diff --git a/.forgejo/setup-trunk/action.yaml b/.forgejo/setup-trunk/action.yaml
index 6007527..85e8a31 100644
--- a/.forgejo/setup-trunk/action.yaml
+++ b/.forgejo/setup-trunk/action.yaml
@@ -10,7 +10,7 @@ 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
diff --git a/.forgejo/workflows/continuous-integration.yaml b/.forgejo/workflows/continuous-integration.yaml
index f3b0130..6b53b3a 100644
--- a/.forgejo/workflows/continuous-integration.yaml
+++ b/.forgejo/workflows/continuous-integration.yaml
@@ -24,6 +24,6 @@ jobs:
       # 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
\ No newline at end of file
+      - run: RUSTFLAGS='-D warnings' cargo test
diff --git a/README.md b/README.md
index cf3e589..16b0d35 100644
--- a/README.md
+++ b/README.md
@@ -12,11 +12,11 @@ Note that currently this is just the barest beginnings of the project, more of a
 
 ### Implementation goals
 
-* Comfortable, intuitive UI
+* Provide a comfortable, intuitive UI
 
-* Able to run in browser (so implemented in WASM-compatible language)
+* Allow execution in browser (so implemented in WASM-compatible language)
 
-* Produce scalable graphics of 3D diagrams, and maybe STL files (or other fabricatable file format) as well.
+* Produce scalable graphics of 3D diagrams, and maybe STL files (or other fabricatable file format) as well
 
 ## Prototype
 
@@ -24,64 +24,66 @@ The latest prototype is in the folder `app-proto`. It includes both a user inter
 
 ### Install the prerequisites
 
-1. Install [`rustup`](https://rust-lang.github.io/rustup/): the officially recommended Rust toolchain manager
-   - It's available on Ubuntu as a [Snap](https://snapcraft.io/rustup)
-2. Call `rustup default stable` to "download the latest stable release of Rust and set it as your default toolchain"
-   - If you forget, the `rustup` [help system](https://github.com/rust-lang/rustup/blob/d9b3601c3feb2e88cf3f8ca4f7ab4fdad71441fd/src/errors.rs#L109-L112) will remind you
-3. Call `rustup target add wasm32-unknown-unknown` to add the [most generic 32-bit WebAssembly target](https://doc.rust-lang.org/nightly/rustc/platform-support/wasm32-unknown-unknown.html)
-4. Call `cargo install wasm-pack` to install the [WebAssembly toolchain](https://rustwasm.github.io/docs/wasm-pack/)
-5. Call `cargo install trunk` to install the [Trunk](https://trunkrs.dev/) web-build tool
+1. Install [`rustup`](https://rust-lang.github.io/rustup/): the officially recommended Rust toolchain manager.
+   - It's available on Ubuntu as a [Snap](https://snapcraft.io/rustup).
+2. Call `rustup default stable` to "download the latest stable release of Rust and set it as your default toolchain".
+   - If you forget, the `rustup` [help system](https://github.com/rust-lang/rustup/blob/d9b3601c3feb2e88cf3f8ca4f7ab4fdad71441fd/src/errors.rs#L109-L112) will remind you.
+3. Call `rustup target add wasm32-unknown-unknown` to add the [most generic 32-bit WebAssembly target](https://doc.rust-lang.org/nightly/rustc/platform-support/wasm32-unknown-unknown.html).
+4. Call `cargo install wasm-pack` to install the [WebAssembly toolchain](https://rustwasm.github.io/docs/wasm-pack/).
+5. Call `cargo install trunk` to install the [Trunk](https://trunkrs.dev/) web-build tool.
+   - In the future, `trunk` can be updated with the same command. (You may need the `--locked` flag if your ambient version of `rustc` does not match that required by `trunk`.)
 6. Add the `.cargo/bin` folder in your home directory to your executable search path
-   - This lets you call Trunk, and other tools installed by Cargo, without specifying their paths
-   - On POSIX systems, the search path is stored in the `PATH` environment variable
+   - This lets you call Trunk, and other tools installed by Cargo, without specifying their paths.
+   - On POSIX systems, the search path is stored in the `PATH` environment variable.
+   - Alternatively, if you don't want to adjust your `PATH`, you can install `trunk` in another directory `DIR` via `cargo install --root DIR trunk`.
 
 ### Play with the prototype
 
-1. From the `app-proto` folder, call `trunk serve --release` to build and serve the prototype
-   - The crates the prototype depends on will be downloaded and served automatically
-   - For a faster build, at the expense of a much slower prototype, you can call `trunk serve` without the `--release` flag
+1. From the `app-proto` folder, call `trunk serve --release` to build and serve the prototype.
+   - The crates the prototype depends on will be downloaded and served automatically.
+   - For a faster build, at the expense of a much slower prototype, you can call `trunk serve` without the `--release` flag.
    - If you want to stay in the top-level folder, you can call `trunk serve --config app-proto [--release]` from there instead.
-3. In a web browser, visit one of the URLs listed under the message `INFO 📡 server listening at:`
-   - Touching any file in the `app-proto` folder will make Trunk rebuild and live-reload the prototype
-4. Press *ctrl+C* in the shell where Trunk is running to stop serving the prototype
+3. In a web browser, visit one of the URLs listed under the message `INFO 📡 server listening at:`.
+   - Touching any file in the `app-proto` folder will make Trunk rebuild and live-reload the prototype.
+4. Press *ctrl+C* in the shell where Trunk is running to stop serving the prototype.
 
 ### Run the engine on some example problems
 
-1. Use `sh` to run the script `tools/run-examples.sh`
-   - The script is location-independent, so you can do this from anywhere in the dyna3 repository
+1. Use `sh` to run the script `tools/run-examples.sh`.
+   - The script is location-independent, so you can do this from anywhere in the dyna3 repository.
    - The call from the top level of the repository is:
-     
+
      ```bash
      sh tools/run-examples.sh
      ```
-   - For each example problem, the engine will print the value of the loss function at each optimization step
-   - The first example that prints is the same as the Irisawa hexlet example from the Julia version of the engine prototype. If you go into `engine-proto/gram-test`, launch Julia, and then
-     
+   - For each example problem, the engine will print the value of the loss function at each optimization step.
+   - The first example that prints is the same as the Irisawa hexlet example from the Julia version of the engine prototype. If you go into `engine-proto/gram-test`, launch Julia, and then execute
+
      ```julia
      include("irisawa-hexlet.jl")
      for (step, scaled_loss) in enumerate(history_alt.scaled_loss)
        println(rpad(step-1, 4), " | ", scaled_loss)
      end
      ```
-     
-     you should see that it prints basically the same loss history until the last few steps, when the lower default precision of the Rust engine really starts to show
+
+     you should see that it prints basically the same loss history until the last few steps, when the lower default precision of the Rust engine really starts to show.
 
 ### Run the automated tests
 
-1. Go into the `app-proto` folder
-2. Call `cargo test`
+1. Go into the `app-proto` folder.
+2. Call `cargo test`.
 
 ### Deploy the prototype
 
-1. From the `app-proto` folder, call `trunk build --release`
-   - Building in [release mode](https://doc.rust-lang.org/cargo/reference/profiles.html#release) produces an executable which is smaller and often much faster, but harder to debug and more time-consuming to build
-   - If you want to stay in the top-level folder, you can call `trunk build --config app-proto --release` from there instead
+1. From the `app-proto` folder, call `trunk build --release`.
+   - Building in [release mode](https://doc.rust-lang.org/cargo/reference/profiles.html#release) produces an executable which is smaller and often much faster, but harder to debug and more time-consuming to build.
+   - If you want to stay in the top-level folder, you can call `trunk build --config app-proto --release` from there instead.
 2. Use `sh` to run the packaging script `tools/package-for-deployment.sh`.
-   - The script is location-independent, so you can do this from anywhere in the dyna3 repository
+   - The script is location-independent, so you can do this from anywhere in the dyna3 repository.
    - The call from the top level of the repository is:
      ```bash
      sh tools/package-for-deployment.sh
      ```
-   - This will overwrite or replace the files in `deploy/dyna3`
+   - This will overwrite or replace the files in `deploy/dyna3`.
 3. Put the contents of `deploy/dyna3` in the folder on your server that the prototype will be served from.
-   - To simplify uploading, you might want to combine these files into an archive called `deploy/dyna3.zip`. Git has been set to ignore this path
\ No newline at end of file
+   - To simplify uploading, you might want to combine these files into an archive called `deploy/dyna3.zip`. Git has been set to ignore this path.
diff --git a/app-proto/Cargo.lock b/app-proto/Cargo.lock
index 4f75c45..731dd84 100644
--- a/app-proto/Cargo.lock
+++ b/app-proto/Cargo.lock
@@ -255,6 +255,7 @@ dependencies = [
  "charming",
  "console_error_panic_hook",
  "dyna3",
+ "enum-iterator",
  "itertools",
  "js-sys",
  "lazy_static",
@@ -271,6 +272,26 @@ version = "1.13.0"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "60b1af1c220855b6ceac025d3f6ecdd2b7c4894bfe9cd9bda4fbb4bc7c0d4cf0"
 
+[[package]]
+name = "enum-iterator"
+version = "2.3.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "a4549325971814bda7a44061bf3fe7e487d447cba01e4220a4b454d630d7a016"
+dependencies = [
+ "enum-iterator-derive",
+]
+
+[[package]]
+name = "enum-iterator-derive"
+version = "1.5.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "685adfa4d6f3d765a26bc5dbc936577de9abf756c1feeb3089b01dd395034842"
+dependencies = [
+ "proc-macro2",
+ "quote",
+ "syn",
+]
+
 [[package]]
 name = "equivalent"
 version = "1.0.1"
diff --git a/app-proto/Cargo.toml b/app-proto/Cargo.toml
index 1230b47..d5221a1 100644
--- a/app-proto/Cargo.toml
+++ b/app-proto/Cargo.toml
@@ -10,6 +10,7 @@ default = ["console_error_panic_hook"]
 dev = []
 
 [dependencies]
+enum-iterator = "2.3.0"
 itertools = "0.13.0"
 js-sys = "0.3.70"
 lazy_static = "1.5.0"
diff --git a/app-proto/examples/irisawa-hexlet.rs b/app-proto/examples/irisawa-hexlet.rs
index 0d710ff..d7dd5fc 100644
--- a/app-proto/examples/irisawa-hexlet.rs
+++ b/app-proto/examples/irisawa-hexlet.rs
@@ -15,9 +15,9 @@ fn main() {
         for k in 4..9 {
             println!("  {} sun", 1.0 / config[(3, k)]);
         }
-        
+
         // print the completed Gram matrix
         print::gram_matrix(&config);
     }
     print::loss_history(&realization.history);
-}
\ No newline at end of file
+}
diff --git a/app-proto/examples/kaleidocycle.rs b/app-proto/examples/kaleidocycle.rs
index ae4eb07..4a9ad7e 100644
--- a/app-proto/examples/kaleidocycle.rs
+++ b/app-proto/examples/kaleidocycle.rs
@@ -14,7 +14,7 @@ fn main() {
         // print the completed Gram matrix and the realized configuration
         print::gram_matrix(&config);
         print::config(&config);
-        
+
         // find the kaleidocycle's twist motion by projecting onto the tangent
         // space
         const N_POINTS: usize = 12;
@@ -29,4 +29,4 @@ fn main() {
         let normalization = 5.0 / twist_motion[(2, 0)];
         println!("\nTwist motion:{}", (normalization * twist_motion).to_string().trim_end());
     }
-}
\ No newline at end of file
+}
diff --git a/app-proto/index.html b/app-proto/index.html
index 4fbe52f..c312b51 100644
--- a/app-proto/index.html
+++ b/app-proto/index.html
@@ -6,7 +6,7 @@
     <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">
-    
+
     <!--
       the Charming visualization crate, which we use to show engine diagnostics,
       depends the ECharts JavaScript package
diff --git a/app-proto/src/assembly.rs b/app-proto/src/assembly.rs
index 669c0d0..9b177b5 100644
--- a/app-proto/src/assembly.rs
+++ b/app-proto/src/assembly.rs
@@ -1,10 +1,11 @@
+use enum_iterator::{all, Sequence};
 use nalgebra::{DMatrix, DVector, DVectorView};
 use std::{
     cell::Cell,
     cmp::Ordering,
     collections::{BTreeMap, BTreeSet},
     fmt,
-    fmt::{Debug, Formatter},
+    fmt::{Debug, Display, Formatter},
     hash::{Hash, Hasher},
     rc::Rc,
     sync::{atomic, atomic::AtomicU64},
@@ -26,6 +27,7 @@ use crate::{
         ConfigSubspace,
         ConstraintProblem,
         DescentHistory,
+        MatrixEntry,
         Realization,
     },
     specified::SpecifiedValue,
@@ -43,7 +45,7 @@ static NEXT_SERIAL: AtomicU64 = AtomicU64::new(0);
 pub trait Serial {
     // a serial number that uniquely identifies this element
     fn serial(&self) -> u64;
-    
+
     // take the next serial number, panicking if that was the last one left
     fn next_serial() -> u64 where Self: Sized {
         // the technique we use to panic on overflow is taken from _Rust Atomics
@@ -84,6 +86,14 @@ impl Ord for dyn Serial {
     }
 }
 
+// Small helper function to generate consistent errors when there
+// are indexing issues in a ProblemPoser
+fn indexing_error(item: &str, name: &str, actor: &str) -> String {
+    format!(
+        "{item} \"{name}\" must be indexed before {actor} writes problem data"
+    )
+}
+
 pub trait ProblemPoser {
     fn pose(&self, problem: &mut ConstraintProblem);
 }
@@ -91,33 +101,33 @@ pub trait ProblemPoser {
 pub trait Element: Serial + ProblemPoser + DisplayItem {
     // the default identifier for an element of this type
     fn default_id() -> String where Self: Sized;
-    
+
     // the default example of an element of this type
     fn default(id: String, id_num: u64) -> Self where Self: Sized;
-    
+
     // the default regulators that come with this element
     fn default_regulators(self: Rc<Self>) -> Vec<Rc<dyn Regulator>> {
         Vec::new()
     }
-    
+
     fn id(&self) -> &String;
     fn label(&self) -> &String;
     fn representation(&self) -> Signal<DVector<f64>>;
     fn ghost(&self) -> Signal<bool>;
-    
+
     // the regulators the element is subject to. the assembly that owns the
     // element is responsible for keeping this set up to date
     fn regulators(&self) -> Signal<BTreeSet<Rc<dyn Regulator>>>;
-    
+
     // project a representation vector for this kind of element onto its
     // normalization variety
     fn project_to_normalized(&self, rep: &mut DVector<f64>);
-    
+
     // the configuration matrix column index that was assigned to the element
     // last time the assembly was realized, or `None` if the element has never
     // been through a realization
     fn column_index(&self) -> Option<usize>;
-    
+
     // assign the element a configuration matrix column index. this method must
     // be used carefully to preserve invariant (1), described in the comment on
     // the `tangent` field of the `Assembly` structure
@@ -125,8 +135,8 @@ pub trait Element: Serial + ProblemPoser + DisplayItem {
 }
 
 impl Debug for dyn Element {
-    fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), fmt::Error> {
-        self.id().fmt(f)
+    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
+        Debug::fmt(&self.id(), f)
     }
 }
 
@@ -169,7 +179,7 @@ pub struct Sphere {
 
 impl Sphere {
     const CURVATURE_COMPONENT: usize = 3;
-    
+
     pub fn new(
         id: String,
         label: String,
@@ -193,7 +203,7 @@ impl Element for Sphere {
     fn default_id() -> String {
         "sphere".to_string()
     }
-    
+
     fn default(id: String, id_num: u64) -> Self {
         Self::new(
             id,
@@ -202,39 +212,39 @@ impl Element for Sphere {
             sphere(0.0, 0.0, 0.0, 1.0),
         )
     }
-    
+
     fn default_regulators(self: Rc<Self>) -> Vec<Rc<dyn Regulator>> {
         vec![Rc::new(HalfCurvatureRegulator::new(self))]
     }
-    
+
     fn id(&self) -> &String {
         &self.id
     }
-    
+
     fn label(&self) -> &String {
         &self.label
     }
-    
+
     fn representation(&self) -> Signal<DVector<f64>> {
         self.representation
     }
-    
+
     fn ghost(&self) -> Signal<bool> {
         self.ghost
     }
-    
+
     fn regulators(&self) -> Signal<BTreeSet<Rc<dyn Regulator>>> {
         self.regulators
     }
-    
+
     fn project_to_normalized(&self, rep: &mut DVector<f64>) {
         project_sphere_to_normalized(rep);
     }
-    
+
     fn column_index(&self) -> Option<usize> {
         self.column_index.get()
     }
-    
+
     fn set_column_index(&self, index: usize) {
         self.column_index.set(Some(index));
     }
@@ -249,8 +259,7 @@ impl Serial for Sphere {
 impl ProblemPoser for Sphere {
     fn pose(&self, problem: &mut ConstraintProblem) {
         let index = self.column_index().expect(
-            format!("Sphere \"{}\" should be indexed before writing problem data", self.id).as_str()
-        );
+            indexing_error("Sphere", &self.id, "it").as_str());
         problem.gram.push_sym(index, index, 1.0);
         problem.guess.set_column(index, &self.representation.get_clone_untracked());
     }
@@ -269,7 +278,8 @@ pub struct Point {
 
 impl Point {
     const WEIGHT_COMPONENT: usize = 3;
-    
+    const NORM_COMPONENT: usize = 4;
+
     pub fn new(
         id: String,
         label: String,
@@ -293,7 +303,7 @@ impl Element for Point {
     fn default_id() -> String {
         "point".to_string()
     }
-    
+
     fn default(id: String, id_num: u64) -> Self {
         Self::new(
             id,
@@ -302,35 +312,44 @@ impl Element for Point {
             point(0.0, 0.0, 0.0),
         )
     }
-    
+
+    fn default_regulators(self: Rc<Self>) -> Vec<Rc<dyn Regulator>> {
+        all::<Axis>()
+            .map(|axis| {
+                Rc::new(PointCoordinateRegulator::new(self.clone(), axis))
+                    as Rc::<dyn Regulator>
+            })
+            .collect()
+    }
+
     fn id(&self) -> &String {
         &self.id
     }
-    
+
     fn label(&self) -> &String {
         &self.label
     }
-    
+
     fn representation(&self) -> Signal<DVector<f64>> {
         self.representation
     }
-    
+
     fn ghost(&self) -> Signal<bool> {
         self.ghost
     }
-    
+
     fn regulators(&self) -> Signal<BTreeSet<Rc<dyn Regulator>>> {
         self.regulators
     }
-    
+
     fn project_to_normalized(&self, rep: &mut DVector<f64>) {
         project_point_to_normalized(rep);
     }
-    
+
     fn column_index(&self) -> Option<usize> {
         self.column_index.get()
     }
-    
+
     fn set_column_index(&self, index: usize) {
         self.column_index.set(Some(index));
     }
@@ -345,8 +364,7 @@ impl Serial for Point {
 impl ProblemPoser for Point {
     fn pose(&self, problem: &mut ConstraintProblem) {
         let index = self.column_index().expect(
-            format!("Point \"{}\" should be indexed before writing problem data", self.id).as_str()
-        );
+            indexing_error("Point", &self.id, "it").as_str());
         problem.gram.push_sym(index, index, 0.0);
         problem.frozen.push(Self::WEIGHT_COMPONENT, index, 0.5);
         problem.guess.set_column(index, &self.representation.get_clone_untracked());
@@ -402,10 +420,10 @@ impl InversiveDistanceRegulator {
                 )
             )
         });
-        
+
         let set_point = create_signal(SpecifiedValue::from_empty_spec());
         let serial = Self::next_serial();
-        
+
         Self { subjects, measurement, set_point, serial }
     }
 }
@@ -414,11 +432,11 @@ impl Regulator for InversiveDistanceRegulator {
     fn subjects(&self) -> Vec<Rc<dyn Element>> {
         self.subjects.clone().into()
     }
-    
+
     fn measurement(&self) -> ReadSignal<f64> {
         self.measurement
     }
-    
+
     fn set_point(&self) -> Signal<SpecifiedValue> {
         self.set_point
     }
@@ -436,8 +454,8 @@ impl ProblemPoser for InversiveDistanceRegulator {
             if let Some(val) = set_pt.value {
                 let [row, col] = self.subjects.each_ref().map(
                     |subj| subj.column_index().expect(
-                        "Subjects should be indexed before inversive distance regulator writes problem data"
-                    )
+                        indexing_error("Subject", subj.id(),
+                            "inversive distance regulator").as_str())
                 );
                 problem.gram.push_sym(row, col, val);
             }
@@ -446,21 +464,21 @@ impl ProblemPoser for InversiveDistanceRegulator {
 }
 
 pub struct HalfCurvatureRegulator {
-    pub subject: Rc<dyn Element>,
+    pub subject: Rc<Sphere>,
     pub measurement: ReadSignal<f64>,
     pub set_point: Signal<SpecifiedValue>,
     serial: u64,
 }
 
 impl HalfCurvatureRegulator {
-    pub fn new(subject: Rc<dyn Element>) -> Self {
+    pub fn new(subject: Rc<Sphere>) -> Self {
         let measurement = subject.representation().map(
             |rep| rep[Sphere::CURVATURE_COMPONENT]
         );
-        
+
         let set_point = create_signal(SpecifiedValue::from_empty_spec());
         let serial = Self::next_serial();
-        
+
         Self { subject, measurement, set_point, serial }
     }
 }
@@ -469,11 +487,11 @@ impl Regulator for HalfCurvatureRegulator {
     fn subjects(&self) -> Vec<Rc<dyn Element>> {
         vec![self.subject.clone()]
     }
-    
+
     fn measurement(&self) -> ReadSignal<f64> {
         self.measurement
     }
-    
+
     fn set_point(&self) -> Signal<SpecifiedValue> {
         self.set_point
     }
@@ -490,14 +508,84 @@ impl ProblemPoser for HalfCurvatureRegulator {
         self.set_point.with_untracked(|set_pt| {
             if let Some(val) = set_pt.value {
                 let col = self.subject.column_index().expect(
-                    "Subject should be indexed before half-curvature regulator writes problem data"
-                );
+                    indexing_error("Subject", &self.subject.id,
+                        "half-curvature regulator").as_str());
                 problem.frozen.push(Sphere::CURVATURE_COMPONENT, col, val);
             }
         });
     }
 }
 
+#[derive(Clone, Copy, Sequence)]
+pub enum Axis { X = 0, Y = 1, Z = 2 }
+impl Axis {
+    fn name(&self) -> &'static str {
+        match self { Axis::X => "X", Axis::Y => "Y", Axis::Z => "Z" }
+    }
+}
+
+impl Display for Axis {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        f.write_str(self.name())
+    }
+}
+
+pub struct PointCoordinateRegulator {
+    pub subject: Rc<Point>,
+    pub axis: Axis,
+    pub measurement: ReadSignal<f64>,
+    pub set_point: Signal<SpecifiedValue>,
+    serial: u64
+}
+
+impl PointCoordinateRegulator {
+    pub fn new(subject: Rc<Point>, axis: Axis) -> Self {
+        let measurement = subject.representation().map(
+            move |rep| rep[axis as usize]
+        );
+        let set_point = create_signal(SpecifiedValue::from_empty_spec());
+        Self { subject, axis, measurement, set_point, serial: Self::next_serial() }
+    }
+}
+
+impl Serial for PointCoordinateRegulator {
+    fn serial(&self) -> u64 { self.serial }
+}
+
+impl Regulator for PointCoordinateRegulator {
+    fn subjects(&self) -> Vec<Rc<dyn Element>> { vec![self.subject.clone()] }
+    fn measurement(&self) -> ReadSignal<f64> { self.measurement }
+    fn set_point(&self) -> Signal<SpecifiedValue> { self.set_point }
+}
+
+impl ProblemPoser for PointCoordinateRegulator {
+    fn pose(&self, problem: &mut ConstraintProblem) {
+        self.set_point.with_untracked(|set_pt| {
+            if let Some(val) = set_pt.value {
+                let col = self.subject.column_index().expect(
+                    indexing_error("Subject", &self.subject.id,
+                        "point-coordinate regulator").as_str());
+                problem.frozen.push(self.axis as usize, col, val);
+                // If all three of the subject's spatial coordinates have been
+                // frozen, then freeze its norm component:
+                let mut coords = [0.0; Axis::CARDINALITY];
+                let mut nset: usize = 0;
+                for &MatrixEntry {index, value} in &(problem.frozen) {
+                    if index.1 == col && index.0 < Axis::CARDINALITY {
+                        nset += 1;
+                        coords[index.0] = value
+                    }
+                }
+                if nset == Axis::CARDINALITY {
+                    let [x, y, z] = coords;
+                    problem.frozen.push(
+                        Point::NORM_COMPONENT, col, point(x,y,z)[Point::NORM_COMPONENT]);
+                }
+            }
+        });
+    }
+}
+
 // the velocity is expressed in uniform coordinates
 pub struct ElementMotion<'a> {
     pub element: Rc<dyn Element>,
@@ -512,7 +600,7 @@ pub struct Assembly {
     // elements and regulators
     pub elements: Signal<BTreeSet<Rc<dyn Element>>>,
     pub regulators: Signal<BTreeSet<Rc<dyn Regulator>>>,
-    
+
     // solution variety tangent space. the basis vectors are stored in
     // configuration matrix format, ordered according to the elements' column
     // indices. when you realize the assembly, every element that's present
@@ -524,13 +612,13 @@ pub struct Assembly {
     //      in that column of the tangent space basis matrices
     //
     pub tangent: Signal<ConfigSubspace>,
-    
+
     // indexing
     pub elements_by_id: Signal<BTreeMap<String, Rc<dyn Element>>>,
-    
+
     // realization control
     pub realization_trigger: Signal<()>,
-    
+
     // realization diagnostics
     pub realization_status: Signal<Result<(), String>>,
     pub descent_history: Signal<DescentHistory>,
@@ -550,7 +638,7 @@ impl Assembly {
             descent_history: create_signal(DescentHistory::new()),
             step: create_signal(SpecifiedValue::from_empty_spec()),
         };
-        
+
         // realize the assembly whenever the element list, the regulator list,
         // a regulator's set point, or the realization trigger is updated
         let assembly_for_realization = assembly.clone();
@@ -564,7 +652,7 @@ impl Assembly {
             assembly_for_realization.realization_trigger.track();
             assembly_for_realization.realize();
         });
-        
+
         // load a configuration from the descent history whenever the active
         // step is updated
         let assembly_for_step_selection = assembly.clone();
@@ -576,12 +664,12 @@ impl Assembly {
                 assembly_for_step_selection.load_config(&config)
             }
         });
-        
+
         assembly
     }
-    
+
     // --- inserting elements and regulators ---
-    
+
     // 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
@@ -591,13 +679,13 @@ impl Assembly {
         let elt_rc = Rc::new(elt);
         self.elements.update(|elts| elts.insert(elt_rc.clone()));
         self.elements_by_id.update(|elts_by_id| elts_by_id.insert(id, elt_rc.clone()));
-        
+
         // create and insert the element's default regulators
         for reg in elt_rc.default_regulators() {
             self.insert_regulator(reg);
         }
     }
-    
+
     pub fn try_insert_element(&self, elt: impl Element + 'static) -> bool {
         let can_insert = self.elements_by_id.with_untracked(
             |elts_by_id| !elts_by_id.contains_key(elt.id())
@@ -607,7 +695,7 @@ impl Assembly {
         }
         can_insert
     }
-    
+
     pub fn insert_element_default<T: Element + 'static>(&self) {
         // find the next unused identifier in the default sequence
         let default_id = T::default_id();
@@ -619,17 +707,17 @@ impl Assembly {
             id_num += 1;
             id = format!("{default_id}{id_num}");
         }
-        
+
         // create and insert the default example of `T`
         let _ = self.insert_element_unchecked(T::default(id, id_num));
     }
-    
+
     pub fn insert_regulator(&self, regulator: Rc<dyn Regulator>) {
         // add the regulator to the assembly's regulator list
         self.regulators.update(
             |regs| regs.insert(regulator.clone())
         );
-        
+
         // add the regulator to each subject's regulator list
         let subject_regulators: Vec<_> = regulator.subjects().into_iter().map(
             |subj| subj.regulators()
@@ -637,7 +725,7 @@ impl Assembly {
         for regulators in subject_regulators {
             regulators.update(|regs| regs.insert(regulator.clone()));
         }
-        
+
         /* DEBUG */
         // print an updated list of regulators
         console_log!("Regulators:");
@@ -660,9 +748,9 @@ impl Assembly {
             }
         });
     }
-    
+
     // --- updating the configuration ---
-    
+
     pub fn load_config(&self, config: &DMatrix<f64>) {
         for elt in self.elements.get_clone_untracked() {
             elt.representation().update(
@@ -670,9 +758,9 @@ impl Assembly {
             );
         }
     }
-    
+
     // --- realization ---
-    
+
     pub fn realize(&self) {
         // index the elements
         self.elements.update_silent(|elts| {
@@ -680,7 +768,7 @@ impl Assembly {
                 elt.set_column_index(index);
             }
         });
-        
+
         // set up the constraint problem
         let problem = self.elements.with_untracked(|elts| {
             let mut problem = ConstraintProblem::new(elts.len());
@@ -694,20 +782,21 @@ impl Assembly {
             });
             problem
         });
-        
+
         /* DEBUG */
         // log the Gram matrix
         console_log!("Gram matrix:\n{}", problem.gram);
-        
+        console_log!("Frozen entries:\n{}", problem.frozen);
+
         /* DEBUG */
         // log the initial configuration matrix
         console_log!("Old configuration:{:>8.3}", problem.guess);
-        
+
         // look for a configuration with the given Gram matrix
         let Realization { result, history } = realize_gram(
             &problem, 1.0e-12, 0.5, 0.9, 1.1, 200, 110
         );
-        
+
         /* DEBUG */
         // report the outcome of the search in the browser console
         if let Err(ref message) = result {
@@ -719,20 +808,20 @@ impl Assembly {
             console_log!("Steps: {}", history.scaled_loss.len() - 1);
             console_log!("Loss: {}", history.scaled_loss.last().unwrap());
         }
-        
+
         // report the descent history
         let step_cnt = history.config.len();
         self.descent_history.set(history);
-        
+
         match result {
             Ok(ConfigNeighborhood { nbhd: tangent, .. }) => {
                 /* DEBUG */
                 // report the tangent dimension
                 console_log!("Tangent dimension: {}", tangent.dim());
-                
+
                 // report the realization status
                 self.realization_status.set(Ok(()));
-                
+
                 // display the last realization step
                 self.step.set(
                     if step_cnt > 0 {
@@ -742,7 +831,7 @@ impl Assembly {
                         SpecifiedValue::from_empty_spec()
                     }
                 );
-                
+
                 // save the tangent space
                 self.tangent.set_silent(tangent);
             },
@@ -752,15 +841,15 @@ impl Assembly {
                 // `Err(message)` we received from the match: we're changing the
                 // `Ok` type from `Realization` to `()`
                 self.realization_status.set(Err(message));
-                
+
                 // display the initial guess
                 self.step.set(SpecifiedValue::from(Some(0.0)));
             },
         }
     }
-    
+
     // --- deformation ---
-    
+
     // project the given motion to the tangent space of the solution variety and
     // move the assembly along it. the implementation is based on invariant (1)
     // from above and the following additional invariant:
@@ -777,7 +866,7 @@ impl Assembly {
         if self.tangent.with(|tan| tan.dim() <= 0 && tan.assembly_dim() > 0) {
             console::log_1(&JsValue::from("The assembly is rigid"));
         }
-        
+
         // give a column index to each moving element that doesn't have one yet.
         // this temporarily breaks invariant (1), but the invariant will be
         // restored when we realize the assembly at the end of the deformation.
@@ -795,7 +884,7 @@ impl Assembly {
             }
             next_column_index
         };
-        
+
         // project the element motions onto the tangent space of the solution
         // variety and sum them to get a deformation of the whole assembly. the
         // matrix `motion_proj` that holds the deformation has extra columns for
@@ -806,7 +895,7 @@ impl Assembly {
             // we can unwrap the column index because we know that every moving
             // element has one at this point
             let column_index = elt_motion.element.column_index().unwrap();
-            
+
             if column_index < realized_dim {
                 // this element had a column index when we started, so by
                 // invariant (1), it's reflected in the tangent space
@@ -824,7 +913,7 @@ impl Assembly {
                 target_column += unif_to_std * elt_motion.velocity;
             }
         }
-        
+
         // step the assembly along the deformation. this changes the elements'
         // normalizations, so we restore those afterward
         for elt in self.elements.get_clone_untracked() {
@@ -842,7 +931,7 @@ impl Assembly {
                 };
             });
         }
-        
+
         // trigger a realization to bring the configuration back onto the
         // solution variety. this also gets the elements' column indices and the
         // saved tangent space back in sync
@@ -853,20 +942,22 @@ impl Assembly {
 #[cfg(test)]
 mod tests {
     use super::*;
-    
+
     use crate::engine;
-    
+
     #[test]
-    #[should_panic(expected = "Sphere \"sphere\" should be indexed before writing problem data")]
+    #[should_panic(expected =
+        "Sphere \"sphere\" must be indexed before it writes problem data")]
     fn unindexed_element_test() {
         let _ = create_root(|| {
             let elt = Sphere::default("sphere".to_string(), 0);
             elt.pose(&mut ConstraintProblem::new(1));
         });
     }
-    
+
     #[test]
-    #[should_panic(expected = "Subjects should be indexed before inversive distance regulator writes problem data")]
+    #[should_panic(expected = "Subject \"sphere1\" must be indexed before \
+        inversive distance regulator writes problem data")]
     fn unindexed_subject_test_inversive_distance() {
         let _ = create_root(|| {
             let subjects = [0, 1].map(
@@ -881,7 +972,7 @@ mod tests {
             }.pose(&mut ConstraintProblem::new(2));
         });
     }
-    
+
     #[test]
     fn curvature_drift_test() {
         const INITIAL_RADIUS: f64 = 0.25;
@@ -901,7 +992,7 @@ mod tests {
                     engine::sphere(0.0, 0.0, 0.0, INITIAL_RADIUS),
                 )
             );
-            
+
             // nudge the sphere repeatedly along the `z` axis
             const STEP_SIZE: f64 = 0.0025;
             const STEP_CNT: usize = 400;
@@ -917,7 +1008,7 @@ mod tests {
                     ]
                 );
             }
-            
+
             // check how much the sphere's curvature has drifted
             const INITIAL_HALF_CURV: f64 = 0.5 / INITIAL_RADIUS;
             const DRIFT_TOL: f64 = 0.015;
@@ -927,4 +1018,4 @@ mod tests {
             assert!((final_half_curv / INITIAL_HALF_CURV - 1.0).abs() < DRIFT_TOL);
         });
     }
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/components/diagnostics.rs b/app-proto/src/components/diagnostics.rs
index 51d58f1..0909f45 100644
--- a/app-proto/src/components/diagnostics.rs
+++ b/app-proto/src/components/diagnostics.rs
@@ -54,7 +54,7 @@ fn StepInput() -> View {
     // get the assembly
     let state = use_context::<AppState>();
     let assembly = state.assembly;
-    
+
     // the `last_step` signal holds the index of the last step
     let last_step = assembly.descent_history.map(
         |history| match history.config.len() {
@@ -63,15 +63,15 @@ fn StepInput() -> View {
         }
     );
     let input_max = last_step.map(|last| last.unwrap_or(0));
-    
+
     // these signals hold the entered step number
     let value = create_signal(String::new());
     let value_as_number = create_signal(0.0);
-    
+
     create_effect(move || {
         value.set(assembly.step.with(|n| n.spec.clone()));
     });
-    
+
     view! {
         div(id = "step-input") {
             label { "Step" }
@@ -98,7 +98,7 @@ fn StepInput() -> View {
                                 |val| val.clamp(0.0, input_max.get() as f64)
                             )
                         );
-                        
+
                         // set the input string and the assembly's active step
                         value.set(step.spec.clone());
                         assembly.step.set(step);
@@ -124,7 +124,7 @@ fn LossHistory() -> View {
     const CONTAINER_ID: &str = "loss-history";
     let state = use_context::<AppState>();
     let renderer = WasmRenderer::new_opt(None, Some(178));
-    
+
     on_mount(move || {
         create_effect(move || {
             // get the loss history
@@ -136,13 +136,13 @@ fn LossHistory() -> View {
                     .map(into_log10_time_point)
                     .collect()
             );
-            
+
             // initialize the chart axes
             let step_axis = Axis::new()
                 .type_(AxisType::Category)
                 .boundary_gap(false);
             let scaled_loss_axis = Axis::new();
-            
+
             // load the chart data. when there's no history, we load the data
             // point (0, None) to clear the chart. it would feel more natural to
             // load empty data vectors, but that turns out not to clear the
@@ -164,7 +164,7 @@ fn LossHistory() -> View {
             renderer.render(CONTAINER_ID, &chart).unwrap();
         });
     });
-    
+
     view! {
         div(id = CONTAINER_ID, class = "diagnostics-chart")
     }
@@ -176,7 +176,7 @@ fn SpectrumHistory() -> View {
     const CONTAINER_ID: &str = "spectrum-history";
     let state = use_context::<AppState>();
     let renderer = WasmRenderer::new(478, 178);
-    
+
     on_mount(move || {
         create_effect(move || {
             // get the spectrum of the Hessian at each step, split into its
@@ -208,13 +208,13 @@ fn SpectrumHistory() -> View {
             ): (Vec<_>, Vec<_>) = hess_eigvals_nonzero
                 .into_iter()
                 .partition(|&(_, val)| val > 0.0);
-            
+
             // initialize the chart axes
             let step_axis = Axis::new()
                 .type_(AxisType::Category)
                 .boundary_gap(false);
             let eigval_axis = Axis::new();
-            
+
             // load the chart data. when there's no history, we load the data
             // point (0, None) to clear the chart. it would feel more natural to
             // load empty data vectors, but that turns out not to clear the
@@ -270,7 +270,7 @@ fn SpectrumHistory() -> View {
             renderer.render(CONTAINER_ID, &chart).unwrap();
         });
     });
-    
+
     view! {
         div(id = CONTAINER_ID, class = "diagnostics-chart")
     }
@@ -302,7 +302,7 @@ pub fn Diagnostics() -> View {
     let diagnostics_state = DiagnosticsState::new("loss".to_string());
     let active_tab = diagnostics_state.active_tab.clone();
     provide_context(diagnostics_state);
-    
+
     view! {
         div(id = "diagnostics") {
             div(id = "diagnostics-bar") {
@@ -317,4 +317,4 @@ pub fn Diagnostics() -> View {
             DiagnosticsPanel(name = "spectrum") { SpectrumHistory {} }
         }
     }
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/components/display.rs b/app-proto/src/components/display.rs
index 98be85e..07882c5 100644
--- a/app-proto/src/components/display.rs
+++ b/app-proto/src/components/display.rs
@@ -48,11 +48,11 @@ impl SceneSpheres {
             highlights: Vec::new(),
         }
     }
-    
+
     fn len_i32(&self) -> i32 {
         self.representations.len().try_into().expect("Number of spheres must fit in a 32-bit integer")
     }
-    
+
     fn push(
         &mut self, representation: DVector<f64>,
         color: ElementColor, opacity: f32, highlight: f32,
@@ -79,7 +79,7 @@ impl ScenePoints {
             selections: Vec::new(),
         }
     }
-    
+
     fn push(
         &mut self, representation: DVector<f64>,
         color: ElementColor, opacity: f32, highlight: f32, selected: bool,
@@ -107,7 +107,7 @@ impl Scene {
 
 pub trait DisplayItem {
     fn show(&self, scene: &mut Scene, selected: bool);
-    
+
     // the smallest positive depth, represented as a multiple of `dir`, where
     // the line generated by `dir` hits the element. returns `None` if the line
     // misses the element
@@ -125,14 +125,14 @@ impl DisplayItem for Sphere {
         const DEFAULT_OPACITY: f32 = 0.5;
         const GHOST_OPACITY: f32 = 0.2;
         const HIGHLIGHT: f32 = 0.2;
-        
+
         let representation = self.representation.get_clone_untracked();
         let color = if selected { self.color.map(|channel| 0.2 + 0.8*channel) } else { self.color };
         let opacity = if self.ghost.get() { GHOST_OPACITY } else { DEFAULT_OPACITY };
         let highlight = if selected { 1.0 } else { HIGHLIGHT };
         scene.spheres.push(representation, color, opacity, highlight);
     }
-    
+
     // this method should be kept synchronized with `sphere_cast` in
     // `spheres.frag`, which does essentially the same thing on the GPU side
     fn cast(
@@ -144,12 +144,12 @@ impl DisplayItem for Sphere {
         // if `a/b` is less than this threshold, we approximate
         // `a*u^2 + b*u + c` by the linear function `b*u + c`
         const DEG_THRESHOLD: f64 = 1e-9;
-        
+
         let rep = self.representation.with_untracked(|rep| assembly_to_world * rep);
         let a = -rep[3] * dir.norm_squared();
         let b = rep.rows_range(..3).dot(&dir);
         let c = -rep[4];
-        
+
         let adjust = 4.0*a*c/(b*b);
         if adjust < 1.0 {
             // as long as `b` is non-zero, the linear approximation of
@@ -184,14 +184,14 @@ impl DisplayItem for Point {
         /* SCAFFOLDING */
         const GHOST_OPACITY: f32 = 0.4;
         const HIGHLIGHT: f32 = 0.5;
-        
+
         let representation = self.representation.get_clone_untracked();
         let color = if selected { self.color.map(|channel| 0.2 + 0.8*channel) } else { self.color };
         let opacity = if self.ghost.get() { GHOST_OPACITY } else { 1.0 };
         let highlight = if selected { 1.0 } else { HIGHLIGHT };
         scene.points.push(representation, color, opacity, highlight, selected);
     }
-    
+
     /* SCAFFOLDING */
     fn cast(
         &self,
@@ -203,16 +203,16 @@ impl DisplayItem for Point {
         if rep[2] < 0.0 {
             // this constant should be kept synchronized with `point.frag`
             const POINT_RADIUS_PX: f64 = 4.0;
-            
+
             // find the radius of the point in screen projection units
             let point_radius_proj = POINT_RADIUS_PX * pixel_size;
-            
+
             // find the squared distance between the screen projections of the
             // ray and the point
             let dir_proj = -dir.fixed_rows::<2>(0) / dir[2];
             let rep_proj = -rep.fixed_rows::<2>(0) / rep[2];
             let dist_sq = (dir_proj - rep_proj).norm_squared();
-            
+
             // if the ray hits the point, return its depth
             if dist_sq < point_radius_proj * point_radius_proj {
                 Some(rep[2] / dir[2])
@@ -254,13 +254,13 @@ fn set_up_program(
         WebGl2RenderingContext::FRAGMENT_SHADER,
         fragment_shader_source,
     );
-    
+
     // create the program and attach the shaders
     let program = context.create_program().unwrap();
     context.attach_shader(&program, &vertex_shader);
     context.attach_shader(&program, &fragment_shader);
     context.link_program(&program);
-    
+
     /* DEBUG */
     // report whether linking succeeded
     let link_status = context
@@ -273,7 +273,7 @@ fn set_up_program(
         "Linking failed"
     };
     console::log_1(&JsValue::from(link_msg));
-    
+
     program
 }
 
@@ -318,7 +318,7 @@ fn load_new_buffer(
     // create a buffer and bind it to ARRAY_BUFFER
     let buffer = context.create_buffer();
     context.bind_buffer(WebGl2RenderingContext::ARRAY_BUFFER, buffer.as_ref());
-    
+
     // load the given data into the buffer. this block is unsafe because
     // `Float32Array::view` creates a raw view into our module's
     // `WebAssembly.Memory` buffer. allocating more memory will change the
@@ -332,7 +332,7 @@ fn load_new_buffer(
             WebGl2RenderingContext::STATIC_DRAW,
         );
     }
-    
+
     buffer
 }
 
@@ -353,11 +353,11 @@ fn event_dir(event: &MouseEvent) -> (Vector3<f64>, f64) {
     let width = rect.width();
     let height = rect.height();
     let shortdim = width.min(height);
-    
+
     // this constant should be kept synchronized with `spheres.frag` and
     // `point.vert`
     const FOCAL_SLOPE: f64 = 0.3;
-    
+
     (
         Vector3::new(
             FOCAL_SLOPE * (2.0*(f64::from(event.client_x()) - rect.left()) - width) / shortdim,
@@ -373,13 +373,13 @@ fn event_dir(event: &MouseEvent) -> (Vector3<f64>, f64) {
 #[component]
 pub fn Display() -> View {
     let state = use_context::<AppState>();
-    
+
     // canvas
     let display = create_node_ref();
-    
+
     // viewpoint
     let assembly_to_world = create_signal(DMatrix::<f64>::identity(5, 5));
-    
+
     // navigation
     let pitch_up = create_signal(0.0);
     let pitch_down = create_signal(0.0);
@@ -390,7 +390,7 @@ pub fn Display() -> View {
     let zoom_in = create_signal(0.0);
     let zoom_out = create_signal(0.0);
     let turntable = create_signal(false); /* BENCHMARKING */
-    
+
     // manipulation
     let translate_neg_x = create_signal(0.0);
     let translate_pos_x = create_signal(0.0);
@@ -400,7 +400,7 @@ pub fn Display() -> View {
     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);
     create_effect(move || {
@@ -413,18 +413,18 @@ pub fn Display() -> View {
         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);
-    
+
     let assembly_for_raf = state.assembly.clone();
     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
@@ -432,18 +432,18 @@ pub fn Display() -> View {
         let mut orientation = DMatrix::<f64>::identity(5, 5);
         let mut rotation = DMatrix::<f64>::identity(5, 5);
         let mut location_z: f64 = 5.0;
-        
+
         // 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 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
@@ -452,28 +452,28 @@ pub fn Display() -> View {
             .unwrap()
             .dyn_into::<WebGl2RenderingContext>()
             .unwrap();
-        
+
         // disable depth testing
         ctx.disable(WebGl2RenderingContext::DEPTH_TEST);
-        
+
         // set blend mode
         ctx.enable(WebGl2RenderingContext::BLEND);
         ctx.blend_func(WebGl2RenderingContext::SRC_ALPHA, WebGl2RenderingContext::ONE_MINUS_SRC_ALPHA);
-        
+
         // set up the sphere rendering program
         let sphere_program = set_up_program(
             &ctx,
             include_str!("identity.vert"),
             include_str!("spheres.frag"),
         );
-        
+
         // set up the point rendering program
         let point_program = set_up_program(
             &ctx,
             include_str!("point.vert"),
             include_str!("point.frag"),
         );
-        
+
         /* DEBUG */
         // print the maximum number of vectors that can be passed as
         // uniforms to a fragment shader. the OpenGL ES 3.0 standard
@@ -490,10 +490,10 @@ pub fn Display() -> View {
             &ctx.get_parameter(WebGl2RenderingContext::MAX_FRAGMENT_UNIFORM_VECTORS).unwrap(),
             &JsValue::from("uniform vectors available"),
         );
-        
+
         // find the sphere program's vertex attribute
         let viewport_position_attr = ctx.get_attrib_location(&sphere_program, "position") as u32;
-        
+
         // find the sphere program's uniforms
         const SPHERE_MAX: usize = 200;
         let sphere_cnt_loc = ctx.get_uniform_location(&sphere_program, "sphere_cnt");
@@ -513,7 +513,7 @@ pub fn Display() -> View {
         let shortdim_loc = ctx.get_uniform_location(&sphere_program, "shortdim");
         let layer_threshold_loc = ctx.get_uniform_location(&sphere_program, "layer_threshold");
         let debug_mode_loc = ctx.get_uniform_location(&sphere_program, "debug_mode");
-        
+
         // load the viewport vertex positions into a new vertex buffer object
         const VERTEX_CNT: usize = 6;
         let viewport_positions: [f32; 3*VERTEX_CNT] = [
@@ -527,20 +527,20 @@ pub fn Display() -> View {
              1.0, -1.0, 0.0,
         ];
         let viewport_position_buffer = load_new_buffer(&ctx, &viewport_positions);
-        
+
         // find the point program's vertex attributes
         let point_position_attr = ctx.get_attrib_location(&point_program, "position") as u32;
         let point_color_attr = ctx.get_attrib_location(&point_program, "color") as u32;
         let point_highlight_attr = ctx.get_attrib_location(&point_program, "highlight") as u32;
         let point_selection_attr = ctx.get_attrib_location(&point_program, "selected") as u32;
-        
+
         // 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();
@@ -551,7 +551,7 @@ pub fn Display() -> View {
             let zoom_in_val = zoom_in.get();
             let zoom_out_val = zoom_out.get();
             let turntable_val = turntable.get(); /* BENCHMARKING */
-            
+
             // get the manipulation state
             let translate_neg_x_val = translate_neg_x.get();
             let translate_pos_x_val = translate_pos_x.get();
@@ -561,7 +561,7 @@ pub fn Display() -> View {
             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 = {
                 let pitch = pitch_up_val - pitch_down_val;
@@ -582,11 +582,11 @@ pub fn Display() -> View {
                 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();
-            
+
             // manipulate the assembly
             /* KLUDGE */
             // to avoid the complexity of making tangent space projection
@@ -642,11 +642,11 @@ pub fn Display() -> View {
                     scene_changed.set(true);
                 }
             }
-            
+
             if scene_changed.get() {
                 const SPACE_DIM: usize = 3;
                 const COLOR_SIZE: usize = 3;
-                
+
                 /* INSTRUMENTS */
                 // measure mean frame interval
                 frames_since_last_sample += 1;
@@ -655,11 +655,11 @@ pub fn Display() -> View {
                     last_sample_time = time;
                     frames_since_last_sample = 0;
                 }
-                
+
                 // --- get the assembly ---
-                
+
                 let mut scene = Scene::new();
-                
+
                 // find the map from assembly space to world space
                 let location = {
                     let u = -location_z;
@@ -672,7 +672,7 @@ pub fn Display() -> View {
                     ])
                 };
                 let asm_to_world = &location * &orientation;
-                
+
                 // set up the scene
                 state.assembly.elements.with_untracked(
                     |elts| for elt in elts {
@@ -681,26 +681,26 @@ pub fn Display() -> View {
                     }
                 );
                 let sphere_cnt = scene.spheres.len_i32();
-                
+
                 // --- draw the spheres ---
-                
+
                 // use the sphere rendering program
                 ctx.use_program(Some(&sphere_program));
-                
+
                 // enable the sphere program's vertex attribute
                 ctx.enable_vertex_attrib_array(viewport_position_attr);
-                
+
                 // write the spheres in world coordinates
                 let sphere_reps_world: Vec<_> = scene.spheres.representations.into_iter().map(
                     |rep| (&asm_to_world * rep).cast::<f32>()
                 ).collect();
-                
+
                 // 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 scene data
                 ctx.uniform1i(sphere_cnt_loc.as_ref(), sphere_cnt);
                 for n in 0..sphere_reps_world.len() {
@@ -722,33 +722,33 @@ pub fn Display() -> View {
                         scene.spheres.highlights[n],
                     );
                 }
-                
+
                 // pass the display parameters
                 ctx.uniform1i(layer_threshold_loc.as_ref(), LAYER_THRESHOLD);
                 ctx.uniform1i(debug_mode_loc.as_ref(), DEBUG_MODE);
-                
+
                 // bind the viewport vertex position buffer to the position
                 // attribute in the vertex shader
                 bind_to_attribute(&ctx, viewport_position_attr, SPACE_DIM as i32, &viewport_position_buffer);
-                
+
                 // draw the scene
                 ctx.draw_arrays(WebGl2RenderingContext::TRIANGLES, 0, VERTEX_CNT as i32);
-                
+
                 // disable the sphere program's vertex attribute
                 ctx.disable_vertex_attrib_array(viewport_position_attr);
-                
+
                 // --- draw the points ---
-                
+
                 if !scene.points.representations.is_empty() {
                     // use the point rendering program
                     ctx.use_program(Some(&point_program));
-                    
+
                     // enable the point program's vertex attributes
                     ctx.enable_vertex_attrib_array(point_position_attr);
                     ctx.enable_vertex_attrib_array(point_color_attr);
                     ctx.enable_vertex_attrib_array(point_highlight_attr);
                     ctx.enable_vertex_attrib_array(point_selection_attr);
-                    
+
                     // write the points in world coordinates
                     let asm_to_world_sp = asm_to_world.rows(0, SPACE_DIM);
                     let point_positions = DMatrix::from_columns(
@@ -756,7 +756,7 @@ pub fn Display() -> View {
                             |rep| &asm_to_world_sp * rep
                         ).collect::<Vec<_>>().as_slice()
                     ).cast::<f32>();
-                    
+
                     // load the point positions and colors into new buffers and
                     // bind them to the corresponding attributes in the vertex
                     // shader
@@ -764,22 +764,22 @@ pub fn Display() -> View {
                     bind_new_buffer_to_attribute(&ctx, point_color_attr, (COLOR_SIZE + 1) as i32, scene.points.colors_with_opacity.concat().as_slice());
                     bind_new_buffer_to_attribute(&ctx, point_highlight_attr, 1 as i32, scene.points.highlights.as_slice());
                     bind_new_buffer_to_attribute(&ctx, point_selection_attr, 1 as i32, scene.points.selections.as_slice());
-                    
+
                     // draw the scene
                     ctx.draw_arrays(WebGl2RenderingContext::POINTS, 0, point_positions.ncols() as i32);
-                    
+
                     // disable the point program's vertex attributes
                     ctx.disable_vertex_attrib_array(point_position_attr);
                     ctx.disable_vertex_attrib_array(point_color_attr);
                     ctx.disable_vertex_attrib_array(point_highlight_attr);
                     ctx.disable_vertex_attrib_array(point_selection_attr);
                 }
-                
+
                 // --- update the display state ---
-                
+
                 // update the viewpoint
                 assembly_to_world.set(asm_to_world);
-                
+
                 // clear the scene change flag
                 scene_changed.set(
                     pitch_up_val != 0.0
@@ -799,7 +799,7 @@ pub fn Display() -> View {
         });
         start_animation_loop();
     });
-    
+
     let set_nav_signal = move |event: &KeyboardEvent, value: f64| {
         let mut navigating = true;
         let shift = event.shift_key();
@@ -819,7 +819,7 @@ pub fn Display() -> View {
             event.prevent_default();
         }
     };
-    
+
     let set_manip_signal = move |event: &KeyboardEvent, value: f64| {
         let mut manipulating = true;
         let shift = event.shift_key();
@@ -838,7 +838,7 @@ pub fn Display() -> View {
             event.prevent_default();
         }
     };
-    
+
     view! {
         /* TO DO */
         // switch back to integer-valued parameters when that becomes possible
@@ -860,7 +860,7 @@ pub fn Display() -> View {
                     yaw_left.set(0.0);
                     pitch_up.set(0.0);
                     pitch_down.set(0.0);
-                    
+
                     // swap manipulation inputs
                     translate_pos_z.set(translate_neg_y.get());
                     translate_neg_z.set(translate_pos_y.get());
@@ -886,7 +886,7 @@ pub fn Display() -> View {
                     roll_ccw.set(0.0);
                     zoom_in.set(0.0);
                     zoom_out.set(0.0);
-                    
+
                     // swap manipulation inputs
                     translate_pos_y.set(translate_neg_z.get());
                     translate_neg_y.set(translate_pos_z.get());
@@ -927,7 +927,7 @@ pub fn Display() -> View {
                         None => (),
                     };
                 }
-                
+
                 // if we clicked something, select it
                 match clicked {
                     Some((elt, _)) => state.select(&elt, event.shift_key()),
@@ -936,4 +936,4 @@ pub fn Display() -> View {
             },
         )
     }
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/components/outline.rs b/app-proto/src/components/outline.rs
index 5355042..fc041c7 100644
--- a/app-proto/src/components/outline.rs
+++ b/app-proto/src/components/outline.rs
@@ -9,6 +9,7 @@ use crate::{
         Element,
         HalfCurvatureRegulator,
         InversiveDistanceRegulator,
+        PointCoordinateRegulator,
         Regulator,
     },
     specified::SpecifiedValue
@@ -20,16 +21,16 @@ fn RegulatorInput(regulator: Rc<dyn Regulator>) -> View {
     // get the regulator's measurement and set point signals
     let measurement = regulator.measurement();
     let set_point = regulator.set_point();
-    
+
     // the `valid` signal tracks whether the last entered value is a valid set
     // point specification
     let valid = create_signal(true);
-    
+
     // the `value` signal holds the current set point specification
     let value = create_signal(
         set_point.with_untracked(|set_pt| set_pt.spec.clone())
     );
-    
+
     // this `reset_value` closure resets the input value to the regulator's set
     // point specification
     let reset_value = move || {
@@ -38,11 +39,11 @@ fn RegulatorInput(regulator: Rc<dyn Regulator>) -> View {
             value.set(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",
@@ -119,6 +120,20 @@ impl OutlineItem for HalfCurvatureRegulator {
     }
 }
 
+impl OutlineItem for PointCoordinateRegulator {
+    fn outline_item(self: Rc<Self>, _element: &Rc<dyn Element>) -> View {
+        let name = format!("{} coordinate", self.axis);
+        view! {
+            li(class = "regulator") {
+                div(class = "regulator-label") // for spacing
+                div(class = "regulator-type") { (name) }
+                RegulatorInput(regulator = self)
+                div(class = "status")
+            }
+        }
+    }
+}
+
 // a list item that shows an element in an outline view of an assembly
 #[component(inline_props)]
 fn ElementOutlineItem(element: Rc<dyn Element>) -> View {
@@ -226,7 +241,7 @@ fn ElementOutlineItem(element: Rc<dyn Element>) -> View {
 #[component]
 pub fn Outline() -> View {
     let state = use_context::<AppState>();
-    
+
     // list the elements alphabetically by ID
     /* TO DO */
     // this code is designed to generalize easily to other sort keys. if we only
@@ -239,7 +254,7 @@ pub fn Outline() -> View {
             .sorted_by_key(|elt| elt.id().clone())
             .collect::<Vec<_>>()
     );
-    
+
     view! {
         ul(
             id = "outline",
@@ -257,4 +272,4 @@ pub fn Outline() -> View {
             )
         }
     }
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/components/point.frag b/app-proto/src/components/point.frag
index 194a072..5e18bb4 100644
--- a/app-proto/src/components/point.frag
+++ b/app-proto/src/components/point.frag
@@ -10,10 +10,10 @@ out vec4 outColor;
 
 void main() {
     float r = total_radius * length(2.*gl_PointCoord - vec2(1.));
-    
+
     const float POINT_RADIUS = 4.;
     float border = smoothstep(POINT_RADIUS - 1., POINT_RADIUS, r);
     float disk = 1. - smoothstep(total_radius - 1., total_radius, r);
     vec4 color = mix(point_color, vec4(1.), border * point_highlight);
     outColor = vec4(vec3(1.), disk) * color;
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/components/point.vert b/app-proto/src/components/point.vert
index 0b76bc1..14eb2e7 100644
--- a/app-proto/src/components/point.vert
+++ b/app-proto/src/components/point.vert
@@ -14,11 +14,11 @@ const float focal_slope = 0.3;
 
 void main() {
     total_radius = 5. + 0.5*selected;
-    
+
     float depth = -focal_slope * position.z;
     gl_Position = vec4(position.xy / depth, 0., 1.);
     gl_PointSize = 2.*total_radius;
-    
+
     point_color = color;
     point_highlight = highlight;
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/components/spheres.frag b/app-proto/src/components/spheres.frag
index fa317a8..172109a 100644
--- a/app-proto/src/components/spheres.frag
+++ b/app-proto/src/components/spheres.frag
@@ -75,7 +75,7 @@ Fragment sphere_shading(vecInv v, vec3 pt, vec4 base_color) {
     // 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.rgb, base_color.a));
@@ -110,7 +110,7 @@ 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
@@ -136,7 +136,7 @@ vec2 sphere_cast(vecInv v, vec3 dir) {
 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];
@@ -144,7 +144,7 @@ void main() {
     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) {
@@ -169,14 +169,14 @@ void main() {
             }
         }
     }
-    
+
     /* 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.);
@@ -186,7 +186,7 @@ void main() {
         outColor = vec4(color, 1.);
         return;
     }
-    
+
     // composite the sphere fragments
     vec3 color = vec3(0.);
     int layer = layer_cnt - 1;
@@ -203,7 +203,7 @@ void main() {
         // load the current fragment
         Fragment frag = frag_next;
         float highlight = highlight_next;
-        
+
         // shade the next fragment
         hit = top_hits[layer];
         sphere_color = color_list[hit.id];
@@ -213,23 +213,23 @@ void main() {
             vec4(hit.dimming * sphere_color.rgb, sphere_color.a)
         );
         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.);
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/components/test_assembly_chooser.rs b/app-proto/src/components/test_assembly_chooser.rs
index 0d387d3..d572362 100644
--- a/app-proto/src/components/test_assembly_chooser.rs
+++ b/app-proto/src/components/test_assembly_chooser.rs
@@ -144,7 +144,7 @@ fn load_low_curvature(assembly: &Assembly) {
             engine::sphere(2.0/3.0, 4.0/3.0 * a, 0.0, 1.0/3.0),
         )
     );
-    
+
     // impose the desired tangencies and make the sides planar
     let index_range = 1..=3;
     let [central, assemb_plane] = ["central", "assemb_plane"].map(
@@ -217,7 +217,7 @@ fn load_pointed(assembly: &Assembly) {
         for index_y in 0..=1 {
             let x = index_x as f64 - 0.5;
             let y = index_y as f64 - 0.5;
-            
+
             let _ = assembly.try_insert_element(
                 Sphere::new(
                     format!("sphere{index_x}{index_y}"),
@@ -226,7 +226,7 @@ fn load_pointed(assembly: &Assembly) {
                     engine::sphere(x, y, 0.0, 1.0),
                 )
             );
-            
+
             let _ = assembly.try_insert_element(
                 Point::new(
                     format!("point{index_x}{index_y}"),
@@ -310,7 +310,7 @@ fn load_tridiminished_icosahedron(assembly: &Assembly) {
     for vertex in vertices {
         let _ = assembly.try_insert_element(vertex);
     }
-    
+
     // create the faces
     const COLOR_FACE: ElementColor = [0.75_f32, 0.75_f32, 0.75_f32];
     let frac_1_sqrt_6 = 1.0 / 6.0_f64.sqrt();
@@ -339,7 +339,7 @@ fn load_tridiminished_icosahedron(assembly: &Assembly) {
         face.ghost().set(true);
         let _ = assembly.try_insert_element(face);
     }
-    
+
     let index_range = 1..=3;
     for j in index_range.clone() {
         // make each face planar
@@ -352,7 +352,7 @@ fn load_tridiminished_icosahedron(assembly: &Assembly) {
         curvature_regulator.set_point().set(
             SpecifiedValue::try_from("0".to_string()).unwrap()
         );
-        
+
         // put each A vertex on the face it belongs to
         let vertex_a = assembly.elements_by_id.with_untracked(
             |elts_by_id| elts_by_id[&format!("a{j}")].clone()
@@ -360,7 +360,7 @@ fn load_tridiminished_icosahedron(assembly: &Assembly) {
         let incidence_a = InversiveDistanceRegulator::new([face.clone(), vertex_a.clone()]);
         incidence_a.set_point.set(SpecifiedValue::try_from("0".to_string()).unwrap());
         assembly.insert_regulator(Rc::new(incidence_a));
-        
+
         // regulate the B-C vertex distances
         let vertices_bc = ["b", "c"].map(
             |series| assembly.elements_by_id.with_untracked(
@@ -370,10 +370,10 @@ fn load_tridiminished_icosahedron(assembly: &Assembly) {
         assembly.insert_regulator(
             Rc::new(InversiveDistanceRegulator::new(vertices_bc))
         );
-        
+
         // get the pair of indices adjacent to `j`
         let adjacent_indices = [j % 3 + 1, (j + 1) % 3 + 1];
-        
+
         for k in adjacent_indices.clone() {
             for series in ["b", "c"] {
                 // put each B and C vertex on the faces it belongs to
@@ -383,14 +383,14 @@ fn load_tridiminished_icosahedron(assembly: &Assembly) {
                 let incidence = InversiveDistanceRegulator::new([face.clone(), vertex.clone()]);
                 incidence.set_point.set(SpecifiedValue::try_from("0".to_string()).unwrap());
                 assembly.insert_regulator(Rc::new(incidence));
-                
+
                 // regulate the A-B and A-C vertex distances
                 assembly.insert_regulator(
                     Rc::new(InversiveDistanceRegulator::new([vertex_a.clone(), vertex]))
                 );
             }
         }
-        
+
         // regulate the A-A and C-C vertex distances
         let adjacent_pairs = ["a", "c"].map(
             |series| adjacent_indices.map(
@@ -422,14 +422,14 @@ fn load_dodecahedral_packing(assembly: &Assembly) {
     let substrate = assembly.elements_by_id.with_untracked(
         |elts_by_id| elts_by_id["substrate"].clone()
     );
-    
+
     // fix the substrate's curvature
     substrate.regulators().with_untracked(
         |regs| regs.first().unwrap().clone()
     ).set_point().set(
         SpecifiedValue::try_from("0.5".to_string()).unwrap()
     );
-    
+
     // add the circles to be packed
     const COLOR_A: ElementColor = [1.00_f32, 0.25_f32, 0.00_f32];
     const COLOR_B: ElementColor = [1.00_f32, 0.00_f32, 0.25_f32];
@@ -445,10 +445,10 @@ fn load_dodecahedral_packing(assembly: &Assembly) {
         for k in 0..2 {
             let small_coord = face_scales[k] * (2.0*(j as f64) - 1.0);
             let big_coord = face_scales[k] * (2.0*(k as f64) - 1.0) * phi;
-            
+
             let id_num = format!("{j}{k}");
             let label_sub = format!("{}{}", subscripts[j], subscripts[k]);
-            
+
             // add the A face
             let id_a = format!("a{id_num}");
             let _ = assembly.try_insert_element(
@@ -464,7 +464,7 @@ fn load_dodecahedral_packing(assembly: &Assembly) {
                     |elts_by_id| elts_by_id[&id_a].clone()
                 )
             );
-            
+
             // add the B face
             let id_b = format!("b{id_num}");
             let _ = assembly.try_insert_element(
@@ -480,7 +480,7 @@ fn load_dodecahedral_packing(assembly: &Assembly) {
                     |elts_by_id| elts_by_id[&id_b].clone()
                 )
             );
-            
+
             // add the C face
             let id_c = format!("c{id_num}");
             let _ = assembly.try_insert_element(
@@ -498,14 +498,14 @@ fn load_dodecahedral_packing(assembly: &Assembly) {
             );
         }
     }
-    
+
     // make each face sphere perpendicular to the substrate
     for face in faces {
         let right_angle = InversiveDistanceRegulator::new([face, substrate.clone()]);
         right_angle.set_point.set(SpecifiedValue::try_from("0".to_string()).unwrap());
         assembly.insert_regulator(Rc::new(right_angle));
     }
-    
+
     // set up the tangencies that define the packing
     for [long_edge_plane, short_edge_plane] in [["a", "b"], ["b", "c"], ["c", "a"]] {
         for k in 0..2 {
@@ -524,14 +524,14 @@ fn load_dodecahedral_packing(assembly: &Assembly) {
                     )
                 )
             );
-            
+
             // set up the short-edge tangency
             let short_tangency = InversiveDistanceRegulator::new(short_edge.clone());
             if k == 0 {
                 short_tangency.set_point.set(SpecifiedValue::try_from("-1".to_string()).unwrap());
             }
             assembly.insert_regulator(Rc::new(short_tangency));
-            
+
             // set up the side tangencies
             for i in 0..2 {
                 for j in 0..2 {
@@ -577,14 +577,14 @@ fn load_balanced(assembly: &Assembly) {
     for sphere in spheres {
         let _ = assembly.try_insert_element(sphere);
     }
-    
+
     // get references to the spheres
     let [outer, a, b] = ["outer", "a", "b"].map(
         |id| assembly.elements_by_id.with_untracked(
             |elts_by_id| elts_by_id[id].clone()
         )
     );
-    
+
     // fix the diameters of the outer, sun, and moon spheres
     for (sphere, radius) in [
         (outer.clone(), R_OUTER),
@@ -599,7 +599,7 @@ fn load_balanced(assembly: &Assembly) {
             SpecifiedValue::try_from(curvature.to_string()).unwrap()
         );
     }
-    
+
     // set the inversive distances between the spheres. as described above, the
     // initial configuration deliberately violates these constraints
     for inner in [a, b] {
@@ -629,14 +629,14 @@ fn load_off_center(assembly: &Assembly) {
             engine::sphere(0.0, 0.0, 0.0, 1.0),
         ),
     );
-    
+
     // get references to the elements
     let point_and_sphere = ["point", "sphere"].map(
         |id| assembly.elements_by_id.with_untracked(
             |elts_by_id| elts_by_id[id].clone()
         )
     );
-    
+
     // put the point on the sphere
     let incidence = InversiveDistanceRegulator::new(point_and_sphere);
     incidence.set_point.set(SpecifiedValue::try_from("0".to_string()).unwrap());
@@ -650,7 +650,7 @@ fn load_off_center(assembly: &Assembly) {
 // inversive distance of 0 between the circumsphere and each vertex
 fn load_radius_ratio(assembly: &Assembly) {
     let index_range = 1..=4;
-    
+
     // create the spheres
     const GRAY: ElementColor = [0.75_f32, 0.75_f32, 0.75_f32];
     let spheres = [
@@ -670,7 +670,7 @@ fn load_radius_ratio(assembly: &Assembly) {
     for sphere in spheres {
         let _ = assembly.try_insert_element(sphere);
     }
-    
+
     // create the vertices
     let vertices = izip!(
         index_range.clone(),
@@ -699,7 +699,7 @@ fn load_radius_ratio(assembly: &Assembly) {
     for vertex in vertices {
         let _ = assembly.try_insert_element(vertex);
     }
-    
+
     // create the faces
     let base_dir = Vector3::new(1.0, 0.75, 1.0).normalize();
     let offset = base_dir.dot(&Vector3::new(-0.6, 0.8, 0.6));
@@ -731,7 +731,7 @@ fn load_radius_ratio(assembly: &Assembly) {
         face.ghost().set(true);
         let _ = assembly.try_insert_element(face);
     }
-    
+
     // impose the constraints
     for j in index_range.clone() {
         let [face_j, vertex_j] = [
@@ -742,7 +742,7 @@ fn load_radius_ratio(assembly: &Assembly) {
                 |elts_by_id| elts_by_id[&id].clone()
             )
         );
-        
+
         // make the faces planar
         let curvature_regulator = face_j.regulators().with_untracked(
             |regs| regs.first().unwrap().clone()
@@ -750,12 +750,12 @@ fn load_radius_ratio(assembly: &Assembly) {
         curvature_regulator.set_point().set(
             SpecifiedValue::try_from("0".to_string()).unwrap()
         );
-        
+
         for k in index_range.clone().filter(|&index| index != j) {
             let vertex_k = assembly.elements_by_id.with_untracked(
                 |elts_by_id| elts_by_id[&format!("v{k}")].clone()
             );
-            
+
             // fix the distances between the vertices
             if j < k {
                 let distance_regulator = InversiveDistanceRegulator::new(
@@ -763,7 +763,7 @@ fn load_radius_ratio(assembly: &Assembly) {
                 );
                 assembly.insert_regulator(Rc::new(distance_regulator));
             }
-            
+
             // put the vertices on the faces
             let incidence_regulator = InversiveDistanceRegulator::new([face_j.clone(), vertex_k.clone()]);
             incidence_regulator.set_point.set(SpecifiedValue::try_from("0".to_string()).unwrap());
@@ -799,7 +799,7 @@ fn load_irisawa_hexlet(assembly: &Assembly) {
         [0.00_f32, 0.25_f32, 1.00_f32],
         [0.25_f32, 0.00_f32, 1.00_f32],
     ].into_iter();
-    
+
     // create the spheres
     let spheres = [
         Sphere::new(
@@ -836,7 +836,7 @@ fn load_irisawa_hexlet(assembly: &Assembly) {
     for sphere in spheres {
         let _ = assembly.try_insert_element(sphere);
     }
-    
+
     // put the outer sphere in ghost mode and fix its curvature
     let outer = assembly.elements_by_id.with_untracked(
         |elts_by_id| elts_by_id["outer"].clone()
@@ -848,7 +848,7 @@ fn load_irisawa_hexlet(assembly: &Assembly) {
     outer_curvature_regulator.set_point().set(
         SpecifiedValue::try_from((1.0 / 3.0).to_string()).unwrap()
     );
-    
+
     // impose the desired tangencies
     let [outer, sun, moon] = ["outer", "sun", "moon"].map(
         |id| assembly.elements_by_id.with_untracked(
@@ -872,11 +872,11 @@ fn load_irisawa_hexlet(assembly: &Assembly) {
             assembly.insert_regulator(Rc::new(tangency));
         }
     }
-    
+
     let outer_sun_tangency = InversiveDistanceRegulator::new([outer.clone(), sun]);
     outer_sun_tangency.set_point.set(SpecifiedValue::try_from("1".to_string()).unwrap());
     assembly.insert_regulator(Rc::new(outer_sun_tangency));
-    
+
     let outer_moon_tangency = InversiveDistanceRegulator::new([outer.clone(), moon]);
     outer_moon_tangency.set_point.set(SpecifiedValue::try_from("1".to_string()).unwrap());
     assembly.insert_regulator(Rc::new(outer_moon_tangency));
@@ -895,18 +895,18 @@ pub fn TestAssemblyChooser() -> View {
         console::log_1(
             &JsValue::from(format!("Showing assembly \"{}\"", name.clone()))
         );
-        
+
         batch(|| {
             let state = use_context::<AppState>();
             let assembly = &state.assembly;
-            
+
             // clear state
             assembly.regulators.update(|regs| regs.clear());
             assembly.elements.update(|elts| elts.clear());
             assembly.elements_by_id.update(|elts_by_id| elts_by_id.clear());
             assembly.descent_history.set(DescentHistory::new());
             state.selection.update(|sel| sel.clear());
-            
+
             // load assembly
             match name.as_str() {
                 "general" => load_general(assembly),
@@ -922,7 +922,7 @@ pub fn TestAssemblyChooser() -> View {
             };
         });
     });
-    
+
     // build the chooser
     view! {
         select(bind:value = assembly_name) {
@@ -938,4 +938,4 @@ pub fn TestAssemblyChooser() -> View {
             option(value = "empty") { "Empty" }
         }
     }
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/engine.rs b/app-proto/src/engine.rs
index ef150a0..35265e5 100644
--- a/app-proto/src/engine.rs
+++ b/app-proto/src/engine.rs
@@ -52,8 +52,8 @@ pub fn project_point_to_normalized(rep: &mut DVector<f64>) {
 // --- partial matrices ---
 
 pub struct MatrixEntry {
-    index: (usize, usize),
-    value: f64,
+    pub index: (usize, usize),
+    pub value: f64,
 }
 
 pub struct PartialMatrix(Vec<MatrixEntry>);
@@ -62,19 +62,19 @@ impl PartialMatrix {
     pub fn new() -> Self {
         Self(Vec::<MatrixEntry>::new())
     }
-    
+
     pub fn push(&mut self, row: usize, col: usize, value: f64) {
         let Self(entries) = self;
         entries.push(MatrixEntry { index: (row, col), value });
     }
-    
+
     pub fn push_sym(&mut self, row: usize, col: usize, value: f64) {
         self.push(row, col, value);
         if row != col {
             self.push(col, row, value);
         }
     }
-    
+
     fn freeze(&self, a: &DMatrix<f64>) -> DMatrix<f64> {
         let mut result = a.clone();
         for &MatrixEntry { index, value } in self {
@@ -82,7 +82,7 @@ impl PartialMatrix {
         }
         result
     }
-    
+
     fn proj(&self, a: &DMatrix<f64>) -> DMatrix<f64> {
         let mut result = DMatrix::<f64>::zeros(a.nrows(), a.ncols());
         for &MatrixEntry { index, .. } in self {
@@ -90,7 +90,7 @@ impl PartialMatrix {
         }
         result
     }
-    
+
     fn sub_proj(&self, rhs: &DMatrix<f64>) -> DMatrix<f64> {
         let mut result = DMatrix::<f64>::zeros(rhs.nrows(), rhs.ncols());
         for &MatrixEntry { index, value } in self {
@@ -112,7 +112,7 @@ impl Display for PartialMatrix {
 impl IntoIterator for PartialMatrix {
     type Item = MatrixEntry;
     type IntoIter = std::vec::IntoIter<Self::Item>;
-    
+
     fn into_iter(self) -> Self::IntoIter {
         let Self(entries) = self;
         entries.into_iter()
@@ -122,7 +122,7 @@ impl IntoIterator for PartialMatrix {
 impl<'a> IntoIterator for &'a PartialMatrix {
     type Item = &'a MatrixEntry;
     type IntoIter = std::slice::Iter<'a, MatrixEntry>;
-    
+
     fn into_iter(self) -> Self::IntoIter {
         let PartialMatrix(entries) = self;
         entries.into_iter()
@@ -146,7 +146,7 @@ impl ConfigSubspace {
             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`
@@ -167,10 +167,10 @@ impl ConfigSubspace {
                 |(λ, v)| (λ.abs() < THRESHOLD).then_some(v)
             ).collect::<Vec<_>>().as_slice()
         );
-        
+
         // 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;
         Self {
@@ -187,15 +187,15 @@ impl ConfigSubspace {
             ).collect(),
         }
     }
-    
+
     pub fn dim(&self) -> usize {
         self.basis_std.len()
     }
-    
+
     pub fn assembly_dim(&self) -> usize {
         self.assembly_dim
     }
-    
+
     // find the projection onto this subspace of the motion where the element
     // with the given column index has velocity `v`. the velocity is given in
     // projection coordinates, and the projection is done with respect to the
@@ -253,7 +253,7 @@ impl ConstraintProblem {
             guess: DMatrix::<f64>::zeros(ELEMENT_DIM, element_count),
         }
     }
-    
+
     #[cfg(feature = "dev")]
     pub fn from_guess(guess_columns: &[DVector<f64>]) -> Self {
         Self {
@@ -377,10 +377,10 @@ pub fn realize_gram(
 ) -> Realization {
     // destructure the problem data
     let ConstraintProblem { gram, guess, frozen } = problem;
-    
+
     // start the descent history
     let mut history = DescentHistory::new();
-    
+
     // handle the case where the assembly is empty. our general realization
     // routine can't handle this case because it builds the Hessian using
     // `DMatrix::from_columns`, which panics when the list of columns is empty
@@ -394,20 +394,20 @@ pub fn realize_gram(
         );
         return Realization { result, history };
     }
-    
+
     // find the dimension of the search space
     let element_dim = guess.nrows();
     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(
         |MatrixEntry { index: (row, col), .. }| col*element_dim + row
     ).collect();
-    
+
     // use a regularized Newton's method with backtracking
     let mut state = SearchState::from_config(gram, frozen.freeze(guess));
     let mut hess = DMatrix::zeros(element_dim, assembly_dim);
@@ -416,7 +416,7 @@ pub fn realize_gram(
         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 {
@@ -435,7 +435,7 @@ pub fn realize_gram(
             }
         }
         hess = DMatrix::from_columns(hess_cols.as_slice());
-        
+
         // regularize the Hessian
         let hess_eigvals = hess.symmetric_eigenvalues();
         let min_eigval = hess_eigvals.min();
@@ -443,7 +443,7 @@ pub fn realize_gram(
             hess -= reg_scale * min_eigval * DMatrix::identity(total_dim, total_dim);
         }
         history.hess_eigvals.push(hess_eigvals);
-        
+
         // project the negative gradient and negative Hessian onto the
         // orthogonal complement of the frozen subspace
         let zero_col = DVector::zeros(total_dim);
@@ -454,12 +454,12 @@ pub fn realize_gram(
             hess.set_column(k, &zero_col);
             hess[(k, k)] = 1.0;
         }
-        
+
         // 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; }
-        
+
         // compute the Newton step
         /* TO DO */
         /*
@@ -479,7 +479,7 @@ pub fn realize_gram(
         let base_step_stacked = hess_cholesky.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
         if let Some((better_state, backoff_steps)) = seek_better_config(
             gram, &state, &base_step, neg_grad.dot(&base_step),
@@ -505,10 +505,10 @@ pub fn realize_gram(
                 .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
         let tangent = ConfigSubspace::symmetric_kernel(hess, unif_to_std, assembly_dim);
-        
+
         Ok(ConfigNeighborhood { #[cfg(feature = "dev")] config: state.config, nbhd: tangent })
     } else {
         Err("Failed to reach target accuracy".to_string())
@@ -521,9 +521,9 @@ pub fn realize_gram(
 #[cfg(feature = "dev")]
 pub mod examples {
     use std::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
@@ -547,40 +547,40 @@ pub mod examples {
                 )
             ).collect::<Vec<_>>().as_slice()
         );
-        
+
         for s in 0..9 {
             // each sphere is represented by a spacelike vector
             problem.gram.push_sym(s, s, 1.0);
-            
+
             // the circumscribing sphere is tangent to all of the other
             // spheres, with matching orientation
             if s > 0 {
                 problem.gram.push_sym(0, s, 1.0);
             }
-            
+
             if s > 2 {
                 // each chain sphere is tangent to the "sun" and "moon"
                 // spheres, with opposing orientation
                 for n in 1..3 {
                     problem.gram.push_sym(s, n, -1.0);
                 }
-                
+
                 // each chain sphere is tangent to the next chain sphere,
                 // with opposing orientation
                 let s_next = 3 + (s-2) % 6;
                 problem.gram.push_sym(s, s_next, -1.0);
             }
         }
-        
+
         // the frozen entries fix the radii of the circumscribing sphere, the
         // "sun" and "moon" spheres, and one of the chain spheres
         for k in 0..4 {
             problem.frozen.push(3, k, problem.guess[(3, k)]);
         }
-        
+
         realize_gram(&problem, 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) -> Realization {
@@ -601,7 +601,7 @@ pub mod examples {
                 }
             ).collect::<Vec<_>>().as_slice()
         );
-        
+
         const N_POINTS: usize = 2 * N_HINGES;
         for block in (0..N_POINTS).step_by(2) {
             let block_next = (block + 2) % N_POINTS;
@@ -610,18 +610,18 @@ pub mod examples {
                 for k in j..2 {
                     problem.gram.push_sym(block + j, block + k, if j == k { 0.0 } else { -0.5 });
                 }
-                
+
                 // non-hinge edges
                 for k in 0..2 {
                     problem.gram.push_sym(block + j, block_next + k, -0.625);
                 }
             }
         }
-        
+
         for k in 0..N_POINTS {
             problem.frozen.push(3, k, problem.guess[(3, k)])
         }
-        
+
         realize_gram(&problem, scaled_tol, 0.5, 0.9, 1.1, 200, 110)
     }
 }
@@ -630,9 +630,9 @@ pub mod examples {
 mod tests {
     use nalgebra::Vector3;
     use std::{f64::consts::{FRAC_1_SQRT_2, PI}, iter};
-    
+
     use super::{*, examples::*};
-    
+
     #[test]
     fn freeze_test() {
         let frozen = PartialMatrix(vec![
@@ -651,7 +651,7 @@ mod tests {
         ]);
         assert_eq!(frozen.freeze(&config), expected_result);
     }
-    
+
     #[test]
     fn sub_proj_test() {
         let target = PartialMatrix(vec![
@@ -670,7 +670,7 @@ mod tests {
         ]);
         assert_eq!(target.sub_proj(&attempt), expected_result);
     }
-    
+
     #[test]
     fn zero_loss_test() {
         let mut gram = PartialMatrix::new();
@@ -690,7 +690,7 @@ mod tests {
         let state = SearchState::from_config(&gram, config);
         assert!(state.loss.abs() < f64::EPSILON);
     }
-    
+
     /* TO DO */
     // at the frozen indices, the optimization steps should have exact zeros,
     // and the realized configuration should have the desired values
@@ -720,13 +720,13 @@ mod tests {
             assert_eq!(config[index], value);
         }
     }
-    
+
     #[test]
     fn irisawa_hexlet_test() {
         // solve Irisawa's problem
         const SCALED_TOL: f64 = 1.0e-12;
         let config = realize_irisawa_hexlet(SCALED_TOL).result.unwrap().config;
-        
+
         // 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];
@@ -734,7 +734,7 @@ mod tests {
             assert!((config[(3, k)] - 1.0 / diam).abs() < entry_tol);
         }
     }
-    
+
     #[test]
     fn tangent_test_three_spheres() {
         const SCALED_TOL: f64 = 1.0e-12;
@@ -758,7 +758,7 @@ mod tests {
         let ConfigNeighborhood { config, nbhd: tangent } = result.unwrap();
         assert_eq!(config, problem.guess);
         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;
@@ -786,11 +786,11 @@ mod tests {
                 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
@@ -802,13 +802,13 @@ mod tests {
             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| {
@@ -819,7 +819,7 @@ mod tests {
             }
         ).collect()
     }
-    
+
     #[test]
     fn tangent_test_kaleidocycle() {
         // set up a kaleidocycle and find its tangent space
@@ -827,7 +827,7 @@ mod tests {
         let Realization { result, history } = realize_kaleidocycle(SCALED_TOL);
         let ConfigNeighborhood { config, nbhd: tangent } = result.unwrap();
         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;
@@ -838,12 +838,12 @@ mod tests {
             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:
@@ -872,11 +872,11 @@ mod tests {
                 ).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
@@ -888,7 +888,7 @@ mod tests {
             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, &[
@@ -899,7 +899,7 @@ mod tests {
                    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]
@@ -919,7 +919,7 @@ mod tests {
         let ConfigNeighborhood { config: config_orig, nbhd: tangent_orig } = result_orig.unwrap();
         assert_eq!(config_orig, problem_orig.guess);
         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 = {
@@ -940,17 +940,17 @@ mod tests {
         let ConfigNeighborhood { config: config_tfm, nbhd: tangent_tfm } = result_tfm.unwrap();
         assert_eq!(config_tfm, problem_tfm.guess);
         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
@@ -964,7 +964,7 @@ mod tests {
         ]);
         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
@@ -972,4 +972,4 @@ mod tests {
         let tol_sq = ((problem_orig.guess.nrows() * problem_orig.guess.ncols()) as f64) * SCALED_TOL_TFM * SCALED_TOL_TFM;
         assert!((motion_proj_tfm - motion_tfm_proj).norm_squared() < tol_sq);
     }
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/main.rs b/app-proto/src/main.rs
index a03b026..d8fb030 100644
--- a/app-proto/src/main.rs
+++ b/app-proto/src/main.rs
@@ -30,7 +30,7 @@ impl AppState {
             selection: create_signal(BTreeSet::default()),
         }
     }
-    
+
     // in single-selection mode, select the given element. in multiple-selection
     // mode, toggle whether the given element is selected
     fn select(&self, element: &Rc<dyn Element>, multi: bool) {
@@ -53,10 +53,10 @@ fn main() {
     // set the console error panic hook
     #[cfg(feature = "console_error_panic_hook")]
     console_error_panic_hook::set_once();
-    
+
     sycamore::render(|| {
         provide_context(AppState::new());
-        
+
         view! {
             div(id = "sidebar") {
                 AddRemove {}
@@ -66,4 +66,4 @@ fn main() {
             Display {}
         }
     });
-}
\ No newline at end of file
+}
diff --git a/app-proto/src/specified.rs b/app-proto/src/specified.rs
index b0f04b5..d54e75c 100644
--- a/app-proto/src/specified.rs
+++ b/app-proto/src/specified.rs
@@ -20,7 +20,7 @@ impl SpecifiedValue {
     pub fn from_empty_spec() -> Self {
         Self { spec: String::new(), value: None }
     }
-    
+
     pub fn is_present(&self) -> bool {
         matches!(self.value, Some(_))
     }
@@ -42,7 +42,7 @@ impl From<Option<f64>> for SpecifiedValue {
 // 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(Self::from_empty_spec())
@@ -52,4 +52,4 @@ impl TryFrom<String> for SpecifiedValue {
             )
         }
     }
-}
\ No newline at end of file
+}
diff --git a/notes/inversive.md b/notes/inversive.md
index 933eb35..1d35e15 100644
--- a/notes/inversive.md
+++ b/notes/inversive.md
@@ -33,7 +33,7 @@ The unification of spheres/planes is indeed attractive for a project like Dyna3.
 Discussed coordinates with Alex Kontorovich. He was suggesting "inversive coordinates" -- for a sphere, that's 1/coradius, 1/radius, center/radius (where coradius is radius of sphere inverted in the unit sphere.) The advantage is tangent to and perpendicular to are linear in these coordinates (in the sense that if one is known, the condition of being tangent to or perpendicular to that one are linear). Planes have 1/radius = 0, and in fact, you can take the coordinates to be (2s, 0, x, y, z) where s is the distance to the origin and x,y,z are the normal direction. (Note the normal direction is only determined up to a scalar multiple. So could always scale so that the first non-zero coordinate is 1, or if you like only allow x, y to vary and let z be determined as sqrt(1-x^2^-y^2^). ) Points can be given by (r^2,1,x,y,z) where x,y,z are the coordinates and r is the distance to the origin. Quadratic form that tells you if something is a sphere/plane, or in the boundary, or up in the hyperbolic plane above. There are some details, but not quite explicit for modeling R^3, at http://sites.math.rutgers.edu/~alexk/files/LetterToDuke.pdf -- all this emphasize need to be agnostic with respect to geometric model so that we can experiment. Not really sure exactly how this relates or not to conformal geometric algebra, and whether it can be combined with geometric algebra. As formulated, there are clear-ish reps for planes/spheres and for points, but not as clear for lines. Have to see how to compute distance and/or specify a given distance. To combine inversive coordinates and geometric algebra, maybe think dually; there should be a lift from a normal vector and distance from origin to the five-vector; bivectors would rep circles/lines; trivectors would rep point pairs/points. What is the signature of this algebra, i.e. how many coordinates square to +1, -1, or 0? But it doesn't seem worth it for three dimensions, because there is a natural representation of points, as follows:
 
 The signature of Q will be (1,4), and in fact Q(I1,I2) = 1/2(ab+ba) - E1\dot E2, where a is the "first" or "coradius" coordinate, "b" is the "second" or "radius" coordinate, and E is the Euclidean part (x,y,z). Then the inversive coordinates of a sphere with center (x,y,z) and radius r will be I = (1/\hat{r},1/r,x/r,y/r,z/r) where \hat{r} = r/(|E|^2 -r^2). These coordinates satisfy Q(I,I) = -1. For this to make sense, of course r > 0, but we get planes by letting the radius of a tangent sphere to the plane go to infinity, and we get I = (2s, 0, x0, y0, z0) where (x0,y0,z0) is the unit normal to the plane and s is the perpendicular distance from the plane to the origin. Still Q(I,I) = -1.
-Since r>0, we can't represent individual points this way. Instead we will use some coordinates J for which Q(J,J) = 0. In particular, if you take for the Euclidean point E = (u,v,w) the coordinates J = (`|E|`^2,1,u,v,w) then Q(J,J) = 0 and moreover it comes out that Q(I,J) = 0 
+Since r>0, we can't represent individual points this way. Instead we will use some coordinates J for which Q(J,J) = 0. In particular, if you take for the Euclidean point E = (u,v,w) the coordinates J = (`|E|`^2,1,u,v,w) then Q(J,J) = 0 and moreover it comes out that Q(I,J) = 0
 whenever E lies on the sphere or plane described by some I with Q(I,I) = -1.
 The condition that two spheres I1 and I2 are tangent seems to be that Q(I1,I2) = 1. So given a fixed sphere, the condition that another sphere be tangent to it is linear in the coordinates of that other sphere.
 This system does seem promising for encoding points, spheres, and planes, and doing basic computations with them. I guess I would just encode a circle as the intersection of the concentric sphere and the containing plane, and a line as either a pair of points or a pair of planes (modulo some equivalence relation, since I can't see any canonical choice of either two planes or two points). Or actually as described below, there is a more canonical choice.
@@ -62,4 +62,4 @@ In the engine's coordinate conventions, a sphere with radius $r > 0$ centered on
 $$I'_s = \left(\frac{P_x}{r}, \frac{P_y}{r}, \frac{P_z}{r}, \frac1{2r}, \frac{\|P\|^2 - r^2}{2r}\right),$$
 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.
\ No newline at end of file
+In the `engine` module, these formulas are encoded in the `sphere` and `point` functions.
diff --git a/notes/theory.md b/notes/theory.md
index 12190f1..0012488 100644
--- a/notes/theory.md
+++ b/notes/theory.md
@@ -24,7 +24,7 @@ His final mathematical advice was reasonably encouraging, however:
 "But still I would consider it all more or less doable. One should very precisely think about a doable scope.
 I think three things are essential for the math no matter what you exactly plan.
 
-1. Think projectively, 
+1. Think projectively.
    Use Projective Geometry, Homogeneous Coordinates (or to a certain extent Quaternions, and Clifford Algebras, which are more or less an elegant way to merge Complex numbers with projective concepts.)
 2. Consider ambient complex spaces.
    The true nature of the objects can only be understood if embedded into a complex ambient space.
@@ -37,10 +37,8 @@ I think three things are essential for the math no matter what you exactly plan.
 
 It would be nice to see how Jürgen handled some of these issues in a 2D system that he designed. Unfortunately, Cinderella was and remains closed-source; it was distributed for profit for some stretch of time. However, (a part of?) it was reimplemented in JavaScript as CindyJS, which is open source. I took a relatively quick look at that source code at one point, and these were my observations:
 
-CindyJS uses very concrete basic objects: 2D points are represented via projective geometry as a list of three floating-point numbers, and everything is done numerically.  There are no symbolic representations or exact algebraic numbers. (Not sure how a point on a circle or line is handled, that would take further investigation.) 
+CindyJS uses very concrete basic objects: 2D points are represented via projective geometry as a list of three floating-point numbers, and everything is done numerically.  There are no symbolic representations or exact algebraic numbers. (Not sure how a point on a circle or line is handled, that would take further investigation.)
 
 Lines are given by explicit coordinates as well (not sure of the internal details/exact coordinatization, or of how a "LineThrough" is represented).
 
 Was unclear to me how the complex parametrization for preserving continuity was handled in the code, even though Jürgen harps on complex ambient spaces; where are the complex numbers? Perhaps that part of Cinderella was never re-implemented?
-
-
diff --git a/src/index.html b/src/index.html
index 0a5fa60..d4b529f 100644
--- a/src/index.html
+++ b/src/index.html
@@ -7,5 +7,3 @@
   <body><script type="module" src="dyna3.js"></script>
   </body>
 </html>
-
-