Add generic cubic splines to bevy_math (#7683)
# Objective - Make cubic splines more flexible and more performant - Remove the existing spline implementation that is generic over many degrees - This is a potential performance footgun and adds type complexity for negligible gain. - Add implementations of: - Bezier splines - Cardinal splines (inc. Catmull-Rom) - B-Splines - Hermite splines https://user-images.githubusercontent.com/2632925/221780519-495d1b20-ab46-45b4-92a3-32c46da66034.mp4 https://user-images.githubusercontent.com/2632925/221780524-2b154016-699f-404f-9c18-02092f589b04.mp4 https://user-images.githubusercontent.com/2632925/221780525-f934f99d-9ad4-4999-bae2-75d675f5644f.mp4 ## Solution - Implements the concept that splines are curve generators (e.g. https://youtu.be/jvPPXbo87ds?t=3488) via the `CubicGenerator` trait. - Common splines are bespoke data types that implement this trait. This gives us flexibility to add custom spline-specific methods on these types, while ultimately all generating a `CubicCurve`. - All splines generate `CubicCurve`s, which are a chain of precomputed polynomial coefficients. This means that all splines have the same evaluation cost, as the calculations for determining position, velocity, and acceleration are all identical. In addition, `CubicCurve`s are simply a list of `CubicSegment`s, which are evaluated from t=0 to t=1. This also means cubic splines of different type can be chained together, as ultimately they all are simply a collection of `CubicSegment`s. - Because easing is an operation on a singe segment of a Bezier curve, we can simply implement easing on `Beziers` that use the `Vec2` type for points. Higher level crates such as `bevy_ui` can wrap this in a more ergonomic interface as needed. ### Performance Measured on a desktop i5 8600K (6-year-old CPU): - easing: 2.7x faster (19ns) - cubic vec2 position sample: 1.5x faster (1.8ns) - cubic vec3 position sample: 1.5x faster (2.6ns) - cubic vec3a position sample: 1.9x faster (1.4ns) On a laptop i7 11800H: - easing: 16ns - cubic vec2 position sample: 1.6ns - cubic vec3 position sample: 2.3ns - cubic vec3a position sample: 1.2ns --- ## Changelog - Added a generic cubic curve trait, and implementation for Cardinal splines (including Catmull-Rom), B-Splines, Beziers, and Hermite Splines. 2D cubic curve segments also implement easing functionality for animation.
This commit is contained in:
Aevyrie
parent
47ddd4a8ce
commit
2ea0061018
@ -1,128 +1,89 @@
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use criterion::{black_box, criterion_group, criterion_main, Criterion};
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use criterion::{black_box, criterion_group, criterion_main, Criterion};
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use bevy_math::*;
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use bevy_math::{prelude::*, *};
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fn easing(c: &mut Criterion) {
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fn easing(c: &mut Criterion) {
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let cubic_bezier = CubicBezierEasing::new(vec2(0.25, 0.1), vec2(0.25, 1.0));
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let cubic_bezier = CubicSegment::new_bezier(vec2(0.25, 0.1), vec2(0.25, 1.0));
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c.bench_function("easing_1000", |b| {
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c.bench_function("easing_1000", |b| {
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b.iter(|| {
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b.iter(|| {
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(0..1000).map(|i| i as f32 / 1000.0).for_each(|t| {
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(0..1000).map(|i| i as f32 / 1000.0).for_each(|t| {
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cubic_bezier.ease(black_box(t));
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black_box(cubic_bezier.ease(black_box(t)));
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})
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})
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});
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});
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});
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});
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}
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}
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fn fifteen_degree(c: &mut Criterion) {
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let bezier = Bezier::<Vec3A, 16>::new([
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[0.0, 0.0, 0.0],
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[0.0, 1.0, 0.0],
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[1.0, 0.0, 0.0],
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[1.0, 1.0, 1.0],
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[0.0, 0.0, 0.0],
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[0.0, 1.0, 0.0],
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[1.0, 0.0, 0.0],
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[1.0, 1.0, 1.0],
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[0.0, 0.0, 0.0],
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[0.0, 1.0, 0.0],
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[1.0, 0.0, 0.0],
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[1.0, 1.0, 1.0],
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[0.0, 0.0, 0.0],
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[0.0, 1.0, 0.0],
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[1.0, 0.0, 0.0],
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[1.0, 1.0, 1.0],
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]);
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c.bench_function("fifteen_degree_position", |b| {
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b.iter(|| bezier.position(black_box(0.5)));
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});
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}
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fn quadratic_2d(c: &mut Criterion) {
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let bezier = QuadraticBezier2d::new([[0.0, 0.0], [0.0, 1.0], [1.0, 1.0]]);
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c.bench_function("quadratic_position_Vec2", |b| {
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b.iter(|| bezier.position(black_box(0.5)));
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});
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}
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fn quadratic(c: &mut Criterion) {
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let bezier = QuadraticBezier3d::new([[0.0, 0.0, 0.0], [0.0, 1.0, 0.0], [1.0, 1.0, 1.0]]);
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c.bench_function("quadratic_position_Vec3A", |b| {
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b.iter(|| bezier.position(black_box(0.5)));
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});
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}
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fn quadratic_vec3(c: &mut Criterion) {
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let bezier = Bezier::<Vec3, 3>::new([[0.0, 0.0, 0.0], [0.0, 1.0, 0.0], [1.0, 1.0, 1.0]]);
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c.bench_function("quadratic_position_Vec3", |b| {
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b.iter(|| bezier.position(black_box(0.5)));
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});
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}
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fn cubic_2d(c: &mut Criterion) {
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fn cubic_2d(c: &mut Criterion) {
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let bezier = CubicBezier2d::new([[0.0, 0.0], [0.0, 1.0], [1.0, 0.0], [1.0, 1.0]]);
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let bezier = Bezier::new([[
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vec2(0.0, 0.0),
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vec2(0.0, 1.0),
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vec2(1.0, 0.0),
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vec2(1.0, 1.0),
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]])
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.to_curve();
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c.bench_function("cubic_position_Vec2", |b| {
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c.bench_function("cubic_position_Vec2", |b| {
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b.iter(|| bezier.position(black_box(0.5)));
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b.iter(|| black_box(bezier.position(black_box(0.5))));
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});
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});
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}
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}
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fn cubic(c: &mut Criterion) {
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fn cubic(c: &mut Criterion) {
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let bezier = CubicBezier3d::new([
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let bezier = Bezier::new([[
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[0.0, 0.0, 0.0],
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vec3a(0.0, 0.0, 0.0),
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[0.0, 1.0, 0.0],
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vec3a(0.0, 1.0, 0.0),
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[1.0, 0.0, 0.0],
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vec3a(1.0, 0.0, 0.0),
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[1.0, 1.0, 1.0],
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vec3a(1.0, 1.0, 1.0),
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]);
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]])
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.to_curve();
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c.bench_function("cubic_position_Vec3A", |b| {
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c.bench_function("cubic_position_Vec3A", |b| {
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b.iter(|| bezier.position(black_box(0.5)));
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b.iter(|| black_box(bezier.position(black_box(0.5))));
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});
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});
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}
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}
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fn cubic_vec3(c: &mut Criterion) {
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fn cubic_vec3(c: &mut Criterion) {
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let bezier = Bezier::<Vec3, 4>::new([
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let bezier = Bezier::new([[
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[0.0, 0.0, 0.0],
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vec3(0.0, 0.0, 0.0),
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[0.0, 1.0, 0.0],
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vec3(0.0, 1.0, 0.0),
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[1.0, 0.0, 0.0],
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vec3(1.0, 0.0, 0.0),
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[1.0, 1.0, 1.0],
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vec3(1.0, 1.0, 1.0),
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]);
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]])
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.to_curve();
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c.bench_function("cubic_position_Vec3", |b| {
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c.bench_function("cubic_position_Vec3", |b| {
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b.iter(|| bezier.position(black_box(0.5)));
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b.iter(|| black_box(bezier.position(black_box(0.5))));
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});
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});
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}
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}
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fn build_pos_cubic(c: &mut Criterion) {
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fn build_pos_cubic(c: &mut Criterion) {
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let bezier = CubicBezier3d::new([
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let bezier = Bezier::new([[
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[0.0, 0.0, 0.0],
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vec3a(0.0, 0.0, 0.0),
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[0.0, 1.0, 0.0],
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vec3a(0.0, 1.0, 0.0),
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[1.0, 0.0, 0.0],
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vec3a(1.0, 0.0, 0.0),
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[1.0, 1.0, 1.0],
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vec3a(1.0, 1.0, 1.0),
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]);
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]])
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.to_curve();
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c.bench_function("build_pos_cubic_100_points", |b| {
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c.bench_function("build_pos_cubic_100_points", |b| {
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b.iter(|| bezier.iter_positions(black_box(100)).collect::<Vec<_>>());
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b.iter(|| black_box(bezier.iter_positions(black_box(100)).collect::<Vec<_>>()));
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});
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});
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}
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}
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fn build_accel_cubic(c: &mut Criterion) {
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fn build_accel_cubic(c: &mut Criterion) {
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let bezier = CubicBezier3d::new([
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let bezier = Bezier::new([[
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[0.0, 0.0, 0.0],
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vec3a(0.0, 0.0, 0.0),
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[0.0, 1.0, 0.0],
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vec3a(0.0, 1.0, 0.0),
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[1.0, 0.0, 0.0],
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vec3a(1.0, 0.0, 0.0),
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[1.0, 1.0, 1.0],
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vec3a(1.0, 1.0, 1.0),
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]);
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]])
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.to_curve();
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c.bench_function("build_accel_cubic_100_points", |b| {
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c.bench_function("build_accel_cubic_100_points", |b| {
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b.iter(|| bezier.iter_positions(black_box(100)).collect::<Vec<_>>());
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b.iter(|| black_box(bezier.iter_positions(black_box(100)).collect::<Vec<_>>()));
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});
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});
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}
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}
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criterion_group!(
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criterion_group!(
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benches,
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benches,
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easing,
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easing,
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fifteen_degree,
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quadratic_2d,
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quadratic,
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quadratic_vec3,
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cubic_2d,
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cubic_2d,
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cubic,
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cubic_vec3,
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cubic_vec3,
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cubic,
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build_pos_cubic,
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build_pos_cubic,
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build_accel_cubic,
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build_accel_cubic,
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);
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);
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@ -1,457 +0,0 @@
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use glam::{Vec2, Vec3, Vec3A};
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use std::{
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fmt::Debug,
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iter::Sum,
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ops::{Add, Mul, Sub},
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};
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/// A point in space of any dimension that supports the mathematical operations needed by
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/// [`Bezier`].
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pub trait Point:
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Mul<f32, Output = Self>
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+ Add<Self, Output = Self>
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+ Sub<Self, Output = Self>
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+ Sum
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+ Default
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+ Debug
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+ Clone
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+ PartialEq
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+ Copy
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{
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}
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impl Point for Vec3 {} // 3D
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impl Point for Vec3A {} // 3D
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impl Point for Vec2 {} // 2D
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impl Point for f32 {} // 1D
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/// A cubic Bezier curve in 2D space
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pub type CubicBezier2d = Bezier<Vec2, 4>;
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/// A cubic Bezier curve in 3D space
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pub type CubicBezier3d = Bezier<Vec3A, 4>;
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/// A quadratic Bezier curve in 2D space
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pub type QuadraticBezier2d = Bezier<Vec2, 3>;
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/// A quadratic Bezier curve in 3D space
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pub type QuadraticBezier3d = Bezier<Vec3A, 3>;
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/// A generic Bezier curve with `N` control points, and dimension defined by `P`.
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///
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/// Consider the following type aliases for most common uses:
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/// - [`CubicBezier2d`]
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/// - [`CubicBezier3d`]
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/// - [`QuadraticBezier2d`]
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/// - [`QuadraticBezier3d`]
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///
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/// The Bezier degree is equal to `N - 1`. For example, a cubic Bezier has 4 control points, and a
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/// degree of 3. The time-complexity of evaluating a Bezier increases superlinearly with the number
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/// of control points. As such, it is recommended to instead use a chain of quadratic or cubic
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/// `Beziers` instead of a high-degree Bezier.
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///
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/// ### About Bezier curves
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///
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/// `Bezier` curves are parametric implicit functions; all that means is they take a parameter `t`,
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/// and output a point in space, like:
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///
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/// > B(t) = (x, y, z)
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///
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/// So, all that is needed to find a point in space along a Bezier curve is the parameter `t`.
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/// Additionally, the values of `t` are straightforward: `t` is 0 at the start of the curve (first
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/// control point) and 1 at the end (last control point).
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///
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/// ```
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/// # use bevy_math::{Bezier, Vec2, vec2};
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/// let p0 = vec2(0.25, 0.1);
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/// let p1 = vec2(0.25, 1.0);
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/// let bezier = Bezier::<Vec2, 2>::new([p0, p1]);
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/// assert_eq!(bezier.position(0.0), p0);
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/// assert_eq!(bezier.position(1.0), p1);
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/// ```
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///
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/// ### Plotting
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///
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/// To plot a Bezier curve, simply plug in a series of values of `t` from zero to one. The functions
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/// to do this are [`Bezier::position`] to sample the curve at a value of `t`, and
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/// [`Bezier::iter_positions`] to iterate over the curve with a number of subdivisions.
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///
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/// ### Velocity and Acceleration
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///
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/// In addition to the position of a point on the Bezier curve, it is also useful to get information
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/// about the curvature. Methods are provided to help with this:
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///
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/// - [`Bezier::velocity`]: the instantaneous velocity vector with respect to `t`. This is a vector
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/// that points in the direction a point is traveling when it is at point `t`. This vector is
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/// tangent to the curve.
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/// - [`Bezier::acceleration`]: the instantaneous acceleration vector with respect to `t`. This is a
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/// vector that points in the direction a point is accelerating towards when it is at point
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/// `t`. This vector will point to the inside of turns, the direction the point is being
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/// pulled toward to change direction.
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#[derive(Clone, Copy, Debug, PartialEq)]
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pub struct Bezier<P: Point, const N: usize>(pub [P; N]);
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impl<P: Point, const N: usize> Default for Bezier<P, N> {
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fn default() -> Self {
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Bezier([P::default(); N])
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}
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}
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impl<P: Point, const N: usize> Bezier<P, N> {
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/// Construct a new Bezier curve.
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pub fn new(control_points: [impl Into<P>; N]) -> Self {
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let control_points = control_points.map(|v| v.into());
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Self(control_points)
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}
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/// Compute the position of a point along the Bezier curve at the supplied parametric value `t`.
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pub fn position(&self, t: f32) -> P {
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generic::position(self.0, t)
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}
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/// Compute the first derivative B'(t) of this Bezier at `t` with respect to t. This is the
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/// instantaneous velocity of a point on the Bezier curve at `t`.
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pub fn velocity(&self, t: f32) -> P {
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generic::velocity(self.0, t)
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}
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/// Compute the second derivative B''(t) of this Bezier at `t` with respect to t. This is the
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/// instantaneous acceleration of a point on the Bezier curve at `t`.
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pub fn acceleration(&self, t: f32) -> P {
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generic::acceleration(self.0, t)
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}
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/// A flexible iterator used to sample [`Bezier`] curves with arbitrary functions.
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///
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/// This splits the Bezier into `subdivisions` of evenly spaced `t` values across the length of
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/// the curve from start (t = 0) to end (t = 1), returning an iterator that evaluates the curve
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/// with the supplied `sample_function` at each `t`.
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///
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/// Given `subdivisions = 2`, this will split the curve into two lines, or three points, and
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/// return an iterator over those three points, one at the start, middle, and end.
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#[inline]
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pub fn iter_samples(
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&self,
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subdivisions: usize,
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sample_function: fn(&Self, f32) -> P,
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) -> impl Iterator<Item = P> + '_ {
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(0..=subdivisions).map(move |i| {
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let t = i as f32 / subdivisions as f32;
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sample_function(self, t)
|
|
||||||
})
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Iterate over the curve split into `subdivisions`, sampling the position at each step.
|
|
||||||
pub fn iter_positions(&self, subdivisions: usize) -> impl Iterator<Item = P> + '_ {
|
|
||||||
self.iter_samples(subdivisions, Self::position)
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Iterate over the curve split into `subdivisions`, sampling the velocity at each step.
|
|
||||||
pub fn iter_velocities(&self, subdivisions: usize) -> impl Iterator<Item = P> + '_ {
|
|
||||||
self.iter_samples(subdivisions, Self::velocity)
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Iterate over the curve split into `subdivisions`, sampling the acceleration at each step.
|
|
||||||
pub fn iter_accelerations(&self, subdivisions: usize) -> impl Iterator<Item = P> + '_ {
|
|
||||||
self.iter_samples(subdivisions, Self::acceleration)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
impl<T: Into<P>, P: Point, const N: usize> From<[T; N]> for Bezier<P, N> {
|
|
||||||
fn from(control_points: [T; N]) -> Self {
|
|
||||||
Bezier::new(control_points)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
/// A 2-dimensional Bezier curve used for easing in animation.
|
|
||||||
///
|
|
||||||
/// A cubic Bezier easing curve has control point `p0` at (0, 0) and `p3` at (1, 1), leaving only
|
|
||||||
/// `p1` and `p2` as the remaining degrees of freedom. The first and last control points are fixed
|
|
||||||
/// to ensure the animation begins at 0, and ends at 1.
|
|
||||||
#[derive(Default, Clone, Copy, Debug, PartialEq)]
|
|
||||||
pub struct CubicBezierEasing {
|
|
||||||
/// Control point P1 of the 2D cubic Bezier curve. Controls the start of the animation.
|
|
||||||
pub p1: Vec2,
|
|
||||||
/// Control point P2 of the 2D cubic Bezier curve. Controls the end of the animation.
|
|
||||||
pub p2: Vec2,
|
|
||||||
}
|
|
||||||
|
|
||||||
impl CubicBezierEasing {
|
|
||||||
/// Construct a cubic Bezier curve for animation easing, with control points `p1` and `p2`.
|
|
||||||
/// These correspond to the two free "handles" of the Bezier curve.
|
|
||||||
///
|
|
||||||
/// This is a very common tool for animations that accelerate and decelerate smoothly. For
|
|
||||||
/// example, the ubiquitous "ease-in-out" is defined as `(0.25, 0.1), (0.25, 1.0)`.
|
|
||||||
pub fn new(p1: impl Into<Vec2>, p2: impl Into<Vec2>) -> Self {
|
|
||||||
Self {
|
|
||||||
p1: p1.into(),
|
|
||||||
p2: p2.into(),
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Maximum allowable error for iterative Bezier solve
|
|
||||||
const MAX_ERROR: f32 = 1e-5;
|
|
||||||
|
|
||||||
/// Maximum number of iterations during Bezier solve
|
|
||||||
const MAX_ITERS: u8 = 8;
|
|
||||||
|
|
||||||
/// Given a `time` within `0..=1`, returns an eased value that follows the cubic Bezier curve
|
|
||||||
/// instead of a straight line. This eased result may be outside the range `0..=1`, however it
|
|
||||||
/// will always start at 0 and end at 1: `ease(0) = 0` and `ease(1) = 1`.
|
|
||||||
///
|
|
||||||
/// ```
|
|
||||||
/// # use bevy_math::CubicBezierEasing;
|
|
||||||
/// let cubic_bezier = CubicBezierEasing::new((0.25, 0.1), (0.25, 1.0));
|
|
||||||
/// assert_eq!(cubic_bezier.ease(0.0), 0.0);
|
|
||||||
/// assert_eq!(cubic_bezier.ease(1.0), 1.0);
|
|
||||||
/// ```
|
|
||||||
///
|
|
||||||
/// # How cubic Bezier easing works
|
|
||||||
///
|
|
||||||
/// Easing is generally accomplished with the help of "shaping functions". These are curves that
|
|
||||||
/// start at (0,0) and end at (1,1). The x-axis of this plot is the current `time` of the
|
|
||||||
/// animation, from 0 to 1. The y-axis is how far along the animation is, also from 0 to 1. You
|
|
||||||
/// can imagine that if the shaping function is a straight line, there is a 1:1 mapping between
|
|
||||||
/// the `time` and how far along your animation is. If the `time` = 0.5, the animation is
|
|
||||||
/// halfway through. This is known as linear interpolation, and results in objects animating
|
|
||||||
/// with a constant velocity, and no smooth acceleration or deceleration at the start or end.
|
|
||||||
///
|
|
||||||
/// ```text
|
|
||||||
/// y
|
|
||||||
/// │ ●
|
|
||||||
/// │ ⬈
|
|
||||||
/// │ ⬈
|
|
||||||
/// │ ⬈
|
|
||||||
/// │ ⬈
|
|
||||||
/// ●─────────── x (time)
|
|
||||||
/// ```
|
|
||||||
///
|
|
||||||
/// Using cubic Beziers, we have a curve that starts at (0,0), ends at (1,1), and follows a path
|
|
||||||
/// determined by the two remaining control points (handles). These handles allow us to define a
|
|
||||||
/// smooth curve. As `time` (x-axis) progresses, we now follow the curve, and use the `y` value
|
|
||||||
/// to determine how far along the animation is.
|
|
||||||
///
|
|
||||||
/// ```text
|
|
||||||
/// y
|
|
||||||
/// ⬈➔●
|
|
||||||
/// │ ⬈
|
|
||||||
/// │ ↑
|
|
||||||
/// │ ↑
|
|
||||||
/// │ ⬈
|
|
||||||
/// ●➔⬈───────── x (time)
|
|
||||||
/// ```
|
|
||||||
///
|
|
||||||
/// To accomplish this, we need to be able to find the position `y` on a Bezier curve, given the
|
|
||||||
/// `x` value. As discussed in the [`Bezier`] documentation, a Bezier curve is an implicit
|
|
||||||
/// parametric function like B(t) = (x,y). To find `y`, we first solve for `t` that corresponds
|
|
||||||
/// to the given `x` (`time`). We use the Newton-Raphson root-finding method to quickly find a
|
|
||||||
/// value of `t` that matches `x`. Once we have this we can easily plug that `t` into our
|
|
||||||
/// Bezier's `position` function, to find the `y` component, which is how far along our
|
|
||||||
/// animation should be. In other words:
|
|
||||||
///
|
|
||||||
/// > Given `time` in `0..=1`
|
|
||||||
///
|
|
||||||
/// > Use Newton's method to find a value of `t` that results in B(t) = (x,y) where `x == time`
|
|
||||||
///
|
|
||||||
/// > Once a solution is found, use the resulting `y` value as the final result
|
|
||||||
///
|
|
||||||
/// # Performance
|
|
||||||
///
|
|
||||||
/// This operation can be used frequently without fear of performance issues. Benchmarks show
|
|
||||||
/// this operation taking on the order of 50 nanoseconds.
|
|
||||||
pub fn ease(&self, time: f32) -> f32 {
|
|
||||||
let x = time.clamp(0.0, 1.0);
|
|
||||||
let t = self.find_t_given_x(x);
|
|
||||||
self.evaluate_y_at(t)
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Compute the x-coordinate of the point along the Bezier curve at `t`.
|
|
||||||
#[inline]
|
|
||||||
fn evaluate_x_at(&self, t: f32) -> f32 {
|
|
||||||
generic::position([0.0, self.p1.x, self.p2.x, 1.0], t)
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Compute the y-coordinate of the point along the Bezier curve at `t`.
|
|
||||||
#[inline]
|
|
||||||
fn evaluate_y_at(&self, t: f32) -> f32 {
|
|
||||||
generic::position([0.0, self.p1.y, self.p2.y, 1.0], t)
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Compute the slope of the line at the given parametric value `t` with respect to t.
|
|
||||||
#[inline]
|
|
||||||
fn dx_dt(&self, t: f32) -> f32 {
|
|
||||||
generic::velocity([0.0, self.p1.x, self.p2.x, 1.0], t)
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Solve for the parametric value `t` that corresponds to the given value of `x` using the
|
|
||||||
/// Newton-Raphson method. See documentation on [`Self::ease`] for more details.
|
|
||||||
#[inline]
|
|
||||||
fn find_t_given_x(&self, x: f32) -> f32 {
|
|
||||||
// PERFORMANCE NOTE:
|
|
||||||
//
|
|
||||||
// I tried pre-solving and caching values along the curve at struct instantiation to give
|
|
||||||
// the solver a better starting guess. This ended up being slightly slower, possibly due to
|
|
||||||
// the increased size of the type. Another option would be to store the last `t`, and use
|
|
||||||
// that, however it's possible this could end up in a bad state where t is very far from the
|
|
||||||
// naive but generally safe guess of x, e.g. after an animation resets.
|
|
||||||
//
|
|
||||||
// Further optimization might not be needed however - benchmarks are showing it takes about
|
|
||||||
// 50 nanoseconds for an ease operation on my modern laptop, which seems sufficiently fast.
|
|
||||||
let mut t_guess = x;
|
|
||||||
for _ in 0..Self::MAX_ITERS {
|
|
||||||
let x_guess = self.evaluate_x_at(t_guess);
|
|
||||||
let error = x_guess - x;
|
|
||||||
if error.abs() <= Self::MAX_ERROR {
|
|
||||||
break;
|
|
||||||
}
|
|
||||||
// Using Newton's method, use the tangent line to estimate a better guess value.
|
|
||||||
let slope = self.dx_dt(t_guess);
|
|
||||||
t_guess -= error / slope;
|
|
||||||
}
|
|
||||||
t_guess.clamp(0.0, 1.0)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
impl<P: Into<Vec2>> From<[P; 2]> for CubicBezierEasing {
|
|
||||||
fn from(points: [P; 2]) -> Self {
|
|
||||||
let [p0, p1] = points;
|
|
||||||
CubicBezierEasing::new(p0, p1)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Generic implementations for sampling Bezier curves. Consider using the methods on
|
|
||||||
/// [`Bezier`](crate::Bezier) for more ergonomic use.
|
|
||||||
pub mod generic {
|
|
||||||
use super::Point;
|
|
||||||
|
|
||||||
/// Compute the Bernstein basis polynomial `i` of degree `n`, at `t`.
|
|
||||||
///
|
|
||||||
/// For more information, see <https://en.wikipedia.org/wiki/Bernstein_polynomial>.
|
|
||||||
#[inline]
|
|
||||||
pub fn bernstein_basis(n: usize, i: usize, t: f32) -> f32 {
|
|
||||||
(1. - t).powi((n - i) as i32) * t.powi(i as i32)
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Efficiently compute the binomial coefficient of `n` choose `k`.
|
|
||||||
#[inline]
|
|
||||||
fn binomial_coeff(n: usize, k: usize) -> usize {
|
|
||||||
let k = usize::min(k, n - k);
|
|
||||||
(0..k).fold(1, |val, i| val * (n - i) / (i + 1))
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Evaluate the Bezier curve B(t) of degree `N-1` at the parametric value `t`.
|
|
||||||
#[inline]
|
|
||||||
pub fn position<P: Point, const N: usize>(control_points: [P; N], t: f32) -> P {
|
|
||||||
let p = control_points;
|
|
||||||
let degree = N - 1;
|
|
||||||
(0..=degree)
|
|
||||||
.map(|i| p[i] * binomial_coeff(degree, i) as f32 * bernstein_basis(degree, i, t))
|
|
||||||
.sum()
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Compute the first derivative B'(t) of Bezier curve B(t) of degree `N-1` at the given
|
|
||||||
/// parametric value `t` with respect to t. Note that the first derivative of a Bezier is also a
|
|
||||||
/// Bezier, of degree `N-2`.
|
|
||||||
#[inline]
|
|
||||||
pub fn velocity<P: Point, const N: usize>(control_points: [P; N], t: f32) -> P {
|
|
||||||
if N <= 1 {
|
|
||||||
return P::default(); // Zero for numeric types
|
|
||||||
}
|
|
||||||
let p = control_points;
|
|
||||||
let degree = N - 1;
|
|
||||||
let degree_vel = N - 2; // the velocity Bezier is one degree lower than the position Bezier
|
|
||||||
(0..=degree_vel)
|
|
||||||
.map(|i| {
|
|
||||||
// Point on the velocity Bezier curve:
|
|
||||||
let p = (p[i + 1] - p[i]) * degree as f32;
|
|
||||||
p * binomial_coeff(degree_vel, i) as f32 * bernstein_basis(degree_vel, i, t)
|
|
||||||
})
|
|
||||||
.sum()
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Compute the second derivative B''(t) of Bezier curve B(t) of degree `N-1` at the given
|
|
||||||
/// parametric value `t` with respect to t. Note that the second derivative of a Bezier is also
|
|
||||||
/// a Bezier, of degree `N-3`.
|
|
||||||
#[inline]
|
|
||||||
pub fn acceleration<P: Point, const N: usize>(control_points: [P; N], t: f32) -> P {
|
|
||||||
if N <= 2 {
|
|
||||||
return P::default(); // Zero for numeric types
|
|
||||||
}
|
|
||||||
let p = control_points;
|
|
||||||
let degree = N - 1;
|
|
||||||
let degree_vel = N - 2; // the velocity Bezier is one degree lower than the position Bezier
|
|
||||||
let degree_accel = N - 3; // the accel Bezier is one degree lower than the velocity Bezier
|
|
||||||
(0..degree_vel)
|
|
||||||
.map(|i| {
|
|
||||||
// Points on the velocity Bezier curve:
|
|
||||||
let p0 = (p[i + 1] - p[i]) * degree as f32;
|
|
||||||
let p1 = (p[i + 2] - p[i + 1]) * degree as f32;
|
|
||||||
// Point on the acceleration Bezier curve:
|
|
||||||
let p = (p1 - p0) * (degree_vel) as f32;
|
|
||||||
p * binomial_coeff(degree_accel, i) as f32 * bernstein_basis(degree_accel, i, t)
|
|
||||||
})
|
|
||||||
.sum()
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
#[cfg(test)]
|
|
||||||
mod tests {
|
|
||||||
use glam::Vec2;
|
|
||||||
|
|
||||||
use crate::{CubicBezier2d, CubicBezierEasing};
|
|
||||||
|
|
||||||
/// How close two floats can be and still be considered equal
|
|
||||||
const FLOAT_EQ: f32 = 1e-5;
|
|
||||||
|
|
||||||
/// Basic cubic Bezier easing test to verify the shape of the curve.
|
|
||||||
#[test]
|
|
||||||
fn easing_simple() {
|
|
||||||
// A curve similar to ease-in-out, but symmetric
|
|
||||||
let bezier = CubicBezierEasing::new([1.0, 0.0], [0.0, 1.0]);
|
|
||||||
assert_eq!(bezier.ease(0.0), 0.0);
|
|
||||||
assert!(bezier.ease(0.2) < 0.2); // tests curve
|
|
||||||
assert_eq!(bezier.ease(0.5), 0.5); // true due to symmetry
|
|
||||||
assert!(bezier.ease(0.8) > 0.8); // tests curve
|
|
||||||
assert_eq!(bezier.ease(1.0), 1.0);
|
|
||||||
}
|
|
||||||
|
|
||||||
/// A curve that forms an upside-down "U", that should extend below 0.0. Useful for animations
|
|
||||||
/// that go beyond the start and end positions, e.g. bouncing.
|
|
||||||
#[test]
|
|
||||||
fn easing_overshoot() {
|
|
||||||
// A curve that forms an upside-down "U", that should extend above 1.0
|
|
||||||
let bezier = CubicBezierEasing::new([0.0, 2.0], [1.0, 2.0]);
|
|
||||||
assert_eq!(bezier.ease(0.0), 0.0);
|
|
||||||
assert!(bezier.ease(0.5) > 1.5);
|
|
||||||
assert_eq!(bezier.ease(1.0), 1.0);
|
|
||||||
}
|
|
||||||
|
|
||||||
/// A curve that forms a "U", that should extend below 0.0. Useful for animations that go beyond
|
|
||||||
/// the start and end positions, e.g. bouncing.
|
|
||||||
#[test]
|
|
||||||
fn easing_undershoot() {
|
|
||||||
let bezier = CubicBezierEasing::new([0.0, -2.0], [1.0, -2.0]);
|
|
||||||
assert_eq!(bezier.ease(0.0), 0.0);
|
|
||||||
assert!(bezier.ease(0.5) < -0.5);
|
|
||||||
assert_eq!(bezier.ease(1.0), 1.0);
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Sweep along the full length of a 3D cubic Bezier, and manually check the position.
|
|
||||||
#[test]
|
|
||||||
fn cubic() {
|
|
||||||
const N_SAMPLES: usize = 1000;
|
|
||||||
let bezier = CubicBezier2d::new([[-1.0, -20.0], [3.0, 2.0], [5.0, 3.0], [9.0, 8.0]]);
|
|
||||||
assert_eq!(bezier.position(0.0), bezier.0[0]); // 0 == Start
|
|
||||||
assert_eq!(bezier.position(1.0), bezier.0[3]); // 1 == End
|
|
||||||
for i in 0..=N_SAMPLES {
|
|
||||||
let t = i as f32 / N_SAMPLES as f32; // Check along entire length
|
|
||||||
assert!(bezier.position(t).distance(cubic_manual(t, bezier)) <= FLOAT_EQ);
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
/// Manual, hardcoded function for computing the position along a cubic bezier.
|
|
||||||
fn cubic_manual(t: f32, bezier: CubicBezier2d) -> Vec2 {
|
|
||||||
let [p0, p1, p2, p3] = bezier.0;
|
|
||||||
p0 * (1.0 - t).powi(3)
|
|
||||||
+ 3.0 * p1 * t * (1.0 - t).powi(2)
|
|
||||||
+ 3.0 * p2 * t.powi(2) * (1.0 - t)
|
|
||||||
+ p3 * t.powi(3)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
602
crates/bevy_math/src/cubic_splines.rs
Normal file
602
crates/bevy_math/src/cubic_splines.rs
Normal file
@ -0,0 +1,602 @@
|
|||||||
|
//! Provides types for building cubic splines for rendering curves and use with animation easing.
|
||||||
|
|
||||||
|
use glam::{Vec2, Vec3, Vec3A};
|
||||||
|
|
||||||
|
use std::{
|
||||||
|
fmt::Debug,
|
||||||
|
iter::Sum,
|
||||||
|
ops::{Add, Mul, Sub},
|
||||||
|
};
|
||||||
|
|
||||||
|
/// A point in space of any dimension that supports the math ops needed for cubic spline
|
||||||
|
/// interpolation.
|
||||||
|
pub trait Point:
|
||||||
|
Mul<f32, Output = Self>
|
||||||
|
+ Add<Self, Output = Self>
|
||||||
|
+ Sub<Self, Output = Self>
|
||||||
|
+ Add<f32, Output = Self>
|
||||||
|
+ Sum
|
||||||
|
+ Default
|
||||||
|
+ Debug
|
||||||
|
+ Clone
|
||||||
|
+ PartialEq
|
||||||
|
+ Copy
|
||||||
|
{
|
||||||
|
}
|
||||||
|
impl Point for Vec3 {}
|
||||||
|
impl Point for Vec3A {}
|
||||||
|
impl Point for Vec2 {}
|
||||||
|
impl Point for f32 {}
|
||||||
|
|
||||||
|
/// A spline composed of a series of cubic Bezier curves.
|
||||||
|
///
|
||||||
|
/// Useful for user-drawn curves with local control, or animation easing. See
|
||||||
|
/// [`CubicSegment::new_bezier`] for use in easing.
|
||||||
|
///
|
||||||
|
/// ### Interpolation
|
||||||
|
/// The curve only passes through the first and last control point in each set of four points.
|
||||||
|
///
|
||||||
|
/// ### Tangency
|
||||||
|
/// Manually defined by the two intermediate control points within each set of four points.
|
||||||
|
///
|
||||||
|
/// ### Continuity
|
||||||
|
/// At minimum C0 continuous, up to C2. Continuity greater than C0 can result in a loss of local
|
||||||
|
/// control over the spline due to the curvature constraints.
|
||||||
|
///
|
||||||
|
/// ### Usage
|
||||||
|
///
|
||||||
|
/// ```
|
||||||
|
/// # use bevy_math::{*, prelude::*};
|
||||||
|
/// let points = [[
|
||||||
|
/// vec2(-1.0, -20.0),
|
||||||
|
/// vec2(3.0, 2.0),
|
||||||
|
/// vec2(5.0, 3.0),
|
||||||
|
/// vec2(9.0, 8.0),
|
||||||
|
/// ]];
|
||||||
|
/// let bezier = Bezier::new(points).to_curve();
|
||||||
|
/// let positions: Vec<_> = bezier.iter_positions(100).collect();
|
||||||
|
/// ```
|
||||||
|
pub struct Bezier<P: Point> {
|
||||||
|
control_points: Vec<[P; 4]>,
|
||||||
|
}
|
||||||
|
|
||||||
|
impl<P: Point> Bezier<P> {
|
||||||
|
/// Create a new Bezier curve from sets of control points.
|
||||||
|
pub fn new(control_points: impl Into<Vec<[P; 4]>>) -> Self {
|
||||||
|
Self {
|
||||||
|
control_points: control_points.into(),
|
||||||
|
}
|
||||||
|
}
|
||||||
|
}
|
||||||
|
impl<P: Point> CubicGenerator<P> for Bezier<P> {
|
||||||
|
#[inline]
|
||||||
|
fn to_curve(&self) -> CubicCurve<P> {
|
||||||
|
let char_matrix = [
|
||||||
|
[1., 0., 0., 0.],
|
||||||
|
[-3., 3., 0., 0.],
|
||||||
|
[3., -6., 3., 0.],
|
||||||
|
[-1., 3., -3., 1.],
|
||||||
|
];
|
||||||
|
|
||||||
|
let segments = self
|
||||||
|
.control_points
|
||||||
|
.iter()
|
||||||
|
.map(|p| CubicCurve::coefficients(*p, 1.0, char_matrix))
|
||||||
|
.collect();
|
||||||
|
|
||||||
|
CubicCurve { segments }
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A spline interpolated continuously between the nearest two control points, with the position and
|
||||||
|
/// velocity of the curve specified at both control points. This curve passes through all control
|
||||||
|
/// points, with the specified velocity which includes direction and parametric speed.
|
||||||
|
///
|
||||||
|
/// Useful for smooth interpolation when you know the position and velocity at two points in time,
|
||||||
|
/// such as network prediction.
|
||||||
|
///
|
||||||
|
/// ### Interpolation
|
||||||
|
/// The curve passes through every control point.
|
||||||
|
///
|
||||||
|
/// ### Tangency
|
||||||
|
/// Explicitly defined at each control point.
|
||||||
|
///
|
||||||
|
/// ### Continuity
|
||||||
|
/// At minimum C0 continuous, up to C1.
|
||||||
|
///
|
||||||
|
/// ### Usage
|
||||||
|
///
|
||||||
|
/// ```
|
||||||
|
/// # use bevy_math::{*, prelude::*};
|
||||||
|
/// let points = [
|
||||||
|
/// vec2(-1.0, -20.0),
|
||||||
|
/// vec2(3.0, 2.0),
|
||||||
|
/// vec2(5.0, 3.0),
|
||||||
|
/// vec2(9.0, 8.0),
|
||||||
|
/// ];
|
||||||
|
/// let tangents = [
|
||||||
|
/// vec2(0.0, 1.0),
|
||||||
|
/// vec2(0.0, 1.0),
|
||||||
|
/// vec2(0.0, 1.0),
|
||||||
|
/// vec2(0.0, 1.0),
|
||||||
|
/// ];
|
||||||
|
/// let hermite = Hermite::new(points, tangents).to_curve();
|
||||||
|
/// let positions: Vec<_> = hermite.iter_positions(100).collect();
|
||||||
|
/// ```
|
||||||
|
pub struct Hermite<P: Point> {
|
||||||
|
control_points: Vec<(P, P)>,
|
||||||
|
}
|
||||||
|
impl<P: Point> Hermite<P> {
|
||||||
|
/// Create a new Hermite curve from sets of control points.
|
||||||
|
pub fn new(
|
||||||
|
control_points: impl IntoIterator<Item = P>,
|
||||||
|
tangents: impl IntoIterator<Item = P>,
|
||||||
|
) -> Self {
|
||||||
|
Self {
|
||||||
|
control_points: control_points
|
||||||
|
.into_iter()
|
||||||
|
.zip(tangents.into_iter())
|
||||||
|
.collect(),
|
||||||
|
}
|
||||||
|
}
|
||||||
|
}
|
||||||
|
impl<P: Point> CubicGenerator<P> for Hermite<P> {
|
||||||
|
#[inline]
|
||||||
|
fn to_curve(&self) -> CubicCurve<P> {
|
||||||
|
let char_matrix = [
|
||||||
|
[1., 0., 0., 0.],
|
||||||
|
[0., 1., 0., 0.],
|
||||||
|
[-3., -2., 3., -1.],
|
||||||
|
[2., 1., -2., 1.],
|
||||||
|
];
|
||||||
|
|
||||||
|
let segments = self
|
||||||
|
.control_points
|
||||||
|
.windows(2)
|
||||||
|
.map(|p| {
|
||||||
|
let (p0, v0, p1, v1) = (p[0].0, p[0].1, p[1].0, p[1].1);
|
||||||
|
CubicCurve::coefficients([p0, v0, p1, v1], 1.0, char_matrix)
|
||||||
|
})
|
||||||
|
.collect();
|
||||||
|
|
||||||
|
CubicCurve { segments }
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A spline interpolated continuously across the nearest four control points, with the position of
|
||||||
|
/// the curve specified at every control point and the tangents computed automatically.
|
||||||
|
///
|
||||||
|
/// **Note** the Catmull-Rom spline is a special case of Cardinal spline where the tension is 0.5.
|
||||||
|
///
|
||||||
|
/// ### Interpolation
|
||||||
|
/// The curve passes through every control point.
|
||||||
|
///
|
||||||
|
/// ### Tangency
|
||||||
|
/// Automatically defined at each control point.
|
||||||
|
///
|
||||||
|
/// ### Continuity
|
||||||
|
/// C1 continuous.
|
||||||
|
///
|
||||||
|
/// ### Usage
|
||||||
|
///
|
||||||
|
/// ```
|
||||||
|
/// # use bevy_math::{*, prelude::*};
|
||||||
|
/// let points = [
|
||||||
|
/// vec2(-1.0, -20.0),
|
||||||
|
/// vec2(3.0, 2.0),
|
||||||
|
/// vec2(5.0, 3.0),
|
||||||
|
/// vec2(9.0, 8.0),
|
||||||
|
/// ];
|
||||||
|
/// let cardinal = CardinalSpline::new(0.3, points).to_curve();
|
||||||
|
/// let positions: Vec<_> = cardinal.iter_positions(100).collect();
|
||||||
|
/// ```
|
||||||
|
pub struct CardinalSpline<P: Point> {
|
||||||
|
tension: f32,
|
||||||
|
control_points: Vec<P>,
|
||||||
|
}
|
||||||
|
|
||||||
|
impl<P: Point> CardinalSpline<P> {
|
||||||
|
/// Build a new Cardinal spline.
|
||||||
|
pub fn new(tension: f32, control_points: impl Into<Vec<P>>) -> Self {
|
||||||
|
Self {
|
||||||
|
tension,
|
||||||
|
control_points: control_points.into(),
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Build a new Catmull-Rom spline, the special case of a Cardinal spline where tension = 1/2.
|
||||||
|
pub fn new_catmull_rom(control_points: impl Into<Vec<P>>) -> Self {
|
||||||
|
Self {
|
||||||
|
tension: 0.5,
|
||||||
|
control_points: control_points.into(),
|
||||||
|
}
|
||||||
|
}
|
||||||
|
}
|
||||||
|
impl<P: Point> CubicGenerator<P> for CardinalSpline<P> {
|
||||||
|
#[inline]
|
||||||
|
fn to_curve(&self) -> CubicCurve<P> {
|
||||||
|
let s = self.tension;
|
||||||
|
let char_matrix = [
|
||||||
|
[0., 1., 0., 0.],
|
||||||
|
[-s, 0., s, 0.],
|
||||||
|
[2. * s, s - 3., 3. - 2. * s, -s],
|
||||||
|
[-s, 2. - s, s - 2., s],
|
||||||
|
];
|
||||||
|
|
||||||
|
let segments = self
|
||||||
|
.control_points
|
||||||
|
.windows(4)
|
||||||
|
.map(|p| CubicCurve::coefficients([p[0], p[1], p[2], p[3]], 1.0, char_matrix))
|
||||||
|
.collect();
|
||||||
|
|
||||||
|
CubicCurve { segments }
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A spline interpolated continuously across the nearest four control points. The curve does not
|
||||||
|
/// pass through any of the control points.
|
||||||
|
///
|
||||||
|
/// ### Interpolation
|
||||||
|
/// The curve does not pass through control points.
|
||||||
|
///
|
||||||
|
/// ### Tangency
|
||||||
|
/// Automatically computed based on the position of control points.
|
||||||
|
///
|
||||||
|
/// ### Continuity
|
||||||
|
/// C2 continuous! The acceleration continuity of this spline makes it useful for camera paths.
|
||||||
|
///
|
||||||
|
/// ### Usage
|
||||||
|
///
|
||||||
|
/// ```
|
||||||
|
/// # use bevy_math::{*, prelude::*};
|
||||||
|
/// let points = [
|
||||||
|
/// vec2(-1.0, -20.0),
|
||||||
|
/// vec2(3.0, 2.0),
|
||||||
|
/// vec2(5.0, 3.0),
|
||||||
|
/// vec2(9.0, 8.0),
|
||||||
|
/// ];
|
||||||
|
/// let b_spline = BSpline::new(points).to_curve();
|
||||||
|
/// let positions: Vec<_> = b_spline.iter_positions(100).collect();
|
||||||
|
/// ```
|
||||||
|
pub struct BSpline<P: Point> {
|
||||||
|
control_points: Vec<P>,
|
||||||
|
}
|
||||||
|
impl<P: Point> BSpline<P> {
|
||||||
|
/// Build a new Cardinal spline.
|
||||||
|
pub fn new(control_points: impl Into<Vec<P>>) -> Self {
|
||||||
|
Self {
|
||||||
|
control_points: control_points.into(),
|
||||||
|
}
|
||||||
|
}
|
||||||
|
}
|
||||||
|
impl<P: Point> CubicGenerator<P> for BSpline<P> {
|
||||||
|
#[inline]
|
||||||
|
fn to_curve(&self) -> CubicCurve<P> {
|
||||||
|
let char_matrix = [
|
||||||
|
[1., 4., 1., 0.],
|
||||||
|
[-3., 0., 3., 0.],
|
||||||
|
[3., -6., 3., 0.],
|
||||||
|
[-1., 3., -3., 1.],
|
||||||
|
];
|
||||||
|
|
||||||
|
let segments = self
|
||||||
|
.control_points
|
||||||
|
.windows(4)
|
||||||
|
.map(|p| CubicCurve::coefficients([p[0], p[1], p[2], p[3]], 1.0 / 6.0, char_matrix))
|
||||||
|
.collect();
|
||||||
|
|
||||||
|
CubicCurve { segments }
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Implement this on cubic splines that can generate a curve from their spline parameters.
|
||||||
|
pub trait CubicGenerator<P: Point> {
|
||||||
|
/// Build a [`CubicCurve`] by computing the interpolation coefficients for each curve segment.
|
||||||
|
fn to_curve(&self) -> CubicCurve<P>;
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A segment of a cubic curve, used to hold precomputed coefficients for fast interpolation.
|
||||||
|
///
|
||||||
|
/// Segments can be chained together to form a longer compound curve.
|
||||||
|
#[derive(Clone, Debug, Default, PartialEq)]
|
||||||
|
pub struct CubicSegment<P: Point> {
|
||||||
|
coeff: [P; 4],
|
||||||
|
}
|
||||||
|
|
||||||
|
impl<P: Point> CubicSegment<P> {
|
||||||
|
/// Instantaneous position of a point at parametric value `t`.
|
||||||
|
#[inline]
|
||||||
|
pub fn position(&self, t: f32) -> P {
|
||||||
|
let [a, b, c, d] = self.coeff;
|
||||||
|
a + b * t + c * t.powi(2) + d * t.powi(3)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Instantaneous velocity of a point at parametric value `t`.
|
||||||
|
#[inline]
|
||||||
|
pub fn velocity(&self, t: f32) -> P {
|
||||||
|
let [_, b, c, d] = self.coeff;
|
||||||
|
b + c * 2.0 * t + d * 3.0 * t.powi(2)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Instantaneous acceleration of a point at parametric value `t`.
|
||||||
|
#[inline]
|
||||||
|
pub fn acceleration(&self, t: f32) -> P {
|
||||||
|
let [_, _, c, d] = self.coeff;
|
||||||
|
c * 2.0 + d * 6.0 * t
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// The `CubicSegment<Vec2>` can be used as a 2-dimensional easing curve for animation.
|
||||||
|
///
|
||||||
|
/// The x-axis of the curve is time, and the y-axis is the output value. This struct provides
|
||||||
|
/// methods for extremely fast solves for y given x.
|
||||||
|
impl CubicSegment<Vec2> {
|
||||||
|
/// Construct a cubic Bezier curve for animation easing, with control points `p1` and `p2`. A
|
||||||
|
/// cubic Bezier easing curve has control point `p0` at (0, 0) and `p3` at (1, 1), leaving only
|
||||||
|
/// `p1` and `p2` as the remaining degrees of freedom. The first and last control points are
|
||||||
|
/// fixed to ensure the animation begins at 0, and ends at 1.
|
||||||
|
///
|
||||||
|
/// This is a very common tool for UI animations that accelerate and decelerate smoothly. For
|
||||||
|
/// example, the ubiquitous "ease-in-out" is defined as `(0.25, 0.1), (0.25, 1.0)`.
|
||||||
|
pub fn new_bezier(p1: impl Into<Vec2>, p2: impl Into<Vec2>) -> Self {
|
||||||
|
let (p0, p3) = (Vec2::ZERO, Vec2::ONE);
|
||||||
|
let bezier = Bezier::new([[p0, p1.into(), p2.into(), p3]]).to_curve();
|
||||||
|
bezier.segments[0].clone()
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Maximum allowable error for iterative Bezier solve
|
||||||
|
const MAX_ERROR: f32 = 1e-5;
|
||||||
|
|
||||||
|
/// Maximum number of iterations during Bezier solve
|
||||||
|
const MAX_ITERS: u8 = 8;
|
||||||
|
|
||||||
|
/// Given a `time` within `0..=1`, returns an eased value that follows the cubic curve instead
|
||||||
|
/// of a straight line. This eased result may be outside the range `0..=1`, however it will
|
||||||
|
/// always start at 0 and end at 1: `ease(0) = 0` and `ease(1) = 1`.
|
||||||
|
///
|
||||||
|
/// ```
|
||||||
|
/// # use bevy_math::prelude::*;
|
||||||
|
/// let cubic_bezier = CubicSegment::new_bezier((0.25, 0.1), (0.25, 1.0));
|
||||||
|
/// assert_eq!(cubic_bezier.ease(0.0), 0.0);
|
||||||
|
/// assert_eq!(cubic_bezier.ease(1.0), 1.0);
|
||||||
|
/// ```
|
||||||
|
///
|
||||||
|
/// # How cubic easing works
|
||||||
|
///
|
||||||
|
/// Easing is generally accomplished with the help of "shaping functions". These are curves that
|
||||||
|
/// start at (0,0) and end at (1,1). The x-axis of this plot is the current `time` of the
|
||||||
|
/// animation, from 0 to 1. The y-axis is how far along the animation is, also from 0 to 1. You
|
||||||
|
/// can imagine that if the shaping function is a straight line, there is a 1:1 mapping between
|
||||||
|
/// the `time` and how far along your animation is. If the `time` = 0.5, the animation is
|
||||||
|
/// halfway through. This is known as linear interpolation, and results in objects animating
|
||||||
|
/// with a constant velocity, and no smooth acceleration or deceleration at the start or end.
|
||||||
|
///
|
||||||
|
/// ```text
|
||||||
|
/// y
|
||||||
|
/// │ ●
|
||||||
|
/// │ ⬈
|
||||||
|
/// │ ⬈
|
||||||
|
/// │ ⬈
|
||||||
|
/// │ ⬈
|
||||||
|
/// ●─────────── x (time)
|
||||||
|
/// ```
|
||||||
|
///
|
||||||
|
/// Using cubic Beziers, we have a curve that starts at (0,0), ends at (1,1), and follows a path
|
||||||
|
/// determined by the two remaining control points (handles). These handles allow us to define a
|
||||||
|
/// smooth curve. As `time` (x-axis) progresses, we now follow the curve, and use the `y` value
|
||||||
|
/// to determine how far along the animation is.
|
||||||
|
///
|
||||||
|
/// ```text
|
||||||
|
/// y
|
||||||
|
/// ⬈➔●
|
||||||
|
/// │ ⬈
|
||||||
|
/// │ ↑
|
||||||
|
/// │ ↑
|
||||||
|
/// │ ⬈
|
||||||
|
/// ●➔⬈───────── x (time)
|
||||||
|
/// ```
|
||||||
|
///
|
||||||
|
/// To accomplish this, we need to be able to find the position `y` on a curve, given the `x`
|
||||||
|
/// value. Cubic curves are implicit parametric functions like B(t) = (x,y). To find `y`, we
|
||||||
|
/// first solve for `t` that corresponds to the given `x` (`time`). We use the Newton-Raphson
|
||||||
|
/// root-finding method to quickly find a value of `t` that is very near the desired value of
|
||||||
|
/// `x`. Once we have this we can easily plug that `t` into our curve's `position` function, to
|
||||||
|
/// find the `y` component, which is how far along our animation should be. In other words:
|
||||||
|
///
|
||||||
|
/// > Given `time` in `0..=1`
|
||||||
|
///
|
||||||
|
/// > Use Newton's method to find a value of `t` that results in B(t) = (x,y) where `x == time`
|
||||||
|
///
|
||||||
|
/// > Once a solution is found, use the resulting `y` value as the final result
|
||||||
|
#[inline]
|
||||||
|
pub fn ease(&self, time: f32) -> f32 {
|
||||||
|
let x = time.clamp(0.0, 1.0);
|
||||||
|
self.find_y_given_x(x)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Find the `y` value of the curve at the given `x` value using the Newton-Raphson method.
|
||||||
|
#[inline]
|
||||||
|
fn find_y_given_x(&self, x: f32) -> f32 {
|
||||||
|
let mut t_guess = x;
|
||||||
|
let mut pos_guess = Vec2::ZERO;
|
||||||
|
for _ in 0..Self::MAX_ITERS {
|
||||||
|
pos_guess = self.position(t_guess);
|
||||||
|
let error = pos_guess.x - x;
|
||||||
|
if error.abs() <= Self::MAX_ERROR {
|
||||||
|
break;
|
||||||
|
}
|
||||||
|
// Using Newton's method, use the tangent line to estimate a better guess value.
|
||||||
|
let slope = self.velocity(t_guess).x; // dx/dt
|
||||||
|
t_guess -= error / slope;
|
||||||
|
}
|
||||||
|
pos_guess.y
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A collection of [`CubicSegment`]s chained into a curve.
|
||||||
|
#[derive(Clone, Debug, Default, PartialEq)]
|
||||||
|
pub struct CubicCurve<P: Point> {
|
||||||
|
segments: Vec<CubicSegment<P>>,
|
||||||
|
}
|
||||||
|
|
||||||
|
impl<P: Point> CubicCurve<P> {
|
||||||
|
/// Compute the position of a point on the cubic curve at the parametric value `t`.
|
||||||
|
///
|
||||||
|
/// Note that `t` varies from `0..=(n_points - 3)`.
|
||||||
|
#[inline]
|
||||||
|
pub fn position(&self, t: f32) -> P {
|
||||||
|
let (segment, t) = self.segment(t);
|
||||||
|
segment.position(t)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Compute the first derivative with respect to t at `t`. This is the instantaneous velocity of
|
||||||
|
/// a point on the cubic curve at `t`.
|
||||||
|
///
|
||||||
|
/// Note that `t` varies from `0..=(n_points - 3)`.
|
||||||
|
#[inline]
|
||||||
|
pub fn velocity(&self, t: f32) -> P {
|
||||||
|
let (segment, t) = self.segment(t);
|
||||||
|
segment.velocity(t)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Compute the second derivative with respect to t at `t`. This is the instantaneous
|
||||||
|
/// acceleration of a point on the cubic curve at `t`.
|
||||||
|
///
|
||||||
|
/// Note that `t` varies from `0..=(n_points - 3)`.
|
||||||
|
#[inline]
|
||||||
|
pub fn acceleration(&self, t: f32) -> P {
|
||||||
|
let (segment, t) = self.segment(t);
|
||||||
|
segment.acceleration(t)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A flexible iterator used to sample curves with arbitrary functions.
|
||||||
|
///
|
||||||
|
/// This splits the curve into `subdivisions` of evenly spaced `t` values across the
|
||||||
|
/// length of the curve from start (t = 0) to end (t = 1), returning an iterator that evaluates
|
||||||
|
/// the curve with the supplied `sample_function` at each `t`.
|
||||||
|
///
|
||||||
|
/// Given `subdivisions = 2`, this will split the curve into two lines, or three points, and
|
||||||
|
/// return an iterator over those three points, one at the start, middle, and end.
|
||||||
|
#[inline]
|
||||||
|
pub fn iter_samples(
|
||||||
|
&self,
|
||||||
|
subdivisions: usize,
|
||||||
|
sample_function: fn(&Self, f32) -> P,
|
||||||
|
) -> impl Iterator<Item = P> + '_ {
|
||||||
|
(0..subdivisions).map(move |i| {
|
||||||
|
let segments = self.segments.len() as f32;
|
||||||
|
let t = i as f32 / subdivisions as f32 * segments;
|
||||||
|
sample_function(self, t)
|
||||||
|
})
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Iterate over the curve split into `subdivisions`, sampling the position at each step.
|
||||||
|
pub fn iter_positions(&self, subdivisions: usize) -> impl Iterator<Item = P> + '_ {
|
||||||
|
self.iter_samples(subdivisions, Self::position)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Iterate over the curve split into `subdivisions`, sampling the velocity at each step.
|
||||||
|
pub fn iter_velocities(&self, subdivisions: usize) -> impl Iterator<Item = P> + '_ {
|
||||||
|
self.iter_samples(subdivisions, Self::velocity)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Iterate over the curve split into `subdivisions`, sampling the acceleration at each step.
|
||||||
|
pub fn iter_accelerations(&self, subdivisions: usize) -> impl Iterator<Item = P> + '_ {
|
||||||
|
self.iter_samples(subdivisions, Self::acceleration)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Returns the [`CubicSegment`] and local `t` value given a spline's global `t` value.
|
||||||
|
#[inline]
|
||||||
|
fn segment(&self, t: f32) -> (&CubicSegment<P>, f32) {
|
||||||
|
if self.segments.len() == 1 {
|
||||||
|
(&self.segments[0], t)
|
||||||
|
} else {
|
||||||
|
let i = (t.floor() as usize).clamp(0, self.segments.len() - 1);
|
||||||
|
(&self.segments[i], t - i as f32)
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
#[inline]
|
||||||
|
fn coefficients(p: [P; 4], multiplier: f32, char_matrix: [[f32; 4]; 4]) -> CubicSegment<P> {
|
||||||
|
let [c0, c1, c2, c3] = char_matrix;
|
||||||
|
// These are the polynomial coefficients, computed by multiplying the characteristic
|
||||||
|
// matrix by the point matrix.
|
||||||
|
let mut coeff = [
|
||||||
|
p[0] * c0[0] + p[1] * c0[1] + p[2] * c0[2] + p[3] * c0[3],
|
||||||
|
p[0] * c1[0] + p[1] * c1[1] + p[2] * c1[2] + p[3] * c1[3],
|
||||||
|
p[0] * c2[0] + p[1] * c2[1] + p[2] * c2[2] + p[3] * c2[3],
|
||||||
|
p[0] * c3[0] + p[1] * c3[1] + p[2] * c3[2] + p[3] * c3[3],
|
||||||
|
];
|
||||||
|
coeff.iter_mut().for_each(|c| *c = *c * multiplier);
|
||||||
|
CubicSegment { coeff }
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
#[cfg(test)]
|
||||||
|
mod tests {
|
||||||
|
use glam::{vec2, Vec2};
|
||||||
|
|
||||||
|
use crate::cubic_splines::{Bezier, CubicGenerator, CubicSegment};
|
||||||
|
|
||||||
|
/// How close two floats can be and still be considered equal
|
||||||
|
const FLOAT_EQ: f32 = 1e-5;
|
||||||
|
|
||||||
|
/// Sweep along the full length of a 3D cubic Bezier, and manually check the position.
|
||||||
|
#[test]
|
||||||
|
fn cubic() {
|
||||||
|
const N_SAMPLES: usize = 1000;
|
||||||
|
let points = [[
|
||||||
|
vec2(-1.0, -20.0),
|
||||||
|
vec2(3.0, 2.0),
|
||||||
|
vec2(5.0, 3.0),
|
||||||
|
vec2(9.0, 8.0),
|
||||||
|
]];
|
||||||
|
let bezier = Bezier::new(points).to_curve();
|
||||||
|
for i in 0..=N_SAMPLES {
|
||||||
|
let t = i as f32 / N_SAMPLES as f32; // Check along entire length
|
||||||
|
assert!(bezier.position(t).distance(cubic_manual(t, points[0])) <= FLOAT_EQ);
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Manual, hardcoded function for computing the position along a cubic bezier.
|
||||||
|
fn cubic_manual(t: f32, points: [Vec2; 4]) -> Vec2 {
|
||||||
|
let p = points;
|
||||||
|
p[0] * (1.0 - t).powi(3)
|
||||||
|
+ 3.0 * p[1] * t * (1.0 - t).powi(2)
|
||||||
|
+ 3.0 * p[2] * t.powi(2) * (1.0 - t)
|
||||||
|
+ p[3] * t.powi(3)
|
||||||
|
}
|
||||||
|
|
||||||
|
/// Basic cubic Bezier easing test to verify the shape of the curve.
|
||||||
|
#[test]
|
||||||
|
fn easing_simple() {
|
||||||
|
// A curve similar to ease-in-out, but symmetric
|
||||||
|
let bezier = CubicSegment::new_bezier([1.0, 0.0], [0.0, 1.0]);
|
||||||
|
assert_eq!(bezier.ease(0.0), 0.0);
|
||||||
|
assert!(bezier.ease(0.2) < 0.2); // tests curve
|
||||||
|
assert_eq!(bezier.ease(0.5), 0.5); // true due to symmetry
|
||||||
|
assert!(bezier.ease(0.8) > 0.8); // tests curve
|
||||||
|
assert_eq!(bezier.ease(1.0), 1.0);
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A curve that forms an upside-down "U", that should extend below 0.0. Useful for animations
|
||||||
|
/// that go beyond the start and end positions, e.g. bouncing.
|
||||||
|
#[test]
|
||||||
|
fn easing_overshoot() {
|
||||||
|
// A curve that forms an upside-down "U", that should extend above 1.0
|
||||||
|
let bezier = CubicSegment::new_bezier([0.0, 2.0], [1.0, 2.0]);
|
||||||
|
assert_eq!(bezier.ease(0.0), 0.0);
|
||||||
|
assert!(bezier.ease(0.5) > 1.5);
|
||||||
|
assert_eq!(bezier.ease(1.0), 1.0);
|
||||||
|
}
|
||||||
|
|
||||||
|
/// A curve that forms a "U", that should extend below 0.0. Useful for animations that go beyond
|
||||||
|
/// the start and end positions, e.g. bouncing.
|
||||||
|
#[test]
|
||||||
|
fn easing_undershoot() {
|
||||||
|
let bezier = CubicSegment::new_bezier([0.0, -2.0], [1.0, -2.0]);
|
||||||
|
assert_eq!(bezier.ease(0.0), 0.0);
|
||||||
|
assert!(bezier.ease(0.5) < -0.5);
|
||||||
|
assert_eq!(bezier.ease(1.0), 1.0);
|
||||||
|
}
|
||||||
|
}
|
||||||
@ -6,14 +6,10 @@
|
|||||||
|
|
||||||
#![warn(missing_docs)]
|
#![warn(missing_docs)]
|
||||||
|
|
||||||
mod bezier;
|
pub mod cubic_splines;
|
||||||
mod ray;
|
mod ray;
|
||||||
mod rect;
|
mod rect;
|
||||||
|
|
||||||
pub use bezier::{
|
|
||||||
generic as generic_bezier, Bezier, CubicBezier2d, CubicBezier3d, CubicBezierEasing,
|
|
||||||
QuadraticBezier2d, QuadraticBezier3d,
|
|
||||||
};
|
|
||||||
pub use ray::Ray;
|
pub use ray::Ray;
|
||||||
pub use rect::Rect;
|
pub use rect::Rect;
|
||||||
|
|
||||||
@ -21,9 +17,9 @@ pub use rect::Rect;
|
|||||||
pub mod prelude {
|
pub mod prelude {
|
||||||
#[doc(hidden)]
|
#[doc(hidden)]
|
||||||
pub use crate::{
|
pub use crate::{
|
||||||
BVec2, BVec3, BVec4, Bezier, CubicBezier2d, CubicBezier3d, CubicBezierEasing, EulerRot,
|
cubic_splines::{BSpline, Bezier, CardinalSpline, CubicGenerator, CubicSegment, Hermite},
|
||||||
IVec2, IVec3, IVec4, Mat2, Mat3, Mat4, QuadraticBezier2d, QuadraticBezier3d, Quat, Ray,
|
BVec2, BVec3, BVec4, EulerRot, IVec2, IVec3, IVec4, Mat2, Mat3, Mat4, Quat, Ray, Rect,
|
||||||
Rect, UVec2, UVec3, UVec4, Vec2, Vec3, Vec4,
|
UVec2, UVec3, UVec4, Vec2, Vec3, Vec4,
|
||||||
};
|
};
|
||||||
}
|
}
|
||||||
|
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user