Implement the uniform inner product for points

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

app-proto/src/engine.rs

@@ -132,6 +132,9 @@ impl ConfigSubspace {
         // orthonormalize the basis with respect to the projection inner product
         let basis_proj_orth = basis_proj.qr().q();
         let basis_std_orth = proj_to_std * &basis_proj_orth;
+        
+        // print the projection basis in projection coordinates
+        #[cfg(all(target_family = "wasm", target_os = "unknown"))]
         console::log_1(&JsValue::from(
             format!("Basis in projection coordinates:{}", basis_proj_orth)
         ));
@@ -236,19 +239,36 @@ fn basis_matrix(index: (usize, usize), nrows: usize, ncols: usize) -> DMatrix<f6
     result
 }
 
-// given a spacelike unit vector `v`, which represents a sphere, build the basis
+// given a normalized vector `v` representing an element, build a basis for the
-// for the configuration space given by the three unit translation motions of
+// element's linear configuration space consisting of:
-// the sphere, the unit shrinking motion of the sphere, and `v`
+// - 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;
     let curv = 2.0*v[3];
-    DMatrix::from_column_slice(ELEMENT_DIM, ELEMENT_DIM, &[
+    if v.dot(&(&*Q * v)) < 0.5 {
-             curv,       0.0,       0.0,       0.0,            v[0],
+        // `v` represents a point. the normalization condition says that the
-              0.0,      curv,       0.0,       0.0,            v[1],
+        // curvature component of `v` is 1/2
-              0.0,       0.0,      curv,       0.0,            v[2],
+        DMatrix::from_column_slice(ELEMENT_DIM, ELEMENT_DIM, &[
-        curv*v[0], curv*v[1], curv*v[2], curv*v[3], curv*v[4] + 1.0,
+            curv,  0.0,  0.0,  0.0, v[0],
-             v[0],      v[1],      v[2],      v[3],            v[4]
+             0.0, curv,  0.0,  0.0, v[1],
-    ])
+             0.0,  0.0, curv,  0.0, v[2],
+            v[0], v[1], v[2], v[3], v[4],
+             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, ELEMENT_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,
+                 v[0],      v[1],      v[2],      v[3],            v[4]
+        ])
+    }
 }
 
 // use backtracking line search to find a better configuration