Add Beziers to bevy_math (#7653)

# Objective

- Adds foundational math for Bezier curves, useful for UI/2D/3D animation and smooth paths.

https://user-images.githubusercontent.com/2632925/218883143-e138f994-1795-40da-8c59-21d779666991.mp4

## Solution

- Adds the generic `Bezier` type, and a `Point` trait. The `Point` trait allows us to use control points of any dimension, as long as they support vector math. I've implemented it for `f32`(1D), `Vec2`(2D), and `Vec3`/`Vec3A`(3D).
- Adds `CubicBezierEasing` on top of `Bezier` with the addition of an implementation of cubic Bezier easing, which is a foundational tool for UI animation.
  - This involves solving for $t$ in the parametric Bezier function $B(t)$ using the Newton-Raphson method to find a value with error $\leq$ 1e-7, capped at 8 iterations.
- Added type aliases for common Bezier curves: `CubicBezier2d`, `CubicBezier3d`, `QuadraticBezier2d`, and `QuadraticBezier3d`. These types use `Vec3A` to represent control points, as this was found to have an 80-90% speedup over using `Vec3`.
- Benchmarking shows quadratic/cubic Bezier evaluations $B(t)$ take \~1.8/2.4ns respectively. Easing, which requires an iterative solve takes \~50ns for cubic Beziers. 

---

## Changelog

- Added `CubicBezier2d`, `CubicBezier3d`, `QuadraticBezier2d`, and `QuadraticBezier3d` types with methods for sampling position, velocity, and acceleration. The generic `Bezier` type is also available, and generic over any degree of Bezier curve.
- Added `CubicBezierEasing`, with additional methods to allow for smooth easing animations.

benches/Cargo.toml

@@ -16,6 +16,7 @@ bevy_ecs = { path = "../crates/bevy_ecs" }
 bevy_reflect = { path = "../crates/bevy_reflect" }
 bevy_tasks = { path = "../crates/bevy_tasks" }
 bevy_utils = { path = "../crates/bevy_utils" }
+bevy_math = { path = "../crates/bevy_math" }
 
 [profile.release]
 opt-level = 3
@@ -50,3 +51,8 @@ harness = false
 name = "iter"
 path = "benches/bevy_tasks/iter.rs"
 harness = false
+
+[[bench]]
+name = "bezier"
+path = "benches/bevy_math/bezier.rs"
+harness = false

benches/benches/bevy_math/bezier.rs

@@ -0,0 +1,129 @@
+use criterion::{black_box, criterion_group, criterion_main, Criterion};
+
+use bevy_math::*;
+
+fn easing(c: &mut Criterion) {
+    let cubic_bezier = CubicBezierEasing::new(vec2(0.25, 0.1), vec2(0.25, 1.0));
+    c.bench_function("easing_1000", |b| {
+        b.iter(|| {
+            (0..1000).map(|i| i as f32 / 1000.0).for_each(|t| {
+                cubic_bezier.ease(black_box(t));
+            })
+        });
+    });
+}
+
+fn fifteen_degree(c: &mut Criterion) {
+    let bezier = Bezier::<Vec3A, 16>::new([
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+    ]);
+    c.bench_function("fifteen_degree_position", |b| {
+        b.iter(|| bezier.position(black_box(0.5)));
+    });
+}
+
+fn quadratic_2d(c: &mut Criterion) {
+    let bezier = QuadraticBezier2d::new([[0.0, 0.0], [0.0, 1.0], [1.0, 1.0]]);
+    c.bench_function("quadratic_position_Vec2", |b| {
+        b.iter(|| bezier.position(black_box(0.5)));
+    });
+}
+
+fn quadratic(c: &mut Criterion) {
+    let bezier = QuadraticBezier3d::new([[0.0, 0.0, 0.0], [0.0, 1.0, 0.0], [1.0, 1.0, 1.0]]);
+    c.bench_function("quadratic_position_Vec3A", |b| {
+        b.iter(|| bezier.position(black_box(0.5)));
+    });
+}
+
+fn quadratic_vec3(c: &mut Criterion) {
+    let bezier = Bezier::<Vec3, 3>::new([[0.0, 0.0, 0.0], [0.0, 1.0, 0.0], [1.0, 1.0, 1.0]]);
+    c.bench_function("quadratic_position_Vec3", |b| {
+        b.iter(|| bezier.position(black_box(0.5)));
+    });
+}
+
+fn cubic_2d(c: &mut Criterion) {
+    let bezier = CubicBezier2d::new([[0.0, 0.0], [0.0, 1.0], [1.0, 0.0], [1.0, 1.0]]);
+    c.bench_function("cubic_position_Vec2", |b| {
+        b.iter(|| bezier.position(black_box(0.5)));
+    });
+}
+
+fn cubic(c: &mut Criterion) {
+    let bezier = CubicBezier3d::new([
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+    ]);
+    c.bench_function("cubic_position_Vec3A", |b| {
+        b.iter(|| bezier.position(black_box(0.5)));
+    });
+}
+
+fn cubic_vec3(c: &mut Criterion) {
+    let bezier = Bezier::<Vec3, 4>::new([
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+    ]);
+    c.bench_function("cubic_position_Vec3", |b| {
+        b.iter(|| bezier.position(black_box(0.5)));
+    });
+}
+
+fn build_pos_cubic(c: &mut Criterion) {
+    let bezier = CubicBezier3d::new([
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+    ]);
+    c.bench_function("build_pos_cubic_100_points", |b| {
+        b.iter(|| bezier.iter_positions(black_box(100)).collect::<Vec<_>>());
+    });
+}
+
+fn build_accel_cubic(c: &mut Criterion) {
+    let bezier = CubicBezier3d::new([
+        [0.0, 0.0, 0.0],
+        [0.0, 1.0, 0.0],
+        [1.0, 0.0, 0.0],
+        [1.0, 1.0, 1.0],
+    ]);
+    c.bench_function("build_accel_cubic_100_points", |b| {
+        b.iter(|| bezier.iter_positions(black_box(100)).collect::<Vec<_>>());
+    });
+}
+
+criterion_group!(
+    benches,
+    easing,
+    fifteen_degree,
+    quadratic_2d,
+    quadratic,
+    quadratic_vec3,
+    cubic_2d,
+    cubic,
+    cubic_vec3,
+    build_pos_cubic,
+    build_accel_cubic,
+);
+criterion_main!(benches);