly, Constraint, Element}};
    engine,
    AppState,
    assembly::{Assembly, Element}
 };
 /* DEBUG */
 // load an example assembly for testing. this code will be removed once we've
@ -194,8 +190,44 @@ pub fn AddRemove() -> View {
                            (subject_vec[0].clone(), subject_vec[1].clone())
                        }
                    );
-                    state.assembly.insert_new_regulator(subjects);
+                    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
--- a/app-proto/src/assembly.rs
+++ b/app-proto/src/assembly.rs
@ -5,14 +5,11 @@ use std::{collections::BTreeSet, sync::atomic::{AtomicU64, Ordering}};
 use sycamore::prelude::*;
 use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
-use crate::{
+use crate::engine::{realize_gram, ConfigSubspace, PartialMatrix, Q};
    engine::{Q, local_unif_to_std, realize_gram, ConfigSubspace, PartialMatrix},
    specified::SpecifiedValue
 };
-// the types of the keys we use to access an assembly's elements and regulators
+// the types of the keys we use to access an assembly's elements and constraints
 pub type ElementKey = usize;
-pub type RegulatorKey = usize;
+pub type ConstraintKey = usize;
 pub type ElementColor = [f32; 3];
@ -29,11 +26,8 @@ pub struct Element {
    pub label: String,
    pub color: ElementColor,
    pub representation: Signal<DVector<f64>>,
-    
+    pub constraints: Signal<BTreeSet<ConstraintKey>>,
-    // All regulators with this element as a subject. The assembly owning
+
    // this element is responsible for keeping this set up to date.
    pub regulators: Signal<BTreeSet<RegulatorKey>>,
    // a serial number, assigned by `Element::new`, that uniquely identifies
    // each element
    pub serial: u64,
@ -67,7 +61,7 @@ impl Element {
            label: label,
            color: color,
            representation: create_signal(representation),
-            regulators: create_signal(BTreeSet::default()),
+            constraints: create_signal(BTreeSet::default()),
            serial: serial,
            column_index: None
        }
@ -117,14 +111,15 @@ impl Element {
    }
 }
-#[derive(Clone, Copy)]
+#[derive(Clone)]
-pub struct Regulator {
+pub struct Constraint {
    pub subjects: (ElementKey, ElementKey),
-    pub measurement: ReadSignal<f64>,
+    pub lorentz_prod: Signal<f64>,
-    pub set_point: Signal<SpecifiedValue>
+    pub lorentz_prod_text: Signal<String>,
    pub lorentz_prod_valid: Signal<bool>,
    pub active: Signal<bool>
 }
 // the velocity is expressed in uniform coordinates
 pub struct ElementMotion<'a> {
    pub key: ElementKey,
    pub velocity: DVectorView<'a, f64>
@ -135,9 +130,9 @@ type AssemblyMotion<'a> = Vec<ElementMotion<'a>>;
 // a complete, view-independent description of an assembly
 #[derive(Clone)]
 pub struct Assembly {
-    // elements and regulators
+    // elements and constraints
    pub elements: Signal<Slab<Element>>,
-    pub regulators: Signal<Slab<Regulator>>,
+    pub constraints: Signal<Slab<Constraint>>,
    // solution variety tangent space. the basis vectors are stored in
    // configuration matrix format, ordered according to the elements' column
@ -159,13 +154,13 @@ impl Assembly {
    pub fn new() -> Assembly {
        Assembly {
            elements: create_signal(Slab::new()),
-            regulators: create_signal(Slab::new()),
+            constraints: create_signal(Slab::new()),
            tangent: create_signal(ConfigSubspace::zero(0)),
            elements_by_id: create_signal(FxHashMap::default())
        }
    }
-    // --- inserting elements and regulators ---
+    // --- 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
@ -208,61 +203,14 @@ impl Assembly {
        );
    }
-    fn insert_regulator(&self, regulator: Regulator) {
+    pub fn insert_constraint(&self, constraint: Constraint) {
-        let subjects = regulator.subjects;
+        let subjects = constraint.subjects;
-        let key = self.regulators.update(|regs| regs.insert(regulator));
+        let key = self.constraints.update(|csts| csts.insert(constraint));
-        let subject_regulators = self.elements.with(
+        let subject_constraints = self.elements.with(
-            |elts| (elts[subjects.0].regulators, elts[subjects.1].regulators)
+            |elts| (elts[subjects.0].constraints, elts[subjects.1].constraints)
        );
-        subject_regulators.0.update(|regs| regs.insert(key));
+        subject_constraints.0.update(|csts| csts.insert(key));
-        subject_regulators.1.update(|regs| regs.insert(key));
+        subject_constraints.1.update(|csts| csts.insert(key));
    }
    pub fn insert_new_regulator(self, subjects: (ElementKey, ElementKey)) {
        // create and insert a new regulator
        let measurement = self.elements.map(
            move |elts| {
                let reps = (
                    elts[subjects.0].representation.get_clone(),
                    elts[subjects.1].representation.get_clone()
                );
                reps.0.dot(&(&*Q * reps.1))
            }
        );
        let set_point = create_signal(SpecifiedValue::from_empty_spec());
        self.insert_regulator(Regulator {
            subjects: subjects,
            measurement: measurement,
            set_point: set_point
        });
        /* DEBUG */
        // print an updated list of regulators
        console::log_1(&JsValue::from("Regulators:"));
        self.regulators.with(|regs| {
            for (_, reg) in regs.into_iter() {
                console::log_5(
                    &JsValue::from(" "),
                    &JsValue::from(reg.subjects.0),
                    &JsValue::from(reg.subjects.1),
                    &JsValue::from(":"),
                    &reg.set_point.with_untracked(
                        |set_pt| JsValue::from(set_pt.spec.as_str())
                    )
                );
            }
        });
        // update the realization when the regulator becomes a constraint, or is
        // edited while acting as a constraint
        create_effect(move || {
            console::log_1(&JsValue::from(
                format!("Updated constraint with subjects ({}, {})", subjects.0, subjects.1)
            ));
            if set_point.with(|set_pt| set_pt.is_present()) {
                self.realize();
            }
        });
    }
    // --- realization ---
@ -279,16 +227,14 @@ impl Assembly {
        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.regulators.with_untracked(|regs| {
+            self.constraints.with_untracked(|csts| {
-                for (_, reg) in regs {
+                for (_, cst) in csts {
-                    reg.set_point.with_untracked(|set_pt| {
+                    if cst.active.get_untracked() && cst.lorentz_prod_valid.get_untracked() {
-                        if let Some(val) = set_pt.value {
+                        let subjects = cst.subjects;
-                            let subjects = reg.subjects;
+                        let row = elts[subjects.0].column_index.unwrap();
-                            let row = elts[subjects.0].column_index.unwrap();
+                        let col = elts[subjects.1].column_index.unwrap();
-                            let col = elts[subjects.1].column_index.unwrap();
+                        gram_to_be.push_sym(row, col, cst.lorentz_prod.get_untracked());
-                            gram_to_be.push_sym(row, col, val);
+                    }
                        }
                    });
                }
            });
@ -413,19 +359,12 @@ impl Assembly {
                // this element didn't have a column index when we started, so
                // by invariant (2), it's unconstrained
                let mut target_column = motion_proj.column_mut(column_index);
-                let unif_to_std = self.elements.with_untracked(
+                target_column += elt_motion.velocity;
                    |elts| {
                        elts[elt_motion.key].representation.with_untracked(
                            |rep| local_unif_to_std(rep.as_view())
                        )
                    }
                );
                target_column += unif_to_std * elt_motion.velocity;
            }
        }
-        // step the assembly along the deformation. this changes the elements'
+        // step each element along the mass shell geodesic that matches its
-        // normalizations, so we restore those afterward
+        // velocity in the deformation found above
        /* KLUDGE */
        // since our test assemblies only include spheres, we assume that every
        // element is on the 1 mass shell
@ -433,16 +372,9 @@ impl Assembly {
            elt.representation.update_silent(|rep| {
                match elt.column_index {
                    Some(column_index) => {
-                        // step the assembly along the deformation
+                        let rep_next = &*rep + motion_proj.column(column_index);
-                        *rep += motion_proj.column(column_index);
+                        let normalizer = rep_next.dot(&(&*Q * &rep_next));
-                        
+                        rep.set_column(0, &(rep_next / normalizer));
                        // restore normalization by contracting toward the last
                        // coordinate axis
                        let q_sp = rep.fixed_rows::<3>(0).norm_squared();
                        let half_q_lt = -2.0 * rep[3] * rep[4];
                        let half_q_lt_sq = half_q_lt * half_q_lt;
                        let scaling = half_q_lt + (q_sp + half_q_lt_sq).sqrt();
                        rep.fixed_rows_mut::<4>(0).scale_mut(1.0 / scaling);
                    },
                    None => {
                        console::log_1(&JsValue::from(
--- a/app-proto/src/display.rs
+++ b/app-proto/src/display.rs
@ -130,8 +130,6 @@ pub fn Display() -> View {
    let translate_pos_y = create_signal(0.0);
    let translate_neg_z = create_signal(0.0);
    let translate_pos_z = create_signal(0.0);
    let shrink_neg = create_signal(0.0);
    let shrink_pos = create_signal(0.0);
    // change listener
    let scene_changed = create_signal(true);
@ -166,7 +164,6 @@ pub fn Display() -> View {
        // manipulation
        const TRANSLATION_SPEED: f64 = 0.15; // in length units per second
        const SHRINKING_SPEED: f64 = 0.15; // in length units per second
        // display parameters
        const OPACITY: f32 = 0.5; /* SCAFFOLDING */
@ -295,8 +292,6 @@ pub fn Display() -> View {
            let translate_pos_y_val = translate_pos_y.get();
            let translate_neg_z_val = translate_neg_z.get();
            let translate_pos_z_val = translate_pos_z.get();
            let shrink_neg_val = shrink_neg.get();
            let shrink_pos_val = shrink_pos.get();
            // update the assembly's orientation
            let ang_vel = {
@ -328,27 +323,24 @@ pub fn Display() -> View {
                let sel = state.selection.with(
                    |sel| *sel.into_iter().next().unwrap()
                );
                let rep = state.assembly.elements.with_untracked(
                    |elts| elts[sel].representation.get_clone_untracked()
                );
                let translate_x = translate_pos_x_val - translate_neg_x_val;
                let translate_y = translate_pos_y_val - translate_neg_y_val;
                let translate_z = translate_pos_z_val - translate_neg_z_val;
-                let shrink = shrink_pos_val - shrink_neg_val;
+                if translate_x != 0.0 || translate_y != 0.0 || translate_z != 0.0 {
-                let translating =
+                    let vel_field = {
-                    translate_x != 0.0
+                        let u = Vector3::new(translate_x, translate_y, translate_z).normalize();
-                    || translate_y != 0.0
+                        DMatrix::from_column_slice(5, 5, &[
-                    || translate_z != 0.0;
+                                 0.0,      0.0,      0.0, 0.0, u[0],
-                if translating || shrink != 0.0 {
+                                 0.0,      0.0,      0.0, 0.0, u[1],
-                    let elt_motion = {
+                                 0.0,      0.0,      0.0, 0.0, u[2],
-                        let u = if translating {
+                            2.0*u[0], 2.0*u[1], 2.0*u[2], 0.0,  0.0,
-                            TRANSLATION_SPEED * Vector3::new(
+                                 0.0,      0.0,      0.0, 0.0,  0.0
-                                translate_x, translate_y, translate_z
+                        ])
                            ).normalize()
                        } else {
                            Vector3::zeros()
                        };
                        time_step * DVector::from_column_slice(
                            &[u[0], u[1], u[2], SHRINKING_SPEED * shrink]
                        )
                    };
                    let elt_motion: DVector<f64> = time_step * TRANSLATION_SPEED * vel_field * rep;
                    assembly_for_raf.deform(
                        vec![
                            ElementMotion {
@ -509,8 +501,6 @@ pub fn Display() -> View {
            "s" | "S" if shift => translate_pos_z.set(value),
            "w" | "W" => translate_pos_y.set(value),
            "s" | "S" => translate_neg_y.set(value),
            "]" | "}" => shrink_neg.set(value),
            "[" | "{" => shrink_pos.set(value),
            _ => manipulating = false
        };
        if manipulating {
--- a/app-proto/src/engine.rs
+++ b/app-proto/src/engine.rs
@ -90,34 +90,32 @@ impl PartialMatrix {
 #[derive(Clone)]
 pub struct ConfigSubspace {
    assembly_dim: usize,
-    basis_std: Vec<DMatrix<f64>>,
+    basis: Vec<DMatrix<f64>>
    basis_proj: Vec<DMatrix<f64>>
 }
 impl ConfigSubspace {
    pub fn zero(assembly_dim: usize) -> ConfigSubspace {
        ConfigSubspace {
            assembly_dim: assembly_dim,
-            basis_proj: Vec::new(),
+            basis: Vec::new()
            basis_std: Vec::new()
        }
    }
    // approximate the kernel of a symmetric endomorphism of the configuration
    // space for `assembly_dim` elements. we consider an eigenvector to be part
    // of the kernel if its eigenvalue is smaller than the constant `THRESHOLD`
-    fn symmetric_kernel(a: DMatrix<f64>, proj_to_std: DMatrix<f64>, assembly_dim: usize) -> ConfigSubspace {
+    fn symmetric_kernel(a: DMatrix<f64>, assembly_dim: usize) -> ConfigSubspace {
-        // find a basis for the kernel. the basis is expressed in the projection
+        const ELEMENT_DIM: usize = 5;
-        // coordinates, and it's orthonormal with respect to the projection
+        const THRESHOLD: f64 = 1.0e-4;
-        // inner product
+        let eig = SymmetricEigen::new(a);
        const THRESHOLD: f64 = 0.1;
        let eig = SymmetricEigen::new(proj_to_std.tr_mul(&a) * &proj_to_std);
        let eig_vecs = eig.eigenvectors.column_iter();
        let eig_pairs = eig.eigenvalues.iter().zip(eig_vecs);
-        let basis_proj = DMatrix::from_columns(
+        let basis = eig_pairs.filter_map(
-            eig_pairs.filter_map(
+            |(λ, v)| (λ.abs() < THRESHOLD).then_some(
-                |(λ, v)| (λ.abs() < THRESHOLD).then_some(v)
+                Into::<DMatrix<f64>>::into(
-            ).collect::<Vec<_>>().as_slice()
+                    v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
                )
            )
        );
        /* DEBUG */
@ -127,45 +125,30 @@ impl ConfigSubspace {
            format!("Eigenvalues used to find kernel:{}", eig.eigenvalues)
        ));
        // express the basis in the standard coordinates
        let basis_std = proj_to_std * &basis_proj;
        const ELEMENT_DIM: usize = 5;
        const UNIFORM_DIM: usize = 4;
        ConfigSubspace {
            assembly_dim: assembly_dim,
-            basis_std: basis_std.column_iter().map(
+            basis: basis.collect()
                |v| Into::<DMatrix<f64>>::into(
                    v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
                )
            ).collect(),
            basis_proj: basis_proj.column_iter().map(
                |v| Into::<DMatrix<f64>>::into(
                    v.reshape_generic(Dyn(UNIFORM_DIM), Dyn(assembly_dim))
                )
            ).collect()
        }
    }
    pub fn dim(&self) -> usize {
-        self.basis_std.len()
+        self.basis.len()
    }
    pub fn assembly_dim(&self) -> usize {
        self.assembly_dim
    }
-    // find the projection onto this subspace of the motion where the element
+    // find the projection onto this subspace, with respect to the Euclidean
-    // with the given column index has velocity `v`. the velocity is given in
+    // inner product, of the motion where the element with the given column
-    // projection coordinates, and the projection is done with respect to the
+    // index has velocity `v`
    // projection inner product
    pub fn proj(&self, v: &DVectorView<f64>, column_index: usize) -> DMatrix<f64> {
        if self.dim() == 0 {
            const ELEMENT_DIM: usize = 5;
            DMatrix::zeros(ELEMENT_DIM, self.assembly_dim)
        } else {
-            self.basis_proj.iter().zip(self.basis_std.iter()).map(
+            self.basis.iter().map(
-                |(b_proj, b_std)| b_proj.column(column_index).dot(&v) * b_std
+                |b| b.column(column_index).dot(&v) * b
            ).sum()
        }
    }
@ -232,37 +215,6 @@ fn basis_matrix(index: (usize, usize), nrows: usize, ncols: usize) -> DMatrix<f6
    result
 }
 // given a normalized vector `v` representing an element, build a basis for the
 // element's linear configuration space consisting of:
 // - the unit translation motions of the element
 // - the unit shrinking motion of the element, if it's a sphere
 // - one or two vectors whose coefficients vanish on the tangent space of the
 //   normalization variety
 pub fn local_unif_to_std(v: DVectorView<f64>) -> DMatrix<f64> {
    const ELEMENT_DIM: usize = 5;
    const UNIFORM_DIM: usize = 4;
    let curv = 2.0*v[3];
    if v.dot(&(&*Q * v)) < 0.5 {
        // `v` represents a point. the normalization condition says that the
        // curvature component of `v` is 1/2
        DMatrix::from_column_slice(ELEMENT_DIM, UNIFORM_DIM, &[
            curv,  0.0,  0.0,  0.0, v[0],
             0.0, curv,  0.0,  0.0, v[1],
             0.0,  0.0, curv,  0.0, v[2],
             0.0,  0.0,  0.0,  0.0,  1.0
        ])
    } else {
        // `v` represents a sphere. the normalization condition says that the
        // Lorentz product of `v` with itself is 1
        DMatrix::from_column_slice(ELEMENT_DIM, UNIFORM_DIM, &[
                 curv,       0.0,       0.0,       0.0,            v[0],
                  0.0,      curv,       0.0,       0.0,            v[1],
                  0.0,       0.0,      curv,       0.0,            v[2],
            curv*v[0], curv*v[1], curv*v[2], curv*v[3], curv*v[4] + 1.0
        ])
    }
 }
 // use backtracking line search to find a better configuration
 fn seek_better_config(
    gram: &PartialMatrix,
@ -392,19 +344,7 @@ pub fn realize_gram(
    }
    let success = state.loss < tol;
    let tangent = if success {
-        // express the uniform basis in the standard basis
+        ConfigSubspace::symmetric_kernel(hess, assembly_dim)
        const UNIFORM_DIM: usize = 4;
        let total_dim_unif = UNIFORM_DIM * assembly_dim;
        let mut unif_to_std = DMatrix::<f64>::zeros(total_dim, total_dim_unif);
        for n in 0..assembly_dim {
            let block_start = (element_dim * n, UNIFORM_DIM * n);
            unif_to_std
                .view_mut(block_start, (element_dim, UNIFORM_DIM))
                .copy_from(&local_unif_to_std(state.config.column(n)));
        }
        // find the kernel of the Hessian. give it the uniform inner product
        ConfigSubspace::symmetric_kernel(hess, unif_to_std, assembly_dim)
    } else {
        ConfigSubspace::zero(assembly_dim)
    };
@ -413,20 +353,20 @@ pub fn realize_gram(
 // --- tests ---
 // 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/
 //
 #[cfg(feature = "dev")]
-pub mod examples {
+pub mod irisawa {
    use std::{array, f64::consts::PI};
    use super::*;
    // 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/
    //
    pub fn realize_irisawa_hexlet(scaled_tol: f64) -> (DMatrix<f64>, ConfigSubspace, bool, DescentHistory) {
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
@ -480,64 +420,11 @@ pub mod examples {
            scaled_tol, 0.5, 0.9, 1.1, 200, 110
        )
    }
    // set up a kaleidocycle, made of points with fixed distances between them,
    // and find its tangent space
    pub fn realize_kaleidocycle(scaled_tol: f64) -> (DMatrix<f64>, ConfigSubspace, bool, DescentHistory) {
        const N_POINTS: usize = 12;
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            for block in (0..N_POINTS).step_by(2) {
                let block_next = (block + 2) % N_POINTS;
                for j in 0..2 {
                    // diagonal and hinge edges
                    for k in j..2 {
                        gram_to_be.push_sym(block + j, block + k, if j == k { 0.0 } else { -0.5 });
                    }
                    // non-hinge edges
                    for k in 0..2 {
                        gram_to_be.push_sym(block + j, block_next + k, -0.625);
                    }
                }
            }
            gram_to_be
        };
        let guess = {
            const N_HINGES: usize = 6;
            let guess_elts = (0..N_HINGES).step_by(2).flat_map(
                |n| {
                    let ang_hor = (n as f64) * PI/3.0;
                    let ang_vert = ((n + 1) as f64) * PI/3.0;
                    let x_vert = ang_vert.cos();
                    let y_vert = ang_vert.sin();
                    [
                        point(0.0, 0.0, 0.0),
                        point(ang_hor.cos(), ang_hor.sin(), 0.0),
                        point(x_vert, y_vert, -0.5),
                        point(x_vert, y_vert, 0.5)
                    ]
                }
            ).collect::<Vec<_>>();
            DMatrix::from_columns(&guess_elts)
        };
        let frozen: [_; N_POINTS] = array::from_fn(|k| (3, k));
        realize_gram(
            &gram, guess, &frozen,
            scaled_tol, 0.5, 0.9, 1.1, 200, 110
        )
    }
 }
 #[cfg(test)]
 mod tests {
-    use nalgebra::Vector3;
+    use super::{*, irisawa::realize_irisawa_hexlet};
    use std::{f64::consts::{FRAC_1_SQRT_2, PI}, iter};
    use super::{*, examples::*};
    #[test]
    fn sub_proj_test() {
@ -573,7 +460,7 @@ mod tests {
            entries
        });
        let config = {
-            let a = 0.75_f64.sqrt();
+            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),
@ -584,6 +471,75 @@ mod tests {
        assert!(state.loss.abs() < f64::EPSILON);
    }
    #[test]
    fn irisawa_hexlet_test() {
        // solve Irisawa's problem
        const SCALED_TOL: f64 = 1.0e-12;
        let (config, _, _, _) = realize_irisawa_hexlet(SCALED_TOL);
        // check against Irisawa's solution
        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);
        }
    }
    #[test]
    fn tangent_test() {
        const SCALED_TOL: f64 = 1.0e-12;
        const ELEMENT_DIM: usize = 5;
        const ASSEMBLY_DIM: usize = 3;
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            for j in 0..3 {
                for k in j..3 {
                    gram_to_be.push_sym(j, k, if j == k { 1.0 } else { -1.0 });
                }
            }
            gram_to_be
        };
        let guess = DMatrix::from_columns(&[
            sphere(0.0, 0.0, 0.0, -2.0),
            sphere(0.0, 0.0, 1.0, 1.0),
            sphere(0.0, 0.0, -1.0, 1.0)
        ]);
        let frozen: [_; 5] = std::array::from_fn(|k| (k, 0));
        let (config, tangent, success, history) = realize_gram(
            &gram, guess.clone(), &frozen,
            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
        );
        assert_eq!(config, guess);
        assert_eq!(success, true);
        assert_eq!(history.scaled_loss.len(), 1);
        // confirm that the tangent space has dimension five or less
        let ConfigSubspace(ref tangent_basis) = tangent;
        assert_eq!(tangent_basis.len(), 5);
        // confirm that the tangent space contains all the motions we expect it
        // to. since we've already bounded the dimension of the tangent space,
        // this confirms that the tangent space is what we expect it to be
        let tangent_motions = vec![
            basis_matrix((0, 1), ELEMENT_DIM, ASSEMBLY_DIM),
            basis_matrix((1, 1), ELEMENT_DIM, ASSEMBLY_DIM),
            basis_matrix((0, 2), ELEMENT_DIM, ASSEMBLY_DIM),
            basis_matrix((1, 2), ELEMENT_DIM, ASSEMBLY_DIM),
            DMatrix::<f64>::from_column_slice(ELEMENT_DIM, 3, &[
                0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, -1.0, -0.25, -1.0,
                0.0, 0.0, -1.0, 0.25, 1.0
            ])
        ];
        let tol_sq = ((ELEMENT_DIM * ASSEMBLY_DIM) as f64) * SCALED_TOL * SCALED_TOL;
        for motion in tangent_motions {
            let motion_proj: DMatrix<_> = motion.column_iter().enumerate().map(
                |(k, v)| tangent.proj(&v, k)
            ).sum();
            assert!((motion - motion_proj).norm_squared() < tol_sq);
        }
    }
    // at the frozen indices, the optimization steps should have exact zeros,
    // and the realized configuration should match the initial guess
    #[test]
@ -617,259 +573,4 @@ mod tests {
            assert_eq!(config[index], guess[index]);
        }
    }
    #[test]
    fn irisawa_hexlet_test() {
        // solve Irisawa's problem
        const SCALED_TOL: f64 = 1.0e-12;
        let (config, _, _, _) = realize_irisawa_hexlet(SCALED_TOL);
        // check against Irisawa's solution
        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);
        }
    }
    #[test]
    fn tangent_test_three_spheres() {
        const SCALED_TOL: f64 = 1.0e-12;
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            for j in 0..3 {
                for k in j..3 {
                    gram_to_be.push_sym(j, k, if j == k { 1.0 } else { -1.0 });
                }
            }
            gram_to_be
        };
        let guess = DMatrix::from_columns(&[
            sphere(0.0, 0.0, 0.0, -2.0),
            sphere(0.0, 0.0, 1.0, 1.0),
            sphere(0.0, 0.0, -1.0, 1.0)
        ]);
        let frozen: [_; 5] = std::array::from_fn(|k| (k, 0));
        let (config, tangent, success, history) = realize_gram(
            &gram, guess.clone(), &frozen,
            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
        );
        assert_eq!(config, guess);
        assert_eq!(success, true);
        assert_eq!(history.scaled_loss.len(), 1);
        // list some motions that should form a basis for the tangent space of
        // the solution variety
        const UNIFORM_DIM: usize = 4;
        let element_dim = guess.nrows();
        let assembly_dim = guess.ncols();
        let tangent_motions_unif = vec![
            basis_matrix((0, 1), UNIFORM_DIM, assembly_dim),
            basis_matrix((1, 1), UNIFORM_DIM, assembly_dim),
            basis_matrix((0, 2), UNIFORM_DIM, assembly_dim),
            basis_matrix((1, 2), UNIFORM_DIM, assembly_dim),
            DMatrix::<f64>::from_column_slice(UNIFORM_DIM, assembly_dim, &[
                0.0, 0.0,  0.0,  0.0,
                0.0, 0.0, -0.5, -0.5,
                0.0, 0.0, -0.5,  0.5
            ])
        ];
        let tangent_motions_std = vec![
            basis_matrix((0, 1), element_dim, assembly_dim),
            basis_matrix((1, 1), element_dim, assembly_dim),
            basis_matrix((0, 2), element_dim, assembly_dim),
            basis_matrix((1, 2), element_dim, assembly_dim),
            DMatrix::<f64>::from_column_slice(element_dim, assembly_dim, &[
                0.0, 0.0,  0.0,  0.00,  0.0,
                0.0, 0.0, -1.0, -0.25, -1.0,
                0.0, 0.0, -1.0,  0.25,  1.0
            ])
        ];
        // confirm that the dimension of the tangent space is no greater than
        // expected
        assert_eq!(tangent.basis_std.len(), tangent_motions_std.len());
        // confirm that the tangent space contains all the motions we expect it
        // to. since we've already bounded the dimension of the tangent space,
        // this confirms that the tangent space is what we expect it to be
        let tol_sq = ((element_dim * assembly_dim) as f64) * SCALED_TOL * SCALED_TOL;
        for (motion_unif, motion_std) in tangent_motions_unif.into_iter().zip(tangent_motions_std) {
            let motion_proj: DMatrix<_> = motion_unif.column_iter().enumerate().map(
                |(k, v)| tangent.proj(&v, k)
            ).sum();
            assert!((motion_std - motion_proj).norm_squared() < tol_sq);
        }
    }
    fn translation_motion_unif(vel: &Vector3<f64>, assembly_dim: usize) -> Vec<DVector<f64>> {
        let mut elt_motion = DVector::zeros(4);
        elt_motion.fixed_rows_mut::<3>(0).copy_from(vel);
        iter::repeat(elt_motion).take(assembly_dim).collect()
    }
    fn rotation_motion_unif(ang_vel: &Vector3<f64>, points: Vec<DVectorView<f64>>) -> Vec<DVector<f64>> {
        points.into_iter().map(
            |pt| {
                let vel = ang_vel.cross(&pt.fixed_rows::<3>(0));
                let mut elt_motion = DVector::zeros(4);
                elt_motion.fixed_rows_mut::<3>(0).copy_from(&vel);
                elt_motion
            }
        ).collect()
    }
    #[test]
    fn tangent_test_kaleidocycle() {
        // set up a kaleidocycle and find its tangent space
        const SCALED_TOL: f64 = 1.0e-12;
        let (config, tangent, success, history) = realize_kaleidocycle(SCALED_TOL);
        assert_eq!(success, true);
        assert_eq!(history.scaled_loss.len(), 1);
        // list some motions that should form a basis for the tangent space of
        // the solution variety
        const N_HINGES: usize = 6;
        let element_dim = config.nrows();
        let assembly_dim = config.ncols();
        let tangent_motions_unif = vec![
            // the translations along the coordinate axes
            translation_motion_unif(&Vector3::new(1.0, 0.0, 0.0), assembly_dim),
            translation_motion_unif(&Vector3::new(0.0, 1.0, 0.0), assembly_dim),
            translation_motion_unif(&Vector3::new(0.0, 0.0, 1.0), assembly_dim),
            // the rotations about the coordinate axes
            rotation_motion_unif(&Vector3::new(1.0, 0.0, 0.0), config.column_iter().collect()),
            rotation_motion_unif(&Vector3::new(0.0, 1.0, 0.0), config.column_iter().collect()),
            rotation_motion_unif(&Vector3::new(0.0, 0.0, 1.0), config.column_iter().collect()),
            // the twist motion. more precisely: a motion that keeps the center
            // of mass stationary and preserves the distances between the
            // vertices to first order. this has to be the twist as long as:
            //   - twisting is the kaleidocycle's only internal degree of
            //     freedom
            //   - every first-order motion of the kaleidocycle comes from an
            //     actual motion
            (0..N_HINGES).step_by(2).flat_map(
                |n| {
                    let ang_vert = ((n + 1) as f64) * PI/3.0;
                    let vel_vert_x = 4.0 * ang_vert.cos();
                    let vel_vert_y = 4.0 * ang_vert.sin();
                    [
                        DVector::from_column_slice(&[0.0, 0.0, 5.0, 0.0]),
                        DVector::from_column_slice(&[0.0, 0.0, 1.0, 0.0]),
                        DVector::from_column_slice(&[-vel_vert_x, -vel_vert_y, -3.0, 0.0]),
                        DVector::from_column_slice(&[vel_vert_x, vel_vert_y, -3.0, 0.0])
                    ]
                }
            ).collect::<Vec<_>>()
        ];
        let tangent_motions_std = tangent_motions_unif.iter().map(
            |motion| DMatrix::from_columns(
                &config.column_iter().zip(motion).map(
                    |(v, elt_motion)| local_unif_to_std(v) * elt_motion
                ).collect::<Vec<_>>()
            )
        ).collect::<Vec<_>>();
        // confirm that the dimension of the tangent space is no greater than
        // expected
        assert_eq!(tangent.basis_std.len(), tangent_motions_unif.len());
        // confirm that the tangent space contains all the motions we expect it
        // to. since we've already bounded the dimension of the tangent space,
        // this confirms that the tangent space is what we expect it to be
        let tol_sq = ((element_dim * assembly_dim) as f64) * SCALED_TOL * SCALED_TOL;
        for (motion_unif, motion_std) in tangent_motions_unif.into_iter().zip(tangent_motions_std) {
            let motion_proj: DMatrix<_> = motion_unif.into_iter().enumerate().map(
                |(k, v)| tangent.proj(&v.as_view(), k)
            ).sum();
            assert!((motion_std - motion_proj).norm_squared() < tol_sq);
        }
    }
    fn translation(dis: Vector3<f64>) -> DMatrix<f64> {
        const ELEMENT_DIM: usize = 5;
        DMatrix::from_column_slice(ELEMENT_DIM, ELEMENT_DIM, &[
                   1.0,        0.0,        0.0, 0.0,             dis[0],
                   0.0,        1.0,        0.0, 0.0,             dis[1],
                   0.0,        0.0,        1.0, 0.0,             dis[2],
            2.0*dis[0], 2.0*dis[1], 2.0*dis[2], 1.0, dis.norm_squared(),
                   0.0,        0.0,        0.0, 0.0,                1.0
        ])
    }
    // confirm that projection onto a configuration subspace is equivariant with
    // respect to Euclidean motions
    #[test]
    fn proj_equivar_test() {
        // find a pair of spheres that meet at 120°
        const SCALED_TOL: f64 = 1.0e-12;
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            gram_to_be.push_sym(0, 0, 1.0);
            gram_to_be.push_sym(1, 1, 1.0);
            gram_to_be.push_sym(0, 1, 0.5);
            gram_to_be
        };
        let guess_orig = DMatrix::from_columns(&[
            sphere(0.0, 0.0, 0.5, 1.0),
            sphere(0.0, 0.0, -0.5, 1.0)
        ]);
        let (config_orig, tangent_orig, success_orig, history_orig) = realize_gram(
            &gram, guess_orig.clone(), &[],
            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
        );
        assert_eq!(config_orig, guess_orig);
        assert_eq!(success_orig, true);
        assert_eq!(history_orig.scaled_loss.len(), 1);
        // find another pair of spheres that meet at 120°. we'll think of this
        // solution as a transformed version of the original one
        let guess_tfm = {
            let a = 0.5 * FRAC_1_SQRT_2;
            DMatrix::from_columns(&[
                sphere(a, 0.0, 7.0 + a, 1.0),
                sphere(-a, 0.0, 7.0 - a, 1.0)
            ])
        };
        let (config_tfm, tangent_tfm, success_tfm, history_tfm) = realize_gram(
            &gram, guess_tfm.clone(), &[],
            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
        );
        assert_eq!(config_tfm, guess_tfm);
        assert_eq!(success_tfm, true);
        assert_eq!(history_tfm.scaled_loss.len(), 1);
        // project a nudge to the tangent space of the solution variety at the
        // original solution
        let motion_orig = DVector::from_column_slice(&[0.0, 0.0, 1.0, 0.0]);
        let motion_orig_proj = tangent_orig.proj(&motion_orig.as_view(), 0);
        // project the equivalent nudge to the tangent space of the solution
        // variety at the transformed solution
        let motion_tfm = DVector::from_column_slice(&[FRAC_1_SQRT_2, 0.0, FRAC_1_SQRT_2, 0.0]);
        let motion_tfm_proj = tangent_tfm.proj(&motion_tfm.as_view(), 0);
        // take the transformation that sends the original solution to the
        // transformed solution and apply it to the motion that the original
        // solution makes in response to the nudge
        const ELEMENT_DIM: usize = 5;
        let rot = DMatrix::from_column_slice(ELEMENT_DIM, ELEMENT_DIM, &[
            FRAC_1_SQRT_2, 0.0, -FRAC_1_SQRT_2, 0.0, 0.0,
                      0.0, 1.0,            0.0, 0.0, 0.0,
            FRAC_1_SQRT_2, 0.0,  FRAC_1_SQRT_2, 0.0, 0.0,
                      0.0, 0.0,            0.0, 1.0, 0.0,
                      0.0, 0.0,            0.0, 0.0, 1.0
        ]);
        let transl = translation(Vector3::new(0.0, 0.0, 7.0));
        let motion_proj_tfm = transl * rot * motion_orig_proj;
        // confirm that the projection of the nudge is equivariant. we loosen
        // the comparison tolerance because the transformation seems to
        // introduce some numerical error
        const SCALED_TOL_TFM: f64 = 1.0e-9;
        let tol_sq = ((guess_orig.nrows() * guess_orig.ncols()) as f64) * SCALED_TOL_TFM * SCALED_TOL_TFM;
        assert!((motion_proj_tfm - motion_tfm_proj).norm_squared() < tol_sq);
    }
 }
--- a/app-proto/src/main.rs
+++ b/app-proto/src/main.rs
@ -3,10 +3,6 @@ mod assembly;
 mod display;
 mod engine;
 mod outline;
 mod specified;
 #[cfg(test)]
 mod tests;
 use rustc_hash::FxHashSet;
 use sycamore::prelude::*;
--- a/app-proto/src/outline.rs
+++ b/app-proto/src/outline.rs
@ -1,97 +1,56 @@
 use itertools::Itertools;
 use sycamore::prelude::*;
 use web_sys::{
    Event,
    HtmlInputElement,
    KeyboardEvent,
    MouseEvent,
    wasm_bindgen::JsCast
 };
-use crate::{
+use crate::{AppState, assembly, assembly::{Constraint, ConstraintKey, ElementKey}};
    AppState,
    assembly,
    assembly::{ElementKey, Regulator, RegulatorKey},
    specified::SpecifiedValue
 };
-// an editable view of a regulator
+// an editable view of the Lorentz product representing a constraint
 #[component(inline_props)]
-fn RegulatorInput(regulator: Regulator) -> View {
+fn LorentzProductInput(constraint: Constraint) -> View {
    let valid = create_signal(true);
    let value = create_signal(
        regulator.set_point.with_untracked(|set_pt| set_pt.spec.clone())
    );
    // this closure resets the input value to the regulator's set point
    // specification
    let reset_value = move || {
        batch(|| {
            valid.set(true);
            value.set(regulator.set_point.with(|set_pt| set_pt.spec.clone()));
        })
    };
    // reset the input value whenever the regulator's set point specification
    // is updated
    create_effect(reset_value);
    view! {
        input(
            r#type="text",
-            class=move || {
+            bind:value=constraint.lorentz_prod_text,
-                if valid.get() {
+            on:change=move |event: Event| {
-                    regulator.set_point.with(|set_pt| {
+                let target: HtmlInputElement = event.target().unwrap().unchecked_into();
-                        if set_pt.is_present() {
+                match target.value().parse::<f64>() {
-                            "regulator-input constraint"
+                    Ok(lorentz_prod) => batch(|| {
-                        } else {
+                        constraint.lorentz_prod.set(lorentz_prod);
-                            "regulator-input"
+                        constraint.lorentz_prod_valid.set(true);
-                        }
+                    }),
-                    })
+                    Err(_) => constraint.lorentz_prod_valid.set(false)
-                } else {
+                };
                    "regulator-input invalid"
                }
            },
            placeholder=regulator.measurement.with(|result| result.to_string()),
            bind:value=value,
            on:change=move |_| {
                valid.set(
                    match SpecifiedValue::try_from(value.get_clone_untracked()) {
                        Ok(set_pt) => {
                            regulator.set_point.set(set_pt);
                            true
                        }
                        Err(_) => false
                    }
                )
            },
            on:keydown={
                move |event: KeyboardEvent| {
                    match event.key().as_str() {
                        "Escape" => reset_value(),
                        _ => ()
                    }
                }
            }
        )
    }
 }
-// a list item that shows a regulator in an outline view of an element
+// a list item that shows a constraint in an outline view of an element
 #[component(inline_props)]
-fn RegulatorOutlineItem(regulator_key: RegulatorKey, element_key: ElementKey) -> View {
+fn ConstraintOutlineItem(constraint_key: ConstraintKey, element_key: ElementKey) -> View {
    let state = use_context::<AppState>();
    let assembly = &state.assembly;
-    let regulator = assembly.regulators.with(|regs| regs[regulator_key]);
+    let constraint = assembly.constraints.with(|csts| csts[constraint_key].clone());
-    let other_subject = if regulator.subjects.0 == element_key {
+    let other_subject = if constraint.subjects.0 == element_key {
-        regulator.subjects.1
+        constraint.subjects.1
    } else {
-        regulator.subjects.0
+        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="regulator") {
+        li(class=class.get()) {
-            div(class="regulator-label") { (other_subject_label) }
+            input(r#type="checkbox", bind:checked=constraint.active)
-            div(class="regulator-type") { "Inversive distance" }
+            div(class="constraint-label") { (other_subject_label) }
-            RegulatorInput(regulator=regulator)
+            LorentzProductInput(constraint=constraint)
            div(class="status")
        }
    }
@ -105,19 +64,14 @@ fn ElementOutlineItem(key: ElementKey, element: assembly::Element) -> View {
        move |sel| if sel.contains(&key) { "selected" } else { "" }
    );
    let label = element.label.clone();
-    let rep_components = move || {
+    let rep_components = element.representation.map(
-        element.representation.with(
+        |rep| rep.iter().map(
-            |rep| rep.iter().map(
+            |u| format!("{:.3}", u).replace("-", "\u{2212}")
-                |u| {
+        ).collect()
-                    let u_str = format!("{:.3}", u).replace("-", "\u{2212}");
+    );
-                    view! { div { (u_str) } }
+    let constrained = element.constraints.map(|csts| csts.len() > 0);
-                }
+    let constraint_list = element.constraints.map(
-            ).collect::<Vec<_>>()
+        |csts| csts.clone().into_iter().collect()
        )
    };
    let regulated = element.regulators.map(|regs| regs.len() > 0);
    let regulator_list = element.regulators.map(
        |regs| regs.clone().into_iter().collect()
    );
    let details_node = create_node_ref();
    view! {
@ -132,7 +86,7 @@ fn ElementOutlineItem(key: ElementKey, element: assembly::Element) -> View {
                                    state.select(key, event.shift_key());
                                    event.prevent_default();
                                },
-                                "ArrowRight" if regulated.get() => {
+                                "ArrowRight" if constrained.get() => {
                                    let _ = details_node
                                        .get()
                                        .unchecked_into::<web_sys::Element>()
@ -175,20 +129,27 @@ fn ElementOutlineItem(key: ElementKey, element: assembly::Element) -> View {
                        }
                    ) {
                        div(class="element-label") { (label) }
-                        div(class="element-representation") { (rep_components) }
+                        div(class="element-representation") {
                            Indexed(
                                list=rep_components,
                                view=|coord_str| view! {
                                    div { (coord_str) }
                                }
                            )
                        }
                        div(class="status")
                    }
                }
-                ul(class="regulators") {
+                ul(class="constraints") {
                    Keyed(
-                        list=regulator_list,
+                        list=constraint_list,
-                        view=move |reg_key| view! {
+                        view=move |cst_key| view! {
-                            RegulatorOutlineItem(
+                            ConstraintOutlineItem(
-                                regulator_key=reg_key,
+                                constraint_key=cst_key,
                                element_key=key
                            )
                        },
-                        key=|reg_key| reg_key.clone()
+                        key=|cst_key| cst_key.clone()
                    )
                }
            }
@ -196,9 +157,9 @@ fn ElementOutlineItem(key: ElementKey, element: assembly::Element) -> View {
    }
 }
-// a component that lists the elements of the current assembly, showing each
+// a component that lists the elements of the current assembly, showing the
-// element's regulators in a collapsible sub-list. its implementation is based
+// constraints on each element as a collapsible sub-list. its implementation
-// on Kate Morley's HTML + CSS tree views:
+// is based on Kate Morley's HTML + CSS tree views:
 //
 //   https://iamkate.com/code/tree-views/
 //
--- a/app-proto/src/specified.rs
+++ b/app-proto/src/specified.rs
@ -1,44 +0,0 @@
 use std::num::ParseFloatError;
 // a real number described by a specification string. since the structure is
 // read-only, we can guarantee that `spec` always specifies `value` in the
 // following format
 //   ┌──────────────────────────────────────────────────────┬───────────┐
 //   │ `spec`                                               │ `value`   │
 //   ┝━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┿━━━━━━━━━━━┥
 //   │ a string that parses to the floating-point value `x` │ `Some(x)` │
 //   ├──────────────────────────────────────────────────────┼───────────┤
 //   │ the empty string                                     │ `None`    │
 //   └──────────────────────────────────────────────────────┴───────────┘
 #[readonly::make]
 pub struct SpecifiedValue {
    pub spec: String,
    pub value: Option<f64>
 }
 impl SpecifiedValue {
    pub fn from_empty_spec() -> SpecifiedValue {
        SpecifiedValue { spec: String::new(), value: None }
    }
    pub fn is_present(&self) -> bool {
        matches!(self.value, Some(_))
    }
 }
 // a `SpecifiedValue` can be constructed from a specification string, formatted
 // as described in the comment on the structure definition. the result is `Ok`
 // if the specification is properly formatted, and `Error` if not
 impl TryFrom<String> for SpecifiedValue {
    type Error = ParseFloatError;
    fn try_from(spec: String) -> Result<Self, Self::Error> {
        if spec.is_empty() {
            Ok(SpecifiedValue::from_empty_spec())
        } else {
            spec.parse::<f64>().map(
                |value| SpecifiedValue { spec: spec, value: Some(value) }
            )
        }
    }
 }
--- a/app-proto/src/tests.rs
+++ b/app-proto/src/tests.rs
@ -1,14 +0,0 @@
 use std::process::Command;
 // build and bundle the application, reporting success if there are no errors or
 // warnings. to see this test fail while others succeed, try moving `index.html`
 // or one of the assets that it links to
 #[test]
 fn trunk_build_test() {
    let build_status = Command::new("trunk")
        .arg("build")
        .env("RUSTFLAGS", "-D warnings")
        .status()
        .expect("Call to Trunk failed");
    assert!(build_status.success());
 }