Improve cubic segment bezier functionality (#17645)

# Objective

- Fixes #17642

## Solution

- Implemented method `new_bezier(points: [P; 4]) -> Self` for
`CubicSegment<P>`
- Old implementation of `new_bezier` is now `new_bezier_easing(p1: impl
Into<Vec2>, p2: impl Into<Vec2>) -> Self` (**breaking change**)
- ~~added method `new_bezier_with_anchor`, which can make a bezier curve
between two points with one control anchor~~
- added methods `iter_positions`, `iter_velocities`,
`iter_accelerations`, the same as in `CubicCurve` (**copied code,
potentially can be reduced)**
- bezier creation logic is moved from `CubicCurve` to `CubicSegment`,
removing the unneeded allocation

## Testing

- Did you test these changes? If so, how?
  - Run tests inside `crates/bevy_math/`
  - Tested the functionality in my project
- Are there any parts that need more testing?
  - Did not run `cargo test` on the whole bevy directory because of OOM
- Performance improvements are expected when creating `CubicCurve` with
`new_bezier` and `new_bezier_easing`, but not tested
- How can other people (reviewers) test your changes? Is there anything
specific they need to know?
  - Use in any code that works created `CubicCurve::new_bezier`
- If relevant, what platforms did you test these changes on, and are
there any important ones you can't test?
  - I don't think relevant

---

## Showcase

```rust
// Imagine a car goes towards a local target

// Create a simple `CubicSegment`, without using heap
let planned_path = CubicSegment::new_bezier([
    car_pos,
    car_pos + car_dir * turn_radius,
    target_point - target_dir * turn_radius,
    target_point,
]);

// Check if the planned path itersect other entities
for pos in planned_path.iter_positions(8) {
   // do some collision checks
}
```

## Migration Guide

> This section is optional. If there are no breaking changes, you can
delete this section.

- Replace `CubicCurve::new_bezier` with `CubicCurve::new_bezier_easing`

benches/benches/bevy_math/bezier.rs

@@ -8,7 +8,10 @@ use criterion::{
 criterion_group!(benches, segment_ease, curve_position, curve_iter_positions);
 
 fn segment_ease(c: &mut Criterion) {
-    let segment = black_box(CubicSegment::new_bezier(vec2(0.25, 0.1), vec2(0.25, 1.0)));
+    let segment = black_box(CubicSegment::new_bezier_easing(
+        vec2(0.25, 0.1),
+        vec2(0.25, 1.0),
+    ));
 
     c.bench_function(bench!("segment_ease"), |b| {
         let mut t = 0;

crates/bevy_math/src/cubic_splines/mod.rs

@@ -15,7 +15,7 @@ use {alloc::vec, alloc::vec::Vec, core::iter::once, itertools::Itertools};
 /// A spline composed of a single cubic Bezier curve.
 ///
 /// Useful for user-drawn curves with local control, or animation easing. See
-/// [`CubicSegment::new_bezier`] for use in easing.
+/// [`CubicSegment::new_bezier_easing`] for use in easing.
 ///
 /// ### Interpolation
 ///
@@ -73,20 +73,10 @@ impl<P: VectorSpace> CubicGenerator<P> for CubicBezier<P> {
 
     #[inline]
     fn to_curve(&self) -> Result<CubicCurve<P>, Self::Error> {
-        // A derivation for this matrix can be found in "General Matrix Representations for B-splines" by Kaihuai Qin.
-        // <https://xiaoxingchen.github.io/2020/03/02/bspline_in_so3/general_matrix_representation_for_bsplines.pdf>
-        // See section 4.2 and equation 11.
-        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| CubicSegment::coefficients(*p, char_matrix))
+            .map(|p| CubicSegment::new_bezier(*p))
             .collect_vec();
 
         if segments.is_empty() {
@@ -993,14 +983,21 @@ impl<P: VectorSpace> CubicSegment<P> {
         c * 2.0 + d * 6.0 * t
     }
 
+    /// Creates a cubic segment from four points, representing a Bezier curve.
+    pub fn new_bezier(points: [P; 4]) -> Self {
+        // A derivation for this matrix can be found in "General Matrix Representations for B-splines" by Kaihuai Qin.
+        // <https://xiaoxingchen.github.io/2020/03/02/bspline_in_so3/general_matrix_representation_for_bsplines.pdf>
+        // See section 4.2 and equation 11.
+        let char_matrix = [
+            [1., 0., 0., 0.],
+            [-3., 3., 0., 0.],
+            [3., -6., 3., 0.],
+            [-1., 3., -3., 1.],
+        ];
+        Self::coefficients(points, char_matrix)
+    }
+
     /// Calculate polynomial coefficients for the cubic curve using a characteristic matrix.
-    #[cfg_attr(
-        not(feature = "alloc"),
-        expect(
-            dead_code,
-            reason = "Method only used when `alloc` feature is enabled."
-        )
-    )]
     #[inline]
     fn coefficients(p: [P; 4], char_matrix: [[f32; 4]; 4]) -> Self {
         let [c0, c1, c2, c3] = char_matrix;
@@ -1014,6 +1011,46 @@ impl<P: VectorSpace> CubicSegment<P> {
         ];
         Self { coeff }
     }
+
+    /// 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 = n), where `n = self.segment_count()`,
+    /// returning an iterator evaluating the curve with the supplied `sample_function` at each `t`.
+    ///
+    /// For `subdivisions = 2`, this will split the curve into two lines, or three points, and
+    /// return an iterator with 3 items, the three points, one at the start, middle, and end.
+    #[inline]
+    pub fn iter_samples<'a, 'b: 'a>(
+        &'b self,
+        subdivisions: usize,
+        mut sample_function: impl FnMut(&Self, f32) -> P + 'a,
+    ) -> impl Iterator<Item = P> + 'a {
+        self.iter_uniformly(subdivisions)
+            .map(move |t| sample_function(self, t))
+    }
+
+    /// An iterator that returns values of `t` uniformly spaced over `0..=subdivisions`.
+    #[inline]
+    fn iter_uniformly(&self, subdivisions: usize) -> impl Iterator<Item = f32> {
+        let step = 1.0 / subdivisions as f32;
+        (0..=subdivisions).map(move |i| i as f32 * step)
+    }
+
+    /// 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)
+    }
 }
 
 /// The `CubicSegment<Vec2>` can be used as a 2-dimensional easing curve for animation.
@@ -1029,12 +1066,9 @@ impl CubicSegment<Vec2> {
     /// 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)`.
     #[cfg(feature = "alloc")]
-    pub fn new_bezier(p1: impl Into<Vec2>, p2: impl Into<Vec2>) -> Self {
+    pub fn new_bezier_easing(p1: impl Into<Vec2>, p2: impl Into<Vec2>) -> Self {
         let (p0, p3) = (Vec2::ZERO, Vec2::ONE);
-        let bezier = CubicBezier::new([[p0, p1.into(), p2.into(), p3]])
+        Self::new_bezier([p0, p1.into(), p2.into(), p3])
-            .to_curve()
-            .unwrap(); // Succeeds because resulting curve is guaranteed to have one segment
-        bezier.segments[0]
     }
 
     /// Maximum allowable error for iterative Bezier solve
@@ -1051,7 +1085,7 @@ impl CubicSegment<Vec2> {
     /// # use bevy_math::prelude::*;
     /// # #[cfg(feature = "alloc")]
     /// # {
-    /// let cubic_bezier = CubicSegment::new_bezier((0.25, 0.1), (0.25, 1.0));
+    /// let cubic_bezier = CubicSegment::new_bezier_easing((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);
     /// # }
@@ -1656,7 +1690,7 @@ mod tests {
     #[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]);
+        let bezier = CubicSegment::new_bezier_easing([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
@@ -1669,7 +1703,7 @@ mod tests {
     #[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]);
+        let bezier = CubicSegment::new_bezier_easing([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);
@@ -1679,7 +1713,7 @@ mod tests {
     /// 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]);
+        let bezier = CubicSegment::new_bezier_easing([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);