From 303507eb08c973ee534ebbc8c97cad16382c75bb Mon Sep 17 00:00:00 2001
From: Aaron Fenyes <aaron.fenyes@fareycircles.ooo>
Date: Tue, 21 Jan 2025 12:05:20 -0800
Subject: [PATCH] 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 | 81 ++++++++++++++++++++++++++++++
 app-proto/run-examples             |  1 +
 2 files changed, 82 insertions(+)
 create mode 100644 app-proto/examples/kaleidocycle.rs

diff --git a/app-proto/examples/kaleidocycle.rs b/app-proto/examples/kaleidocycle.rs
new file mode 100644
index 0000000..effb226
--- /dev/null
+++ b/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);
+}
\ No newline at end of file
diff --git a/app-proto/run-examples b/app-proto/run-examples
index bc7e933..52173b0 100755
--- a/app-proto/run-examples
+++ b/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
\ No newline at end of file