Add kaleidocycle example

The engine successfully captures the kaleidocycle's twisting motion.
However, nudging doesn't work quite the way we want. This is probably to
be expected, since nudges don't commute with Euclidean motions in our
current implementation.

app-proto/examples/kaleidocycle.rs

@@ -0,0 +1,81 @@
+use nalgebra::DMatrix;
+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 twist_motion: DMatrix<_> = (0..N_POINTS).step_by(4).flat_map(
+        |n| {
+            let up_field = {
+                DMatrix::from_column_slice(5, 5, &[
+                    0.0, 0.0, 0.0, 0.0, 0.0,
+                    0.0, 0.0, 0.0, 0.0, 0.0,
+                    0.0, 0.0, 0.0, 0.0, 1.0,
+                    0.0, 0.0, 2.0, 0.0, 0.0,
+                    0.0, 0.0, 0.0, 0.0, 0.0
+                ])
+            };
+            [
+                tangent.proj(&(&up_field * config.column(n)).as_view(), n),
+                tangent.proj(&(-&up_field * config.column(n+1)).as_view(), n+1)
+            ]
+        }
+    ).sum();
+    let normalization = 5.0 / twist_motion[(2, 0)];
+    print!("Twist motion:{}", normalization * twist_motion);
+}

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