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Switch to Euclidean-invariant projection onto tangent space of solution variety #34

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Vectornaut wants to merge 4 commits from uniform-projection into main
pull from: uniform-projection
merge into: glen:main
Conversation 1 Commits 4 Files Changed 5 +97 -33
3 changed files with 97 additions and 33 deletions
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Showing only changes of commit 03da831c9a - Show all commits

12
app-proto/src/assembly.rs
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@ -5,7 +5,7 @@ use std::{collections::BTreeSet, sync::atomic::{AtomicU64, Ordering}};
use sycamore::prelude::*; use sycamore::prelude::*;
use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */ 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 // the types of the keys we use to access an assembly's elements and constraints
pub type ElementKey = usize; pub type ElementKey = usize;
@ -120,6 +120,7 @@ pub struct Constraint {
pub active: Signal<bool> pub active: Signal<bool>
} }
// the velocity is expressed in uniform coordinates
pub struct ElementMotion<'a> { pub struct ElementMotion<'a> {
pub key: ElementKey, pub key: ElementKey,
pub velocity: DVectorView<'a, f64> pub velocity: DVectorView<'a, f64>
@ -359,7 +360,14 @@ impl Assembly {
// this element didn't have a column index when we started, so // this element didn't have a column index when we started, so
// by invariant (2), it's unconstrained // by invariant (2), it's unconstrained
let mut target_column = motion_proj.column_mut(column_index); 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;
} }
} }

38
app-proto/src/display.rs
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@ -130,6 +130,8 @@ pub fn Display() -> View {
let translate_pos_y = create_signal(0.0); let translate_pos_y = create_signal(0.0);
let translate_neg_z = create_signal(0.0); let translate_neg_z = create_signal(0.0);
let translate_pos_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 // change listener
let scene_changed = create_signal(true); let scene_changed = create_signal(true);
@ -164,6 +166,7 @@ pub fn Display() -> View {
// manipulation // manipulation
const TRANSLATION_SPEED: f64 = 0.15; // in length units per second const TRANSLATION_SPEED: f64 = 0.15; // in length units per second
const SHRINKING_SPEED: f64 = 0.15; // in length units per second
// display parameters // display parameters
const OPACITY: f32 = 0.5; /* SCAFFOLDING */ 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_pos_y_val = translate_pos_y.get();
let translate_neg_z_val = translate_neg_z.get(); let translate_neg_z_val = translate_neg_z.get();
let translate_pos_z_val = translate_pos_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 // update the assembly's orientation
let ang_vel = { let ang_vel = {
@ -323,24 +328,27 @@ pub fn Display() -> View {
let sel = state.selection.with( let sel = state.selection.with(
|sel| *sel.into_iter().next().unwrap() |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_x = translate_pos_x_val - translate_neg_x_val;
let translate_y = translate_pos_y_val - translate_neg_y_val; let translate_y = translate_pos_y_val - translate_neg_y_val;
let translate_z = translate_pos_z_val - translate_neg_z_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, 0.0]
)
}; };
let elt_motion: DVector<f64> = time_step * TRANSLATION_SPEED * vel_field * rep;
assembly_for_raf.deform( assembly_for_raf.deform(
vec![ vec![
ElementMotion { ElementMotion {
@ -501,6 +509,8 @@ pub fn Display() -> View {
"s" | "S" if shift => translate_pos_z.set(value), "s" | "S" if shift => translate_pos_z.set(value),
"w" | "W" => translate_pos_y.set(value), "w" | "W" => translate_pos_y.set(value),
"s" | "S" => translate_neg_y.set(value), "s" | "S" => translate_neg_y.set(value),
"]" | "}" => shrink_neg.set(value),
"[" | "{" => shrink_pos.set(value),
_ => manipulating = false _ => manipulating = false
}; };
if manipulating { if manipulating {

80
app-proto/src/engine.rs
View File

@ -90,32 +90,33 @@ impl PartialMatrix {
#[derive(Clone)] #[derive(Clone)]
pub struct ConfigSubspace { pub struct ConfigSubspace {
assembly_dim: usize, assembly_dim: usize,
basis: Vec<DMatrix<f64>> basis_std: Vec<DMatrix<f64>>,
basis_proj: Vec<DMatrix<f64>>
} }
impl ConfigSubspace { impl ConfigSubspace {
pub fn zero(assembly_dim: usize) -> ConfigSubspace { pub fn zero(assembly_dim: usize) -> ConfigSubspace {
ConfigSubspace { ConfigSubspace {
assembly_dim: assembly_dim, 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 // approximate the kernel of a symmetric endomorphism of the configuration
// space for `assembly_dim` elements. we consider an eigenvector to be part // space for `assembly_dim` elements. we consider an eigenvector to be part
// of the kernel if its eigenvalue is smaller than the constant `THRESHOLD` // 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 {
// find a basis for the kernel, expressed in the standard coordinates
const ELEMENT_DIM: usize = 5; const ELEMENT_DIM: usize = 5;
const THRESHOLD: f64 = 1.0e-4; const THRESHOLD: f64 = 1.0e-4;
let eig = SymmetricEigen::new(a); let eig = SymmetricEigen::new(a);
let eig_vecs = eig.eigenvectors.column_iter(); let eig_vecs = eig.eigenvectors.column_iter();
let eig_pairs = eig.eigenvalues.iter().zip(eig_vecs); let eig_pairs = eig.eigenvalues.iter().zip(eig_vecs);
let basis = eig_pairs.filter_map( let basis_std = 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 */ /* DEBUG */
@ -125,30 +126,50 @@ impl ConfigSubspace {
format!("Eigenvalues used to find kernel:{}", eig.eigenvalues) format!("Eigenvalues used to find kernel:{}", eig.eigenvalues)
)); ));
// express the basis in the projection coordinates
let basis_proj = proj_to_std.clone().qr().solve(&basis_std).unwrap();
// 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;
console::log_1(&JsValue::from(
format!("Basis in projection coordinates:{}", basis_proj_orth)
));
ConfigSubspace { ConfigSubspace {
assembly_dim: assembly_dim, assembly_dim: assembly_dim,
basis: basis.collect() basis_std: basis_std_orth.column_iter().map(
|v| Into::<DMatrix<f64>>::into(
v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
)
).collect(),
basis_proj: basis_proj_orth.column_iter().map(
|v| Into::<DMatrix<f64>>::into(
v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
)
).collect()
} }
} }
pub fn dim(&self) -> usize { pub fn dim(&self) -> usize {
self.basis.len() self.basis_std.len()
} }
pub fn assembly_dim(&self) -> usize { pub fn assembly_dim(&self) -> usize {
self.assembly_dim 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> { pub fn proj(&self, v: &DVectorView<f64>, column_index: usize) -> DMatrix<f64> {
if self.dim() == 0 { if self.dim() == 0 {
const ELEMENT_DIM: usize = 5; const ELEMENT_DIM: usize = 5;
DMatrix::zeros(ELEMENT_DIM, self.assembly_dim) DMatrix::zeros(ELEMENT_DIM, self.assembly_dim)
} else { } 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() ).sum()
} }
} }
@ -215,6 +236,21 @@ fn basis_matrix(index: (usize, usize), nrows: usize, ncols: usize) -> DMatrix<f6
result result
} }
// given a spacelike unit vector `v`, which represents a sphere, build the basis
// for the configuration space given by the three unit translation motions of
// the sphere, the unit shrinking motion of the sphere, and `v`
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, &[
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 // use backtracking line search to find a better configuration
fn seek_better_config( fn seek_better_config(
gram: &PartialMatrix, gram: &PartialMatrix,
@ -344,7 +380,17 @@ pub fn realize_gram(
} }
let success = state.loss < tol; let success = state.loss < tol;
let tangent = if success { let tangent = if success {
ConfigSubspace::symmetric_kernel(hess, assembly_dim) // express the uniform basis in the standard basis
let mut unif_to_std = DMatrix::<f64>::zeros(total_dim, total_dim);
for n in 0..assembly_dim {
let block_start = element_dim * n;
unif_to_std
.view_mut((block_start, block_start), (element_dim, element_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 { } else {
ConfigSubspace::zero(assembly_dim) ConfigSubspace::zero(assembly_dim)
}; };