a788e31ad5
# Objective [Rust 1.72.0](https://blog.rust-lang.org/2023/08/24/Rust-1.72.0.html) is now stable. # Notes - `let-else` formatting has arrived! - I chose to allow `explicit_iter_loop` due to https://github.com/rust-lang/rust-clippy/issues/11074. We didn't hit any of the false positives that prevent compilation, but fixing this did produce a lot of the "symbol soup" mentioned, e.g. `for image in &mut *image_events {`. Happy to undo this if there's consensus the other way. --------- Co-authored-by: François <mockersf@gmail.com>
617 lines
21 KiB
Rust
617 lines
21 KiB
Rust
//! Provides types for building cubic splines for rendering curves and use with animation easing.
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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 math ops needed for cubic spline
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/// interpolation.
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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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+ Add<f32, 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 {}
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impl Point for Vec3A {}
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impl Point for Vec2 {}
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impl Point for f32 {}
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/// A spline composed of a series of cubic Bezier curves.
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///
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/// Useful for user-drawn curves with local control, or animation easing. See
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/// [`CubicSegment::new_bezier`] for use in easing.
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///
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/// ### Interpolation
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/// The curve only passes through the first and last control point in each set of four points.
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///
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/// ### Tangency
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/// Manually defined by the two intermediate control points within each set of four points.
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///
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/// ### Continuity
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/// At minimum C0 continuous, up to C2. Continuity greater than C0 can result in a loss of local
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/// control over the spline due to the curvature constraints.
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///
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/// ### Usage
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///
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/// ```
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/// # use bevy_math::{*, prelude::*};
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/// let points = [[
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/// vec2(-1.0, -20.0),
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/// vec2(3.0, 2.0),
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/// vec2(5.0, 3.0),
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/// vec2(9.0, 8.0),
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/// ]];
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/// let bezier = Bezier::new(points).to_curve();
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/// let positions: Vec<_> = bezier.iter_positions(100).collect();
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/// ```
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pub struct Bezier<P: Point> {
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control_points: Vec<[P; 4]>,
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}
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impl<P: Point> Bezier<P> {
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/// Create a new Bezier curve from sets of control points.
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pub fn new(control_points: impl Into<Vec<[P; 4]>>) -> Self {
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Self {
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control_points: control_points.into(),
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}
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}
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}
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impl<P: Point> CubicGenerator<P> for Bezier<P> {
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#[inline]
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fn to_curve(&self) -> CubicCurve<P> {
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let char_matrix = [
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[1., 0., 0., 0.],
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[-3., 3., 0., 0.],
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[3., -6., 3., 0.],
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[-1., 3., -3., 1.],
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];
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let segments = self
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.control_points
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.iter()
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.map(|p| CubicCurve::coefficients(*p, 1.0, char_matrix))
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.collect();
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CubicCurve { segments }
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}
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}
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/// A spline interpolated continuously between the nearest two control points, with the position and
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/// velocity of the curve specified at both control points. This curve passes through all control
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/// points, with the specified velocity which includes direction and parametric speed.
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///
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/// Useful for smooth interpolation when you know the position and velocity at two points in time,
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/// such as network prediction.
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///
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/// ### Interpolation
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/// The curve passes through every control point.
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///
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/// ### Tangency
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/// Explicitly defined at each control point.
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///
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/// ### Continuity
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/// At minimum C0 continuous, up to C1.
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///
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/// ### Usage
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///
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/// ```
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/// # use bevy_math::{*, prelude::*};
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/// let points = [
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/// vec2(-1.0, -20.0),
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/// vec2(3.0, 2.0),
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/// vec2(5.0, 3.0),
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/// vec2(9.0, 8.0),
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/// ];
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/// let tangents = [
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/// vec2(0.0, 1.0),
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/// vec2(0.0, 1.0),
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/// vec2(0.0, 1.0),
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/// vec2(0.0, 1.0),
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/// ];
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/// let hermite = Hermite::new(points, tangents).to_curve();
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/// let positions: Vec<_> = hermite.iter_positions(100).collect();
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/// ```
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pub struct Hermite<P: Point> {
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control_points: Vec<(P, P)>,
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}
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impl<P: Point> Hermite<P> {
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/// Create a new Hermite curve from sets of control points.
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pub fn new(
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control_points: impl IntoIterator<Item = P>,
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tangents: impl IntoIterator<Item = P>,
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) -> Self {
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Self {
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control_points: control_points.into_iter().zip(tangents).collect(),
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}
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}
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}
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impl<P: Point> CubicGenerator<P> for Hermite<P> {
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#[inline]
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fn to_curve(&self) -> CubicCurve<P> {
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let char_matrix = [
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[1., 0., 0., 0.],
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[0., 1., 0., 0.],
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[-3., -2., 3., -1.],
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[2., 1., -2., 1.],
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];
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let segments = self
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.control_points
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.windows(2)
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.map(|p| {
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let (p0, v0, p1, v1) = (p[0].0, p[0].1, p[1].0, p[1].1);
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CubicCurve::coefficients([p0, v0, p1, v1], 1.0, char_matrix)
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})
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.collect();
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CubicCurve { segments }
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}
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}
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/// A spline interpolated continuously across the nearest four control points, with the position of
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/// the curve specified at every control point and the tangents computed automatically.
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///
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/// **Note** the Catmull-Rom spline is a special case of Cardinal spline where the tension is 0.5.
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///
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/// ### Interpolation
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/// The curve passes through every control point.
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///
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/// ### Tangency
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/// Automatically defined at each control point.
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///
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/// ### Continuity
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/// C1 continuous.
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///
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/// ### Usage
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///
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/// ```
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/// # use bevy_math::{*, prelude::*};
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/// let points = [
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/// vec2(-1.0, -20.0),
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/// vec2(3.0, 2.0),
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/// vec2(5.0, 3.0),
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/// vec2(9.0, 8.0),
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/// ];
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/// let cardinal = CardinalSpline::new(0.3, points).to_curve();
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/// let positions: Vec<_> = cardinal.iter_positions(100).collect();
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/// ```
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pub struct CardinalSpline<P: Point> {
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tension: f32,
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control_points: Vec<P>,
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}
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impl<P: Point> CardinalSpline<P> {
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/// Build a new Cardinal spline.
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pub fn new(tension: f32, control_points: impl Into<Vec<P>>) -> Self {
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Self {
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tension,
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control_points: control_points.into(),
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}
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}
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/// Build a new Catmull-Rom spline, the special case of a Cardinal spline where tension = 1/2.
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pub fn new_catmull_rom(control_points: impl Into<Vec<P>>) -> Self {
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Self {
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tension: 0.5,
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control_points: control_points.into(),
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}
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}
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}
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impl<P: Point> CubicGenerator<P> for CardinalSpline<P> {
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#[inline]
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fn to_curve(&self) -> CubicCurve<P> {
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let s = self.tension;
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let char_matrix = [
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[0., 1., 0., 0.],
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[-s, 0., s, 0.],
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[2. * s, s - 3., 3. - 2. * s, -s],
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[-s, 2. - s, s - 2., s],
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];
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let segments = self
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.control_points
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.windows(4)
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.map(|p| CubicCurve::coefficients([p[0], p[1], p[2], p[3]], 1.0, char_matrix))
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.collect();
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CubicCurve { segments }
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}
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}
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/// A spline interpolated continuously across the nearest four control points. The curve does not
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/// pass through any of the control points.
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///
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/// ### Interpolation
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/// The curve does not pass through control points.
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///
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/// ### Tangency
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/// Automatically computed based on the position of control points.
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///
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/// ### Continuity
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/// C2 continuous! The acceleration continuity of this spline makes it useful for camera paths.
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///
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/// ### Usage
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///
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/// ```
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/// # use bevy_math::{*, prelude::*};
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/// let points = [
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/// vec2(-1.0, -20.0),
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/// vec2(3.0, 2.0),
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/// vec2(5.0, 3.0),
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/// vec2(9.0, 8.0),
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/// ];
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/// let b_spline = BSpline::new(points).to_curve();
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/// let positions: Vec<_> = b_spline.iter_positions(100).collect();
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/// ```
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pub struct BSpline<P: Point> {
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control_points: Vec<P>,
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}
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impl<P: Point> BSpline<P> {
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/// Build a new Cardinal spline.
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pub fn new(control_points: impl Into<Vec<P>>) -> Self {
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Self {
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control_points: control_points.into(),
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}
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}
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}
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impl<P: Point> CubicGenerator<P> for BSpline<P> {
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#[inline]
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fn to_curve(&self) -> CubicCurve<P> {
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let char_matrix = [
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[1., 4., 1., 0.],
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[-3., 0., 3., 0.],
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[3., -6., 3., 0.],
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[-1., 3., -3., 1.],
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];
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let segments = self
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.control_points
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.windows(4)
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.map(|p| CubicCurve::coefficients([p[0], p[1], p[2], p[3]], 1.0 / 6.0, char_matrix))
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.collect();
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CubicCurve { segments }
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}
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}
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/// Implement this on cubic splines that can generate a curve from their spline parameters.
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pub trait CubicGenerator<P: Point> {
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/// Build a [`CubicCurve`] by computing the interpolation coefficients for each curve segment.
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fn to_curve(&self) -> CubicCurve<P>;
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}
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/// A segment of a cubic curve, used to hold precomputed coefficients for fast interpolation.
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///
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/// Segments can be chained together to form a longer compound curve.
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#[derive(Clone, Debug, Default, PartialEq)]
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pub struct CubicSegment<P: Point> {
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coeff: [P; 4],
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}
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impl<P: Point> CubicSegment<P> {
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/// Instantaneous position of a point at parametric value `t`.
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#[inline]
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pub fn position(&self, t: f32) -> P {
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let [a, b, c, d] = self.coeff;
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a + b * t + c * t.powi(2) + d * t.powi(3)
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}
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/// Instantaneous velocity of a point at parametric value `t`.
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#[inline]
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pub fn velocity(&self, t: f32) -> P {
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let [_, b, c, d] = self.coeff;
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b + c * 2.0 * t + d * 3.0 * t.powi(2)
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}
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/// Instantaneous acceleration of a point at parametric value `t`.
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#[inline]
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pub fn acceleration(&self, t: f32) -> P {
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let [_, _, c, d] = self.coeff;
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c * 2.0 + d * 6.0 * t
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}
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}
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/// The `CubicSegment<Vec2>` can be used as a 2-dimensional easing curve for animation.
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///
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/// The x-axis of the curve is time, and the y-axis is the output value. This struct provides
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/// methods for extremely fast solves for y given x.
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impl CubicSegment<Vec2> {
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/// Construct a cubic Bezier curve for animation easing, with control points `p1` and `p2`. A
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/// cubic Bezier easing curve has control point `p0` at (0, 0) and `p3` at (1, 1), leaving only
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/// `p1` and `p2` as the remaining degrees of freedom. The first and last control points are
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/// fixed to ensure the animation begins at 0, and ends at 1.
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///
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/// This is a very common tool for UI animations that accelerate and decelerate smoothly. For
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/// example, the ubiquitous "ease-in-out" is defined as `(0.25, 0.1), (0.25, 1.0)`.
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pub fn new_bezier(p1: impl Into<Vec2>, p2: impl Into<Vec2>) -> Self {
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let (p0, p3) = (Vec2::ZERO, Vec2::ONE);
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let bezier = Bezier::new([[p0, p1.into(), p2.into(), p3]]).to_curve();
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bezier.segments[0].clone()
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}
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/// Maximum allowable error for iterative Bezier solve
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const MAX_ERROR: f32 = 1e-5;
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/// Maximum number of iterations during Bezier solve
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const MAX_ITERS: u8 = 8;
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/// Given a `time` within `0..=1`, returns an eased value that follows the cubic curve instead
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/// of a straight line. This eased result may be outside the range `0..=1`, however it will
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/// always start at 0 and end at 1: `ease(0) = 0` and `ease(1) = 1`.
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///
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/// ```
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/// # use bevy_math::prelude::*;
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/// let cubic_bezier = CubicSegment::new_bezier((0.25, 0.1), (0.25, 1.0));
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/// assert_eq!(cubic_bezier.ease(0.0), 0.0);
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/// assert_eq!(cubic_bezier.ease(1.0), 1.0);
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/// ```
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///
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/// # How cubic easing works
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///
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/// Easing is generally accomplished with the help of "shaping functions". These are curves that
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/// start at (0,0) and end at (1,1). The x-axis of this plot is the current `time` of the
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/// animation, from 0 to 1. The y-axis is how far along the animation is, also from 0 to 1. You
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/// can imagine that if the shaping function is a straight line, there is a 1:1 mapping between
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/// the `time` and how far along your animation is. If the `time` = 0.5, the animation is
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/// halfway through. This is known as linear interpolation, and results in objects animating
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/// with a constant velocity, and no smooth acceleration or deceleration at the start or end.
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///
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/// ```text
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/// y
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/// │ ●
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/// │ ⬈
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/// │ ⬈
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/// │ ⬈
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/// │ ⬈
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/// ●─────────── x (time)
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/// ```
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///
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/// Using cubic Beziers, we have a curve that starts at (0,0), ends at (1,1), and follows a path
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/// determined by the two remaining control points (handles). These handles allow us to define a
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/// smooth curve. As `time` (x-axis) progresses, we now follow the curve, and use the `y` value
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/// to determine how far along the animation is.
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///
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/// ```text
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/// y
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/// ⬈➔●
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/// │ ⬈
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/// │ ↑
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/// │ ↑
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/// │ ⬈
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/// ●➔⬈───────── x (time)
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/// ```
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///
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/// To accomplish this, we need to be able to find the position `y` on a curve, given the `x`
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/// value. Cubic curves are implicit parametric functions like B(t) = (x,y). To find `y`, we
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/// first solve for `t` that corresponds to the given `x` (`time`). We use the Newton-Raphson
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/// root-finding method to quickly find a value of `t` that is very near the desired value of
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/// `x`. Once we have this we can easily plug that `t` into our curve's `position` function, to
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/// find the `y` component, which is how far along our animation should be. In other words:
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///
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/// > Given `time` in `0..=1`
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///
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/// > Use Newton's method to find a value of `t` that results in B(t) = (x,y) where `x == time`
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///
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/// > Once a solution is found, use the resulting `y` value as the final result
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#[inline]
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pub fn ease(&self, time: f32) -> f32 {
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let x = time.clamp(0.0, 1.0);
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self.find_y_given_x(x)
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}
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/// Find the `y` value of the curve at the given `x` value using the Newton-Raphson method.
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#[inline]
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fn find_y_given_x(&self, x: f32) -> f32 {
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let mut t_guess = x;
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let mut pos_guess = Vec2::ZERO;
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for _ in 0..Self::MAX_ITERS {
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pos_guess = self.position(t_guess);
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let error = pos_guess.x - x;
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if error.abs() <= Self::MAX_ERROR {
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break;
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}
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// Using Newton's method, use the tangent line to estimate a better guess value.
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let slope = self.velocity(t_guess).x; // dx/dt
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t_guess -= error / slope;
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}
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pos_guess.y
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}
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}
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/// A collection of [`CubicSegment`]s chained into a curve.
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#[derive(Clone, Debug, Default, PartialEq)]
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pub struct CubicCurve<P: Point> {
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segments: Vec<CubicSegment<P>>,
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}
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impl<P: Point> CubicCurve<P> {
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/// Compute the position of a point on the cubic curve at the parametric value `t`.
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///
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/// Note that `t` varies from `0..=(n_points - 3)`.
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#[inline]
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pub fn position(&self, t: f32) -> P {
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let (segment, t) = self.segment(t);
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segment.position(t)
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}
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/// Compute the first derivative with respect to t at `t`. This is the instantaneous velocity of
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/// a point on the cubic curve at `t`.
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///
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/// Note that `t` varies from `0..=(n_points - 3)`.
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#[inline]
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pub fn velocity(&self, t: f32) -> P {
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let (segment, t) = self.segment(t);
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segment.velocity(t)
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}
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/// Compute the second derivative with respect to t at `t`. This is the instantaneous
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/// acceleration of a point on the cubic curve at `t`.
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///
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/// Note that `t` varies from `0..=(n_points - 3)`.
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#[inline]
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pub fn acceleration(&self, t: f32) -> P {
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let (segment, t) = self.segment(t);
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segment.acceleration(t)
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}
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/// A flexible iterator used to sample curves with arbitrary functions.
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///
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/// This splits the curve into `subdivisions` of evenly spaced `t` values across the
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|
/// 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`.
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|
#[inline]
|
|
fn iter_uniformly(&self, subdivisions: usize) -> impl Iterator<Item = f32> {
|
|
let segments = self.segments.len() as f32;
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|
let step = segments / subdivisions as f32;
|
|
(0..=subdivisions).map(move |i| i as f32 * step)
|
|
}
|
|
|
|
/// The list of segments contained in this `CubicCurve`.
|
|
///
|
|
/// This spline's global `t` value is equal to how many segments it has.
|
|
///
|
|
/// All method accepting `t` on `CubicCurve` depends on the global `t`.
|
|
/// When sampling over the entire curve, you should either use one of the
|
|
/// `iter_*` methods or account for the segment count using `curve.segments().len()`.
|
|
#[inline]
|
|
pub fn segments(&self) -> &[CubicSegment<P>] {
|
|
&self.segments
|
|
}
|
|
|
|
/// 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);
|
|
}
|
|
}
|