dyna3/app-proto/src/engine.rs

use lazy_static::lazy_static;
use nalgebra::{Const, DMatrix, DVector, DVectorView, Dyn, SymmetricEigen};
use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */

// --- elements ---

#[cfg(feature = "dev")]
pub fn point(x: f64, y: f64, z: f64) -> DVector<f64> {
    DVector::from_column_slice(&[x, y, z, 0.5, 0.5*(x*x + y*y + z*z)])
}

// the sphere with the given center and radius, with inward-pointing normals
pub fn sphere(center_x: f64, center_y: f64, center_z: f64, radius: f64) -> DVector<f64> {
    let center_norm_sq = center_x * center_x + center_y * center_y + center_z * center_z;
    DVector::from_column_slice(&[
        center_x / radius,
        center_y / radius,
        center_z / radius,
        0.5 / radius,
        0.5 * (center_norm_sq / radius - radius)
    ])
}

// the sphere of curvature `curv` whose closest point to the origin has position
// `off * dir` and normal `dir`, where `dir` is a unit vector. setting the
// curvature to zero gives a plane
pub fn sphere_with_offset(dir_x: f64, dir_y: f64, dir_z: f64, off: f64, curv: f64) -> DVector<f64> {
    let norm_sp = 1.0 + off * curv;
    DVector::from_column_slice(&[
        norm_sp * dir_x,
        norm_sp * dir_y,
        norm_sp * dir_z,
        0.5 * curv,
        off * (1.0 + 0.5 * off * curv)
    ])
}

// --- partial matrices ---

struct MatrixEntry {
    index: (usize, usize),
    value: f64
}

pub struct PartialMatrix(Vec<MatrixEntry>);

impl PartialMatrix {
    pub fn new() -> PartialMatrix {
        PartialMatrix(Vec::<MatrixEntry>::new())
    }
    
    pub fn push_sym(&mut self, row: usize, col: usize, value: f64) {
        let PartialMatrix(entries) = self;
        entries.push(MatrixEntry { index: (row, col), value: value });
        if row != col {
            entries.push(MatrixEntry { index: (col, row), value: value });
        }
    }
    
    /* DEBUG */
    pub fn log_to_console(&self) {
        let PartialMatrix(entries) = self;
        for ent in entries {
            let ent_str = format!("  {} {} {}", ent.index.0, ent.index.1, ent.value);
            console::log_1(&JsValue::from(ent_str.as_str()));
        }
    }
    
    fn proj(&self, a: &DMatrix<f64>) -> DMatrix<f64> {
        let mut result = DMatrix::<f64>::zeros(a.nrows(), a.ncols());
        let PartialMatrix(entries) = self;
        for ent in entries {
            result[ent.index] = a[ent.index];
        }
        result
    }
    
    fn sub_proj(&self, rhs: &DMatrix<f64>) -> DMatrix<f64> {
        let mut result = DMatrix::<f64>::zeros(rhs.nrows(), rhs.ncols());
        let PartialMatrix(entries) = self;
        for ent in entries {
            result[ent.index] = ent.value - rhs[ent.index];
        }
        result
    }
}

// --- configuration subspaces ---

#[derive(Clone)]
pub struct ConfigSubspace {
    assembly_dim: usize,
    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_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>, proj_to_std: DMatrix<f64>, assembly_dim: usize) -> ConfigSubspace {
        // find a basis for the kernel. the basis is expressed in the projection
        // coordinates, and it's orthonormal with respect to the projection
        // 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_proj = DMatrix::from_columns(
            eig_pairs.filter_map(
                |(λ, v)| (λ.abs() < THRESHOLD).then_some(v)
            ).collect::<Vec<_>>().as_slice()
        );
        
        /* DEBUG */
        // print the eigenvalues
        #[cfg(all(target_family = "wasm", target_os = "unknown"))]
        console::log_1(&JsValue::from(
            format!("Eigenvalues used to find kernel:{}", eig.eigenvalues)
        ));
        
        // express the basis in the standard coordinates
        let basis_std = proj_to_std * &basis_proj;
        
        const ELEMENT_DIM: usize = 5;
        const UNIFORM_DIM: usize = 4;
        ConfigSubspace {
            assembly_dim: assembly_dim,
            basis_std: basis_std.column_iter().map(
                |v| Into::<DMatrix<f64>>::into(
                    v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
                )
            ).collect(),
            basis_proj: basis_proj.column_iter().map(
                |v| Into::<DMatrix<f64>>::into(
                    v.reshape_generic(Dyn(UNIFORM_DIM), Dyn(assembly_dim))
                )
            ).collect()
        }
    }
    
    pub fn dim(&self) -> usize {
        self.basis_std.len()
    }
    
    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
    // projection inner product
    pub fn proj(&self, v: &DVectorView<f64>, column_index: usize) -> DMatrix<f64> {
        if self.dim() == 0 {
            const ELEMENT_DIM: usize = 5;
            DMatrix::zeros(ELEMENT_DIM, self.assembly_dim)
        } else {
            self.basis_proj.iter().zip(self.basis_std.iter()).map(
                |(b_proj, b_std)| b_proj.column(column_index).dot(&v) * b_std
            ).sum()
        }
    }
}

// --- descent history ---

pub struct DescentHistory {
    pub config: Vec<DMatrix<f64>>,
    pub scaled_loss: Vec<f64>,
    pub neg_grad: Vec<DMatrix<f64>>,
    pub min_eigval: Vec<f64>,
    pub base_step: Vec<DMatrix<f64>>,
    pub backoff_steps: Vec<i32>
}

impl DescentHistory {
    fn new() -> DescentHistory {
        DescentHistory {
            config: Vec::<DMatrix<f64>>::new(),
            scaled_loss: Vec::<f64>::new(),
            neg_grad: Vec::<DMatrix<f64>>::new(),
            min_eigval: Vec::<f64>::new(),
            base_step: Vec::<DMatrix<f64>>::new(),
            backoff_steps: Vec::<i32>::new(),
        }
    }
}

// --- gram matrix realization ---

// the Lorentz form
lazy_static! {
    pub static ref Q: DMatrix<f64> = DMatrix::from_row_slice(5, 5, &[
        1.0, 0.0, 0.0,  0.0,  0.0,
        0.0, 1.0, 0.0,  0.0,  0.0,
        0.0, 0.0, 1.0,  0.0,  0.0,
        0.0, 0.0, 0.0,  0.0, -2.0,
        0.0, 0.0, 0.0, -2.0,  0.0
    ]);
}

struct SearchState {
    config: DMatrix<f64>,
    err_proj: DMatrix<f64>,
    loss: f64
}

impl SearchState {
    fn from_config(gram: &PartialMatrix, config: DMatrix<f64>) -> SearchState {
        let err_proj = gram.sub_proj(&(config.tr_mul(&*Q) * &config));
        let loss = err_proj.norm_squared();
        SearchState {
            config: config,
            err_proj: err_proj,
            loss: loss
        }
    }
}

fn basis_matrix(index: (usize, usize), nrows: usize, ncols: usize) -> DMatrix<f64> {
    let mut result = DMatrix::<f64>::zeros(nrows, ncols);
    result[index] = 1.0;
    result
}

// 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,
    state: &SearchState,
    base_step: &DMatrix<f64>,
    base_target_improvement: f64,
    min_efficiency: f64,
    backoff: f64,
    max_backoff_steps: i32
) -> Option<(SearchState, i32)> {
    let mut rate = 1.0;
    for backoff_steps in 0..max_backoff_steps {
        let trial_config = &state.config + rate * base_step;
        let trial_state = SearchState::from_config(gram, trial_config);
        let improvement = state.loss - trial_state.loss;
        if improvement >= min_efficiency * rate * base_target_improvement {
            return Some((trial_state, backoff_steps));
        }
        rate *= backoff;
    }
    None
}

// seek a matrix `config` for which `config' * Q * config` matches the partial
// matrix `gram`. use gradient descent starting from `guess`
pub fn realize_gram(
    gram: &PartialMatrix,
    guess: DMatrix<f64>,
    frozen: &[(usize, usize)],
    scaled_tol: f64,
    min_efficiency: f64,
    backoff: f64,
    reg_scale: f64,
    max_descent_steps: i32,
    max_backoff_steps: i32
) -> (DMatrix<f64>, ConfigSubspace, bool, DescentHistory) {
    // start the descent history
    let mut history = DescentHistory::new();
    
    // find the dimension of the search space
    let element_dim = guess.nrows();
    let assembly_dim = guess.ncols();
    let total_dim = element_dim * assembly_dim;
    
    // scale the tolerance
    let scale_adjustment = (gram.0.len() as f64).sqrt();
    let tol = scale_adjustment * scaled_tol;
    
    // convert the frozen indices to stacked format
    let frozen_stacked: Vec<usize> = frozen.into_iter().map(
        |index| index.1*element_dim + index.0
    ).collect();
    
    // use Newton's method with backtracking and gradient descent backup
    let mut state = SearchState::from_config(gram, guess);
    let mut hess = DMatrix::zeros(element_dim, assembly_dim);
    for _ in 0..max_descent_steps {
        // find the negative gradient of the loss function
        let neg_grad = 4.0 * &*Q * &state.config * &state.err_proj;
        let mut neg_grad_stacked = neg_grad.clone().reshape_generic(Dyn(total_dim), Const::<1>);
        history.neg_grad.push(neg_grad.clone());
        
        // find the negative Hessian of the loss function
        let mut hess_cols = Vec::<DVector<f64>>::with_capacity(total_dim);
        for col in 0..assembly_dim {
            for row in 0..element_dim {
                let index = (row, col);
                let basis_mat = basis_matrix(index, element_dim, assembly_dim);
                let neg_d_err =
                    basis_mat.tr_mul(&*Q) * &state.config
                    + state.config.tr_mul(&*Q) * &basis_mat;
                let neg_d_err_proj = gram.proj(&neg_d_err);
                let deriv_grad = 4.0 * &*Q * (
                    -&basis_mat * &state.err_proj
                    + &state.config * &neg_d_err_proj
                );
                hess_cols.push(deriv_grad.reshape_generic(Dyn(total_dim), Const::<1>));
            }
        }
        hess = DMatrix::from_columns(hess_cols.as_slice());
        
        // regularize the Hessian
        let min_eigval = hess.symmetric_eigenvalues().min();
        if min_eigval <= 0.0 {
            hess -= reg_scale * min_eigval * DMatrix::identity(total_dim, total_dim);
        }
        history.min_eigval.push(min_eigval);
        
        // project the negative gradient and negative Hessian onto the
        // orthogonal complement of the frozen subspace
        let zero_col = DVector::zeros(total_dim);
        let zero_row = zero_col.transpose();
        for &k in &frozen_stacked {
            neg_grad_stacked[k] = 0.0;
            hess.set_row(k, &zero_row);
            hess.set_column(k, &zero_col);
            hess[(k, k)] = 1.0;
        }
        
        // 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
        /*
          we need to either handle or eliminate the case where the minimum
          eigenvalue of the Hessian is zero, so the regularized Hessian is
          singular. right now, this causes the Cholesky decomposition to return
          `None`, leading to a panic when we unrap
        */
        let base_step_stacked = hess.clone().cholesky().unwrap().solve(&neg_grad_stacked);
        let base_step = base_step_stacked.reshape_generic(Dyn(element_dim), Dyn(assembly_dim));
        history.base_step.push(base_step.clone());
        
        // use backtracking line search to find a better configuration
        match seek_better_config(
            gram, &state, &base_step, neg_grad.dot(&base_step),
            min_efficiency, backoff, max_backoff_steps
        ) {
            Some((better_state, backoff_steps)) => {
                state = better_state;
                history.backoff_steps.push(backoff_steps);
            },
            None => return (state.config, ConfigSubspace::zero(assembly_dim), false, history)
        };
    }
    let success = state.loss < tol;
    let tangent = if success {
        // 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)
    };
    (state.config, tangent, success, history)
}

// --- tests ---

// this problem is from a sangaku by Irisawa Shintarō Hiroatsu. the article
// below includes a nice translation of the problem statement, which was
// recorded in Uchida Itsumi's book _Kokon sankan_ (_Mathematics, Past and
// Present_)
//
//   "Japan's 'Wasan' Mathematical Tradition", by Abe Haruki
//   https://www.nippon.com/en/japan-topics/c12801/
//
#[cfg(feature = "dev")]
pub mod irisawa {
    use std::{array, f64::consts::PI};
    
    use super::*;
    
    pub fn realize_irisawa_hexlet(scaled_tol: f64) -> (DMatrix<f64>, ConfigSubspace, bool, DescentHistory) {
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            for s in 0..9 {
                // each sphere is represented by a spacelike vector
                gram_to_be.push_sym(s, s, 1.0);
                
                // the circumscribing sphere is tangent to all of the other
                // spheres, with matching orientation
                if s > 0 {
                    gram_to_be.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 {
                        gram_to_be.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;
                    gram_to_be.push_sym(s, s_next, -1.0);
                }
            }
            gram_to_be
        };
        
        let guess = DMatrix::from_columns(
            [
                sphere(0.0, 0.0, 0.0, 15.0),
                sphere(0.0, 0.0, -9.0, 5.0),
                sphere(0.0, 0.0, 11.0, 3.0)
            ].into_iter().chain(
                (1..=6).map(
                    |k| {
                        let ang = (k as f64) * PI/3.0;
                        sphere(9.0 * ang.cos(), 9.0 * ang.sin(), 0.0, 2.5)
                    }
                )
            ).collect::<Vec<_>>().as_slice()
        );
        
        // the frozen entries fix the radii of the circumscribing sphere, the
        // "sun" and "moon" spheres, and one of the chain spheres
        let frozen: [(usize, usize); 4] = array::from_fn(|k| (3, k));
        
        realize_gram(
            &gram, guess, &frozen,
            scaled_tol, 0.5, 0.9, 1.1, 200, 110
        )
    }
}

#[cfg(test)]
mod tests {
    use nalgebra::Vector3;
    use std::{array, f64::consts::{FRAC_1_SQRT_2, PI}, iter};
    
    use super::{*, irisawa::realize_irisawa_hexlet};
    
    #[test]
    fn sub_proj_test() {
        let target = PartialMatrix(vec![
            MatrixEntry { index: (0, 0), value: 19.0 },
            MatrixEntry { index: (0, 2), value: 39.0 },
            MatrixEntry { index: (1, 1), value: 59.0 },
            MatrixEntry { index: (1, 2), value: 69.0 }
        ]);
        let attempt = DMatrix::<f64>::from_row_slice(2, 3, &[
            1.0, 2.0, 3.0,
            4.0, 5.0, 6.0
        ]);
        let expected_result = DMatrix::<f64>::from_row_slice(2, 3, &[
            18.0, 0.0, 36.0,
            0.0, 54.0, 63.0
        ]);
        assert_eq!(target.sub_proj(&attempt), expected_result);
    }
    
    #[test]
    fn zero_loss_test() {
        let gram = PartialMatrix({
            let mut entries = Vec::<MatrixEntry>::new();
            for j in 0..3 {
                for k in 0..3 {
                    entries.push(MatrixEntry {
                        index: (j, k),
                        value: if j == k { 1.0 } else { -1.0 }
                    });
                }
            }
            entries
        });
        let config = {
            let a: f64 = 0.75_f64.sqrt();
            DMatrix::from_columns(&[
                sphere(1.0, 0.0, 0.0, a),
                sphere(-0.5, a, 0.0, a),
                sphere(-0.5, -a, 0.0, a)
            ])
        };
        let state = SearchState::from_config(&gram, config);
        assert!(state.loss.abs() < f64::EPSILON);
    }
    
    #[test]
    fn irisawa_hexlet_test() {
        // solve Irisawa's problem
        const SCALED_TOL: f64 = 1.0e-12;
        let (config, _, _, _) = realize_irisawa_hexlet(SCALED_TOL);
        
        // check against Irisawa's solution
        let entry_tol = SCALED_TOL.sqrt();
        let solution_diams = [30.0, 10.0, 6.0, 5.0, 15.0, 10.0, 3.75, 2.5, 2.0 + 8.0/11.0];
        for (k, diam) in solution_diams.into_iter().enumerate() {
            assert!((config[(3, k)] - 1.0 / diam).abs() < entry_tol);
        }
    }
    
    #[test]
    fn tangent_test_three_spheres() {
        const SCALED_TOL: f64 = 1.0e-12;
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            for j in 0..3 {
                for k in j..3 {
                    gram_to_be.push_sym(j, k, if j == k { 1.0 } else { -1.0 });
                }
            }
            gram_to_be
        };
        let guess = DMatrix::from_columns(&[
            sphere(0.0, 0.0, 0.0, -2.0),
            sphere(0.0, 0.0, 1.0, 1.0),
            sphere(0.0, 0.0, -1.0, 1.0)
        ]);
        let frozen: [_; 5] = std::array::from_fn(|k| (k, 0));
        let (config, tangent, success, history) = realize_gram(
            &gram, guess.clone(), &frozen,
            SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
        );
        assert_eq!(config, guess);
        assert_eq!(success, true);
        assert_eq!(history.scaled_loss.len(), 1);
        
        // confirm that the tangent space has dimension five or less
        assert_eq!(tangent.basis_std.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
        const UNIFORM_DIM: usize = 4;
        let element_dim = guess.nrows();
        let assembly_dim = guess.ncols();
        let tangent_motions_unif = vec![
            basis_matrix((0, 1), UNIFORM_DIM, assembly_dim),
            basis_matrix((1, 1), UNIFORM_DIM, assembly_dim),
            basis_matrix((0, 2), UNIFORM_DIM, assembly_dim),
            basis_matrix((1, 2), UNIFORM_DIM, assembly_dim),
            DMatrix::<f64>::from_column_slice(UNIFORM_DIM, assembly_dim, &[
                0.0, 0.0,  0.0,  0.0,
                0.0, 0.0, -0.5, -0.5,
                0.0, 0.0, -0.5,  0.5
            ])
        ];
        let tangent_motions_std = vec![
            basis_matrix((0, 1), element_dim, assembly_dim),
            basis_matrix((1, 1), element_dim, assembly_dim),
            basis_matrix((0, 2), element_dim, assembly_dim),
            basis_matrix((1, 2), element_dim, assembly_dim),
            DMatrix::<f64>::from_column_slice(element_dim, assembly_dim, &[
                0.0, 0.0,  0.0,  0.00,  0.0,
                0.0, 0.0, -1.0, -0.25, -1.0,
                0.0, 0.0, -1.0,  0.25,  1.0
            ])
        ];
        let tol_sq = ((element_dim * assembly_dim) as f64) * SCALED_TOL * SCALED_TOL;
        for (motion_unif, motion_std) in tangent_motions_unif.into_iter().zip(tangent_motions_std) {
            let motion_proj: DMatrix<_> = motion_unif.column_iter().enumerate().map(
                |(k, v)| tangent.proj(&v, k)
            ).sum();
            assert!((motion_std - motion_proj).norm_squared() < tol_sq);
        }
    }
    
    fn translation_motion_unif(vel: &Vector3<f64>, assembly_dim: usize) -> Vec<DVector<f64>> {
        let mut elt_motion = DVector::zeros(4);
        elt_motion.fixed_rows_mut::<3>(0).copy_from(vel);
        iter::repeat(elt_motion).take(assembly_dim).collect()
    }
    
    fn rotation_motion_unif(ang_vel: &Vector3<f64>, points: Vec<DVectorView<f64>>) -> Vec<DVector<f64>> {
        points.into_iter().map(
            |pt| {
                let vel = ang_vel.cross(&pt.fixed_rows::<3>(0));
                let mut elt_motion = DVector::zeros(4);
                elt_motion.fixed_rows_mut::<3>(0).copy_from(&vel);
                elt_motion
            }
        ).collect()
    }
    
    #[test]
    fn tangent_test_kaleidocycle() {
        // set up a kaleidocycle, 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]
    fn frozen_entry_test() {
        let gram = {
            let mut gram_to_be = PartialMatrix::new();
            for j in 0..2 {
                for k in j..2 {
                    gram_to_be.push_sym(j, k, if (j, k) == (1, 1) { 1.0 } else { 0.0 });
                }
            }
            gram_to_be
        };
        let guess = DMatrix::from_columns(&[
            point(0.0, 0.0, 2.0),
            sphere(0.0, 0.0, 0.0, 1.0)
        ]);
        let frozen = [(3, 0), (3, 1)];
        println!();
        let (config, _, success, history) = realize_gram(
            &gram, guess.clone(), &frozen,
            1.0e-12, 0.5, 0.9, 1.1, 200, 110
        );
        assert_eq!(success, true);
        for base_step in history.base_step.into_iter() {
            for index in frozen {
                assert_eq!(base_step[index], 0.0);
            }
        }
        for index in frozen {
            assert_eq!(config[index], guess[index]);
        }
    }
}