app-proto/examples/kaleidocycle.rs

@@ -0,0 +1,72 @@
+use nalgebra::{DMatrix, DVector};
+use std::{array, f64::consts::PI};
+
+use dyna3::engine::{Q, point, realize_gram, PartialMatrix};
+
+fn main() {
+    // set up a kaleidocycle, made of points with fixed distances between them,
+    // and find its tangent space
+    const N_POINTS: usize = 12;
+    let gram = {
+        let mut gram_to_be = PartialMatrix::new();
+        for block in (0..N_POINTS).step_by(2) {
+            let block_next = (block + 2) % N_POINTS;
+            for j in 0..2 {
+                // diagonal and hinge edges
+                for k in j..2 {
+                    gram_to_be.push_sym(block + j, block + k, if j == k { 0.0 } else { -0.5 });
+                }
+                
+                // non-hinge edges
+                for k in 0..2 {
+                    gram_to_be.push_sym(block + j, block_next + k, -0.625);
+                }
+            }
+        }
+        gram_to_be
+    };
+    let guess = {
+        const N_HINGES: usize = 6;
+        let guess_elts = (0..N_HINGES).step_by(2).flat_map(
+            |n| {
+                let ang_hor = (n as f64) * PI/3.0;
+                let ang_vert = ((n + 1) as f64) * PI/3.0;
+                let x_vert = ang_vert.cos();
+                let y_vert = ang_vert.sin();
+                [
+                    point(0.0, 0.0, 0.0),
+                    point(ang_hor.cos(), ang_hor.sin(), 0.0),
+                    point(x_vert, y_vert, -0.5),
+                    point(x_vert, y_vert, 0.5)
+                ]
+            }
+        ).collect::<Vec<_>>();
+        DMatrix::from_columns(&guess_elts)
+    };
+    let frozen: [_; N_POINTS] = array::from_fn(|k| (3, k));
+    let (config, tangent, success, history) = realize_gram(
+        &gram, guess, &frozen,
+        1.0e-12, 0.5, 0.9, 1.1, 200, 110
+    );
+    print!("Completed Gram matrix:{}", config.tr_mul(&*Q) * &config);
+    print!("Configuration:{}", config);
+    if success {
+        println!("Target accuracy achieved!");
+    } else {
+        println!("Failed to reach target accuracy");
+    }
+    println!("Steps: {}", history.scaled_loss.len() - 1);
+    println!("Loss: {}\n", history.scaled_loss.last().unwrap());
+    
+    // find the kaleidocycle's twist motion
+    let up = DVector::from_column_slice(&[0.0, 0.0, 1.0, 0.0]);
+    let down = -&up;
+    let twist_motion: DMatrix<_> = (0..N_POINTS).step_by(4).flat_map(
+        |n| [
+            tangent.proj(&up.as_view(), n),
+            tangent.proj(&down.as_view(), n+1)
+        ]
+    ).sum();
+    let normalization = 5.0 / twist_motion[(2, 0)];
+    print!("Twist motion:{}", normalization * twist_motion);
+}

app-proto/run-examples

@@ -9,3 +9,4 @@
 cargo run --example irisawa-hexlet
 cargo run --example three-spheres
 cargo run --example point-on-sphere
+cargo run --example kaleidocycle

app-proto/src/assembly.rs

@@ -5,7 +5,7 @@ use std::{collections::BTreeSet, sync::atomic::{AtomicU64, Ordering}};
 use sycamore::prelude::*;
 use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
 
-use crate::engine::{realize_gram, ConfigSubspace, PartialMatrix, Q};
+use crate::engine::{realize_gram, local_unif_to_std, ConfigSubspace, PartialMatrix, Q};
 
 // the types of the keys we use to access an assembly's elements and constraints
 pub type ElementKey = usize;
@@ -120,6 +120,7 @@ pub struct Constraint {
     pub active: Signal<bool>
 }
 
+// the velocity is expressed in uniform coordinates
 pub struct ElementMotion<'a> {
     pub key: ElementKey,
     pub velocity: DVectorView<'a, f64>
@@ -359,7 +360,14 @@ impl Assembly {
                 // this element didn't have a column index when we started, so
                 // by invariant (2), it's unconstrained
                 let mut target_column = motion_proj.column_mut(column_index);
-                target_column += elt_motion.velocity;
+                let unif_to_std = self.elements.with_untracked(
+                    |elts| {
+                        elts[elt_motion.key].representation.with_untracked(
+                            |rep| local_unif_to_std(rep.as_view())
+                        )
+                    }
+                );
+                target_column += unif_to_std * elt_motion.velocity;
             }
         }
         

app-proto/src/display.rs

@@ -130,6 +130,8 @@ pub fn Display() -> View {
     let translate_pos_y = create_signal(0.0);
     let translate_neg_z = create_signal(0.0);
     let translate_pos_z = create_signal(0.0);
+    let shrink_neg = create_signal(0.0);
+    let shrink_pos = create_signal(0.0);
     
     // change listener
     let scene_changed = create_signal(true);
@@ -164,6 +166,7 @@ pub fn Display() -> View {
         
         // manipulation
         const TRANSLATION_SPEED: f64 = 0.15; // in length units per second
+        const SHRINKING_SPEED: f64 = 0.15; // in length units per second
         
         // display parameters
         const OPACITY: f32 = 0.5; /* SCAFFOLDING */
@@ -292,6 +295,8 @@ pub fn Display() -> View {
             let translate_pos_y_val = translate_pos_y.get();
             let translate_neg_z_val = translate_neg_z.get();
             let translate_pos_z_val = translate_pos_z.get();
+            let shrink_neg_val = shrink_neg.get();
+            let shrink_pos_val = shrink_pos.get();
             
             // update the assembly's orientation
             let ang_vel = {
@@ -323,24 +328,27 @@ pub fn Display() -> View {
                 let sel = state.selection.with(
                     |sel| *sel.into_iter().next().unwrap()
                 );
-                let rep = state.assembly.elements.with_untracked(
-                    |elts| elts[sel].representation.get_clone_untracked()
-                );
                 let translate_x = translate_pos_x_val - translate_neg_x_val;
                 let translate_y = translate_pos_y_val - translate_neg_y_val;
                 let translate_z = translate_pos_z_val - translate_neg_z_val;
-                if translate_x != 0.0 || translate_y != 0.0 || translate_z != 0.0 {
+                let shrink = shrink_pos_val - shrink_neg_val;
-                    let vel_field = {
+                let translating =
-                        let u = Vector3::new(translate_x, translate_y, translate_z).normalize();
+                    translate_x != 0.0
-                        DMatrix::from_column_slice(5, 5, &[
+                    || translate_y != 0.0
-                                 0.0,      0.0,      0.0, 0.0, u[0],
+                    || translate_z != 0.0;
-                                 0.0,      0.0,      0.0, 0.0, u[1],
+                if translating || shrink != 0.0 {
-                                 0.0,      0.0,      0.0, 0.0, u[2],
+                    let elt_motion = {
-                            2.0*u[0], 2.0*u[1], 2.0*u[2], 0.0,  0.0,
+                        let u = if translating {
-                                 0.0,      0.0,      0.0, 0.0,  0.0
+                            TRANSLATION_SPEED * Vector3::new(
-                        ])
+                                translate_x, translate_y, translate_z
+                            ).normalize()
+                        } else {
+                            Vector3::zeros()
+                        };
+                        time_step * DVector::from_column_slice(
+                            &[u[0], u[1], u[2], SHRINKING_SPEED * shrink]
+                        )
                     };
-                    let elt_motion: DVector<f64> = time_step * TRANSLATION_SPEED * vel_field * rep;
                     assembly_for_raf.deform(
                         vec![
                             ElementMotion {
@@ -501,6 +509,8 @@ pub fn Display() -> View {
             "s" | "S" if shift => translate_pos_z.set(value),
             "w" | "W" => translate_pos_y.set(value),
             "s" | "S" => translate_neg_y.set(value),
+            "]" | "}" => shrink_neg.set(value),
+            "[" | "{" => shrink_pos.set(value),
             _ => manipulating = false
         };
         if manipulating {

app-proto/src/engine.rs

@@ -90,32 +90,34 @@ impl PartialMatrix {
 #[derive(Clone)]
 pub struct ConfigSubspace {
     assembly_dim: usize,
-    basis: Vec<DMatrix<f64>>
+    basis_std: Vec<DMatrix<f64>>,
+    basis_proj: Vec<DMatrix<f64>>
 }
 
 impl ConfigSubspace {
     pub fn zero(assembly_dim: usize) -> ConfigSubspace {
         ConfigSubspace {
             assembly_dim: assembly_dim,
-            basis: Vec::new()
+            basis_proj: Vec::new(),
+            basis_std: Vec::new()
         }
     }
     
     // approximate the kernel of a symmetric endomorphism of the configuration
     // space for `assembly_dim` elements. we consider an eigenvector to be part
     // of the kernel if its eigenvalue is smaller than the constant `THRESHOLD`
-    fn symmetric_kernel(a: DMatrix<f64>, assembly_dim: usize) -> ConfigSubspace {
+    fn symmetric_kernel(a: DMatrix<f64>, proj_to_std: DMatrix<f64>, assembly_dim: usize) -> ConfigSubspace {
-        const ELEMENT_DIM: usize = 5;
+        // find a basis for the kernel. the basis is expressed in the projection
-        const THRESHOLD: f64 = 1.0e-4;
+        // coordinates, and it's orthonormal with respect to the projection
-        let eig = SymmetricEigen::new(a);
+        // inner product
+        const THRESHOLD: f64 = 0.1;
+        let eig = SymmetricEigen::new(proj_to_std.tr_mul(&a) * &proj_to_std);
         let eig_vecs = eig.eigenvectors.column_iter();
         let eig_pairs = eig.eigenvalues.iter().zip(eig_vecs);
-        let basis = eig_pairs.filter_map(
+        let basis_proj = DMatrix::from_columns(
-            |(λ, v)| (λ.abs() < THRESHOLD).then_some(
+            eig_pairs.filter_map(
-                Into::<DMatrix<f64>>::into(
+                |(λ, v)| (λ.abs() < THRESHOLD).then_some(v)
-                    v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
+            ).collect::<Vec<_>>().as_slice()
-                )
-            )
         );
         
         /* DEBUG */
@@ -125,30 +127,45 @@ impl ConfigSubspace {
             format!("Eigenvalues used to find kernel:{}", eig.eigenvalues)
         ));
         
+        // express the basis in the standard coordinates
+        let basis_std = proj_to_std * &basis_proj;
+        
+        const ELEMENT_DIM: usize = 5;
+        const UNIFORM_DIM: usize = 4;
         ConfigSubspace {
             assembly_dim: assembly_dim,
-            basis: basis.collect()
+            basis_std: basis_std.column_iter().map(
+                |v| Into::<DMatrix<f64>>::into(
+                    v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
+                )
+            ).collect(),
+            basis_proj: basis_proj.column_iter().map(
+                |v| Into::<DMatrix<f64>>::into(
+                    v.reshape_generic(Dyn(UNIFORM_DIM), Dyn(assembly_dim))
+                )
+            ).collect()
         }
     }
     
     pub fn dim(&self) -> usize {
-        self.basis.len()
+        self.basis_std.len()
     }
     
     pub fn assembly_dim(&self) -> usize {
         self.assembly_dim
     }
     
-    // find the projection onto this subspace, with respect to the Euclidean
+    // find the projection onto this subspace of the motion where the element
-    // inner product, of the motion where the element with the given column
+    // with the given column index has velocity `v`. the velocity is given in
-    // index has velocity `v`
+    // projection coordinates, and the projection is done with respect to the
+    // projection inner product
     pub fn proj(&self, v: &DVectorView<f64>, column_index: usize) -> DMatrix<f64> {
         if self.dim() == 0 {
             const ELEMENT_DIM: usize = 5;
             DMatrix::zeros(ELEMENT_DIM, self.assembly_dim)
         } else {
-            self.basis.iter().map(
+            self.basis_proj.iter().zip(self.basis_std.iter()).map(
-                |b| b.column(column_index).dot(&v) * b
+                |(b_proj, b_std)| b_proj.column(column_index).dot(&v) * b_std
             ).sum()
         }
     }
@@ -215,6 +232,37 @@ fn basis_matrix(index: (usize, usize), nrows: usize, ncols: usize) -> DMatrix<f6
     result
 }
 
+// given a normalized vector `v` representing an element, build a basis for the
+// element's linear configuration space consisting of:
+// - the unit translation motions of the element
+// - the unit shrinking motion of the element, if it's a sphere
+// - one or two vectors whose coefficients vanish on the tangent space of the
+//   normalization variety
+pub fn local_unif_to_std(v: DVectorView<f64>) -> DMatrix<f64> {
+    const ELEMENT_DIM: usize = 5;
+    const UNIFORM_DIM: usize = 4;
+    let curv = 2.0*v[3];
+    if v.dot(&(&*Q * v)) < 0.5 {
+        // `v` represents a point. the normalization condition says that the
+        // curvature component of `v` is 1/2
+        DMatrix::from_column_slice(ELEMENT_DIM, UNIFORM_DIM, &[
+            curv,  0.0,  0.0,  0.0, v[0],
+             0.0, curv,  0.0,  0.0, v[1],
+             0.0,  0.0, curv,  0.0, v[2],
+             0.0,  0.0,  0.0,  0.0,  1.0
+        ])
+    } else {
+        // `v` represents a sphere. the normalization condition says that the
+        // Lorentz product of `v` with itself is 1
+        DMatrix::from_column_slice(ELEMENT_DIM, UNIFORM_DIM, &[
+                 curv,       0.0,       0.0,       0.0,            v[0],
+                  0.0,      curv,       0.0,       0.0,            v[1],
+                  0.0,       0.0,      curv,       0.0,            v[2],
+            curv*v[0], curv*v[1], curv*v[2], curv*v[3], curv*v[4] + 1.0
+        ])
+    }
+}
+
 // use backtracking line search to find a better configuration
 fn seek_better_config(
     gram: &PartialMatrix,
@@ -344,7 +392,19 @@ pub fn realize_gram(
     }
     let success = state.loss < tol;
     let tangent = if success {
-        ConfigSubspace::symmetric_kernel(hess, assembly_dim)
+        // express the uniform basis in the standard basis
+        const UNIFORM_DIM: usize = 4;
+        let total_dim_unif = UNIFORM_DIM * assembly_dim;
+        let mut unif_to_std = DMatrix::<f64>::zeros(total_dim, total_dim_unif);
+        for n in 0..assembly_dim {
+            let block_start = (element_dim * n, UNIFORM_DIM * n);
+            unif_to_std
+                .view_mut(block_start, (element_dim, UNIFORM_DIM))
+                .copy_from(&local_unif_to_std(state.config.column(n)));
+        }
+        
+        // find the kernel of the Hessian. give it the uniform inner product
+        ConfigSubspace::symmetric_kernel(hess, unif_to_std, assembly_dim)
     } else {
         ConfigSubspace::zero(assembly_dim)
     };
@@ -424,6 +484,9 @@ pub mod irisawa {
 
 #[cfg(test)]
 mod tests {
+    use nalgebra::Vector3;
+    use std::{array, f64::consts::{FRAC_1_SQRT_2, PI}, iter};
+    
     use super::{*, irisawa::realize_irisawa_hexlet};
     
     #[test]
@@ -486,10 +549,8 @@ mod tests {
     }
     
     #[test]
-    fn tangent_test() {
+    fn tangent_test_three_spheres() {
         const SCALED_TOL: f64 = 1.0e-12;
-        const ELEMENT_DIM: usize = 5;
-        const ASSEMBLY_DIM: usize = 3;
         let gram = {
             let mut gram_to_be = PartialMatrix::new();
             for j in 0..3 {
@@ -514,32 +575,258 @@ mod tests {
         assert_eq!(history.scaled_loss.len(), 1);
         
         // confirm that the tangent space has dimension five or less
-        let ConfigSubspace(ref tangent_basis) = tangent;
+        assert_eq!(tangent.basis_std.len(), 5);
-        assert_eq!(tangent_basis.len(), 5);
         
         // confirm that the tangent space contains all the motions we expect it
         // to. since we've already bounded the dimension of the tangent space,
         // this confirms that the tangent space is what we expect it to be
-        let tangent_motions = vec![
+        const UNIFORM_DIM: usize = 4;
-            basis_matrix((0, 1), ELEMENT_DIM, ASSEMBLY_DIM),
+        let element_dim = guess.nrows();
-            basis_matrix((1, 1), ELEMENT_DIM, ASSEMBLY_DIM),
+        let assembly_dim = guess.ncols();
-            basis_matrix((0, 2), ELEMENT_DIM, ASSEMBLY_DIM),
+        let tangent_motions_unif = vec![
-            basis_matrix((1, 2), ELEMENT_DIM, ASSEMBLY_DIM),
+            basis_matrix((0, 1), UNIFORM_DIM, assembly_dim),
-            DMatrix::<f64>::from_column_slice(ELEMENT_DIM, 3, &[
+            basis_matrix((1, 1), UNIFORM_DIM, assembly_dim),
-                0.0, 0.0, 0.0, 0.0, 0.0,
+            basis_matrix((0, 2), UNIFORM_DIM, assembly_dim),
-                0.0, 0.0, -1.0, -0.25, -1.0,
+            basis_matrix((1, 2), UNIFORM_DIM, assembly_dim),
-                0.0, 0.0, -1.0, 0.25, 1.0
+            DMatrix::<f64>::from_column_slice(UNIFORM_DIM, assembly_dim, &[
+                0.0, 0.0,  0.0,  0.0,
+                0.0, 0.0, -0.5, -0.5,
+                0.0, 0.0, -0.5,  0.5
             ])
         ];
-        let tol_sq = ((ELEMENT_DIM * ASSEMBLY_DIM) as f64) * SCALED_TOL * SCALED_TOL;
+        let tangent_motions_std = vec![
-        for motion in tangent_motions {
+            basis_matrix((0, 1), element_dim, assembly_dim),
-            let motion_proj: DMatrix<_> = motion.column_iter().enumerate().map(
+            basis_matrix((1, 1), element_dim, assembly_dim),
+            basis_matrix((0, 2), element_dim, assembly_dim),
+            basis_matrix((1, 2), element_dim, assembly_dim),
+            DMatrix::<f64>::from_column_slice(element_dim, assembly_dim, &[
+                0.0, 0.0,  0.0,  0.00,  0.0,
+                0.0, 0.0, -1.0, -0.25, -1.0,
+                0.0, 0.0, -1.0,  0.25,  1.0
+            ])
+        ];
+        let tol_sq = ((element_dim * assembly_dim) as f64) * SCALED_TOL * SCALED_TOL;
+        for (motion_unif, motion_std) in tangent_motions_unif.into_iter().zip(tangent_motions_std) {
+            let motion_proj: DMatrix<_> = motion_unif.column_iter().enumerate().map(
                 |(k, v)| tangent.proj(&v, k)
             ).sum();
-            assert!((motion - motion_proj).norm_squared() < tol_sq);
+            assert!((motion_std - motion_proj).norm_squared() < tol_sq);
         }
     }
     
+    fn translation_motion_unif(vel: &Vector3<f64>, assembly_dim: usize) -> Vec<DVector<f64>> {
+        let mut elt_motion = DVector::zeros(4);
+        elt_motion.fixed_rows_mut::<3>(0).copy_from(vel);
+        iter::repeat(elt_motion).take(assembly_dim).collect()
+    }
+    
+    fn rotation_motion_unif(ang_vel: &Vector3<f64>, points: Vec<DVectorView<f64>>) -> Vec<DVector<f64>> {
+        points.into_iter().map(
+            |pt| {
+                let vel = ang_vel.cross(&pt.fixed_rows::<3>(0));
+                let mut elt_motion = DVector::zeros(4);
+                elt_motion.fixed_rows_mut::<3>(0).copy_from(&vel);
+                elt_motion
+            }
+        ).collect()
+    }
+    
+    #[test]
+    fn tangent_test_kaleidocycle() {
+        // set up a kaleidocycle, made of points with fixed distances between
+        // them, and find its tangent space
+        const N_POINTS: usize = 12;
+        const N_HINGES: usize = 6;
+        const SCALED_TOL: f64 = 1.0e-12;
+        let gram = {
+            let mut gram_to_be = PartialMatrix::new();
+            for block in (0..N_POINTS).step_by(2) {
+                let block_next = (block + 2) % N_POINTS;
+                for j in 0..2 {
+                    // diagonal and hinge edges
+                    for k in j..2 {
+                        gram_to_be.push_sym(block + j, block + k, if j == k { 0.0 } else { -0.5 });
+                    }
+                    
+                    // non-hinge edges
+                    for k in 0..2 {
+                        gram_to_be.push_sym(block + j, block_next + k, -0.625);
+                    }
+                }
+            }
+            gram_to_be
+        };
+        let guess = {
+            let guess_elts = (0..N_HINGES).step_by(2).flat_map(
+                |n| {
+                    let ang_hor = (n as f64) * PI/3.0;
+                    let ang_vert = ((n + 1) as f64) * PI/3.0;
+                    let x_vert = ang_vert.cos();
+                    let y_vert = ang_vert.sin();
+                    [
+                        point(0.0, 0.0, 0.0),
+                        point(ang_hor.cos(), ang_hor.sin(), 0.0),
+                        point(x_vert, y_vert, -0.5),
+                        point(x_vert, y_vert, 0.5)
+                    ]
+                }
+            ).collect::<Vec<_>>();
+            DMatrix::from_columns(&guess_elts)
+        };
+        let frozen: [_; N_POINTS] = array::from_fn(|k| (3, k));
+        let (config, tangent, success, history) = realize_gram(
+            &gram, guess.clone(), &frozen,
+            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
+        );
+        assert_eq!(config, guess);
+        assert_eq!(success, true);
+        assert_eq!(history.scaled_loss.len(), 1);
+        
+        // list some motions that should form a basis for the tangent space of
+        // the solution variety
+        let element_dim = guess.nrows();
+        let assembly_dim = guess.ncols();
+        let tangent_motions_unif = vec![
+            // the translations along the coordinate axes
+            translation_motion_unif(&Vector3::new(1.0, 0.0, 0.0), assembly_dim),
+            translation_motion_unif(&Vector3::new(0.0, 1.0, 0.0), assembly_dim),
+            translation_motion_unif(&Vector3::new(0.0, 0.0, 1.0), assembly_dim),
+            
+            // the rotations about the coordinate axes
+            rotation_motion_unif(&Vector3::new(1.0, 0.0, 0.0), guess.column_iter().collect()),
+            rotation_motion_unif(&Vector3::new(0.0, 1.0, 0.0), guess.column_iter().collect()),
+            rotation_motion_unif(&Vector3::new(0.0, 0.0, 1.0), guess.column_iter().collect()),
+            
+            // the twist motion. more precisely: a motion that keeps the center
+            // of mass stationary and preserves the distances between the
+            // vertices to first order. this has to be the twist as long as:
+            //   - twisting is the kaleidocycle's only internal degree of
+            //     freedom
+            //   - every first-order motion of the kaleidocycle comes from an
+            //     actual motion
+            (0..N_HINGES).step_by(2).flat_map(
+                |n| {
+                    let ang_vert = ((n + 1) as f64) * PI/3.0;
+                    let vel_vert_x = 4.0 * ang_vert.cos();
+                    let vel_vert_y = 4.0 * ang_vert.sin();
+                    [
+                        DVector::from_column_slice(&[0.0, 0.0, 5.0, 0.0]),
+                        DVector::from_column_slice(&[0.0, 0.0, 1.0, 0.0]),
+                        DVector::from_column_slice(&[-vel_vert_x, -vel_vert_y, -3.0, 0.0]),
+                        DVector::from_column_slice(&[vel_vert_x, vel_vert_y, -3.0, 0.0])
+                    ]
+                }
+            ).collect::<Vec<_>>()
+        ];
+        let tangent_motions_std = tangent_motions_unif.iter().map(
+            |motion| DMatrix::from_columns(
+                &guess.column_iter().zip(motion).map(
+                    |(v, elt_motion)| local_unif_to_std(v) * elt_motion
+                ).collect::<Vec<_>>()
+            )
+        ).collect::<Vec<_>>();
+        
+        // confirm that the dimension of the tangent space is no greater than
+        // expected
+        assert_eq!(tangent.basis_std.len(), tangent_motions_unif.len());
+        
+        // confirm that the tangent space contains all the motions we expect it
+        // to. since we've already bounded the dimension of the tangent space,
+        // this confirms that the tangent space is what we expect it to be
+        let tol_sq = ((element_dim * assembly_dim) as f64) * SCALED_TOL * SCALED_TOL;
+        for (motion_unif, motion_std) in tangent_motions_unif.into_iter().zip(tangent_motions_std) {
+            let motion_proj: DMatrix<_> = motion_unif.into_iter().enumerate().map(
+                |(k, v)| tangent.proj(&v.as_view(), k)
+            ).sum();
+            assert!((motion_std - motion_proj).norm_squared() < tol_sq);
+        }
+    }
+    
+    fn translation(dis: Vector3<f64>) -> DMatrix<f64> {
+        const ELEMENT_DIM: usize = 5;
+        DMatrix::from_column_slice(ELEMENT_DIM, ELEMENT_DIM, &[
+                   1.0,        0.0,        0.0, 0.0,             dis[0],
+                   0.0,        1.0,        0.0, 0.0,             dis[1],
+                   0.0,        0.0,        1.0, 0.0,             dis[2],
+            2.0*dis[0], 2.0*dis[1], 2.0*dis[2], 1.0, dis.norm_squared(),
+                   0.0,        0.0,        0.0, 0.0,                1.0
+        ])
+    }
+    
+    // confirm that projection onto a configuration subspace is equivariant with
+    // respect to Euclidean motions
+    #[test]
+    fn proj_equivar_test() {
+        // find a pair of spheres that meet at 120°
+        const SCALED_TOL: f64 = 1.0e-12;
+        let gram = {
+            let mut gram_to_be = PartialMatrix::new();
+            gram_to_be.push_sym(0, 0, 1.0);
+            gram_to_be.push_sym(1, 1, 1.0);
+            gram_to_be.push_sym(0, 1, 0.5);
+            gram_to_be
+        };
+        let guess_orig = DMatrix::from_columns(&[
+            sphere(0.0, 0.0, 0.5, 1.0),
+            sphere(0.0, 0.0, -0.5, 1.0)
+        ]);
+        let (config_orig, tangent_orig, success_orig, history_orig) = realize_gram(
+            &gram, guess_orig.clone(), &[],
+            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
+        );
+        assert_eq!(config_orig, guess_orig);
+        assert_eq!(success_orig, true);
+        assert_eq!(history_orig.scaled_loss.len(), 1);
+        
+        // find another pair of spheres that meet at 120°. we'll think of this
+        // solution as a transformed version of the original one
+        let guess_tfm = {
+            let a = 0.5 * FRAC_1_SQRT_2;
+            DMatrix::from_columns(&[
+                sphere(a, 0.0, 7.0 + a, 1.0),
+                sphere(-a, 0.0, 7.0 - a, 1.0)
+            ])
+        };
+        let (config_tfm, tangent_tfm, success_tfm, history_tfm) = realize_gram(
+            &gram, guess_tfm.clone(), &[],
+            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
+        );
+        assert_eq!(config_tfm, guess_tfm);
+        assert_eq!(success_tfm, true);
+        assert_eq!(history_tfm.scaled_loss.len(), 1);
+        
+        // project a nudge to the tangent space of the solution variety at the
+        // original solution
+        let motion_orig = DVector::from_column_slice(&[0.0, 0.0, 1.0, 0.0]);
+        let motion_orig_proj = tangent_orig.proj(&motion_orig.as_view(), 0);
+        
+        // project the equivalent nudge to the tangent space of the solution
+        // variety at the transformed solution
+        let motion_tfm = DVector::from_column_slice(&[FRAC_1_SQRT_2, 0.0, FRAC_1_SQRT_2, 0.0]);
+        let motion_tfm_proj = tangent_tfm.proj(&motion_tfm.as_view(), 0);
+        
+        // take the transformation that sends the original solution to the
+        // transformed solution and apply it to the motion that the original
+        // solution makes in response to the nudge
+        const ELEMENT_DIM: usize = 5;
+        let rot = DMatrix::from_column_slice(ELEMENT_DIM, ELEMENT_DIM, &[
+            FRAC_1_SQRT_2, 0.0, -FRAC_1_SQRT_2, 0.0, 0.0,
+                      0.0, 1.0,            0.0, 0.0, 0.0,
+            FRAC_1_SQRT_2, 0.0,  FRAC_1_SQRT_2, 0.0, 0.0,
+                      0.0, 0.0,            0.0, 1.0, 0.0,
+                      0.0, 0.0,            0.0, 0.0, 1.0
+        ]);
+        let transl = translation(Vector3::new(0.0, 0.0, 7.0));
+        let motion_proj_tfm = transl * rot * motion_orig_proj;
+        
+        // confirm that the projection of the nudge is equivariant. we loosen
+        // the comparison tolerance because the transformation seems to
+        // introduce some numerical error
+        const SCALED_TOL_TFM: f64 = 1.0e-9;
+        let tol_sq = ((guess_orig.nrows() * guess_orig.ncols()) as f64) * SCALED_TOL_TFM * SCALED_TOL_TFM;
+        assert!((motion_proj_tfm - motion_tfm_proj).norm_squared() < tol_sq);
+    }
+    
     // at the frozen indices, the optimization steps should have exact zeros,
     // and the realized configuration should match the initial guess
     #[test]