e46e246581
# Objective Stumbled on one of these and went digging for more ## Solution ```diff - word word + word ```
1419 lines
55 KiB
Rust
1419 lines
55 KiB
Rust
//! Provides types for building cubic splines for rendering curves and use with animation easing.
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use std::{fmt::Debug, iter::once};
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use crate::{Vec2, VectorSpace};
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use thiserror::Error;
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#[cfg(feature = "bevy_reflect")]
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use bevy_reflect::{std_traits::ReflectDefault, Reflect};
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/// A spline composed of a single cubic Bezier curve.
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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. The curve
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/// is divided into "segments" by every fourth control point.
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///
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/// ### Tangency
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/// Tangents are manually defined by the two intermediate control points within each set of four points.
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/// You can think of the control points the curve passes through as "anchors", and as the intermediate
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/// control points as the anchors displaced along their tangent vectors
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///
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/// ### Continuity
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/// A Bezier curve is at minimum C0 continuous, meaning it has no holes or jumps. Each curve segment is
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/// C2, meaning the tangent vector changes smoothly between each set of four control points, but this
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/// doesn't hold at the control points between segments. Making the whole curve C1 or C2 requires moving
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/// the intermediate control points to align the tangent vectors between segments, and can result in a
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/// loss of local control.
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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 = CubicBezier::new(points).to_curve();
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/// let positions: Vec<_> = bezier.iter_positions(100).collect();
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/// ```
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#[derive(Clone, Debug)]
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#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
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pub struct CubicBezier<P: VectorSpace> {
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/// The control points of the Bezier curve
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pub control_points: Vec<[P; 4]>,
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}
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impl<P: VectorSpace> CubicBezier<P> {
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/// Create a new cubic 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: VectorSpace> CubicGenerator<P> for CubicBezier<P> {
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#[inline]
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fn to_curve(&self) -> CubicCurve<P> {
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// A derivation for this matrix can be found in "General Matrix Representations for B-splines" by Kaihuai Qin.
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// <https://xiaoxingchen.github.io/2020/03/02/bspline_in_so3/general_matrix_representation_for_bsplines.pdf>
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// See section 4.2 and equation 11.
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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| CubicSegment::coefficients(*p, 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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/// Tangents are explicitly defined at each control point.
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///
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/// ### Continuity
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/// The curve is at minimum C0 continuous, meaning it has no holes or jumps. It is also C1, meaning the
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/// tangent vector has no sudden jumps.
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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 = CubicHermite::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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#[derive(Clone, Debug)]
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#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
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pub struct CubicHermite<P: VectorSpace> {
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/// The control points of the Hermite curve
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pub control_points: Vec<(P, P)>,
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}
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impl<P: VectorSpace> CubicHermite<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: VectorSpace> CubicGenerator<P> for CubicHermite<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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CubicSegment::coefficients([p0, v0, p1, v1], 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. The associated [`CubicCurve`]
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/// has one segment between each pair of adjacent control points.
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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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/// Tangents are automatically computed based on the positions of control points.
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///
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/// ### Continuity
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/// The curve is at minimum C1, meaning that it is continuous (it has no holes or jumps), and its tangent
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/// vector is also well-defined everywhere, without sudden jumps.
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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 = CubicCardinalSpline::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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#[derive(Clone, Debug)]
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#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
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pub struct CubicCardinalSpline<P: VectorSpace> {
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/// Tension
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pub tension: f32,
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/// The control points of the Cardinal spline
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pub control_points: Vec<P>,
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}
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impl<P: VectorSpace> CubicCardinalSpline<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: VectorSpace> CubicGenerator<P> for CubicCardinalSpline<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 length = self.control_points.len();
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// Early return to avoid accessing an invalid index
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if length < 2 {
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return CubicCurve { segments: vec![] };
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}
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// Extend the list of control points by mirroring the last second-to-last control points on each end;
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// this allows tangents for the endpoints to be provided, and the overall effect is that the tangent
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// at an endpoint is proportional to twice the vector between it and its adjacent control point.
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//
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// The expression used here is P_{-1} := P_0 - (P_1 - P_0) = 2P_0 - P_1. (Analogously at the other end.)
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let mirrored_first = self.control_points[0] * 2. - self.control_points[1];
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let mirrored_last = self.control_points[length - 1] * 2. - self.control_points[length - 2];
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let extended_control_points = once(&mirrored_first)
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.chain(self.control_points.iter())
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.chain(once(&mirrored_last))
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.collect::<Vec<_>>();
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let segments = extended_control_points
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.windows(4)
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.map(|p| CubicSegment::coefficients([*p[0], *p[1], *p[2], *p[3]], 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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/// Tangents are automatically computed based on the position of control points.
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///
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/// ### Continuity
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/// The curve is C2 continuous, meaning it has no holes or jumps, and the tangent vector changes smoothly along
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/// the entire curve length. 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 = CubicBSpline::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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#[derive(Clone, Debug)]
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#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
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pub struct CubicBSpline<P: VectorSpace> {
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/// The control points of the spline
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pub control_points: Vec<P>,
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}
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impl<P: VectorSpace> CubicBSpline<P> {
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/// Build a new B-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: VectorSpace> CubicGenerator<P> for CubicBSpline<P> {
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#[inline]
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fn to_curve(&self) -> CubicCurve<P> {
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// A derivation for this matrix can be found in "General Matrix Representations for B-splines" by Kaihuai Qin.
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// <https://xiaoxingchen.github.io/2020/03/02/bspline_in_so3/general_matrix_representation_for_bsplines.pdf>
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// See section 4.1 and equations 7 and 8.
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let mut char_matrix = [
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[1.0, 4.0, 1.0, 0.0],
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[-3.0, 0.0, 3.0, 0.0],
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[3.0, -6.0, 3.0, 0.0],
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[-1.0, 3.0, -3.0, 1.0],
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];
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char_matrix
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.iter_mut()
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.for_each(|r| r.iter_mut().for_each(|c| *c /= 6.0));
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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| CubicSegment::coefficients([p[0], p[1], p[2], p[3]], 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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/// Error during construction of [`CubicNurbs`]
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#[derive(Clone, Debug, Error)]
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pub enum CubicNurbsError {
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/// Provided the wrong number of knots.
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#[error("Wrong number of knots: expected {expected}, provided {provided}")]
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KnotsNumberMismatch {
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/// Expected number of knots
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expected: usize,
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/// Provided number of knots
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provided: usize,
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},
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/// The provided knots had a descending knot pair. Subsequent knots must
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/// either increase or stay the same.
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#[error("Invalid knots: contains descending knot pair")]
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DescendingKnots,
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/// The provided knots were all equal. Knots must contain at least one increasing pair.
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#[error("Invalid knots: all knots are equal")]
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ConstantKnots,
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/// Provided a different number of weights and control points.
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#[error("Incorrect number of weights: expected {expected}, provided {provided}")]
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WeightsNumberMismatch {
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/// Expected number of weights
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expected: usize,
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/// Provided number of weights
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provided: usize,
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},
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/// The number of control points provided is less than 4.
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#[error("Not enough control points, at least 4 are required, {provided} were provided")]
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NotEnoughControlPoints {
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/// The number of control points provided
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provided: usize,
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},
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}
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/// Non-uniform Rational B-Splines (NURBS) are a powerful generalization of the [`CubicBSpline`] which can
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/// represent a much more diverse class of curves (like perfect circles and ellipses).
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///
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/// ### Non-uniformity
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/// The 'NU' part of NURBS stands for "Non-Uniform". This has to do with a parameter called 'knots'.
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/// The knots are a non-decreasing sequence of floating point numbers. The first and last three pairs of
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/// knots control the behavior of the curve as it approaches its endpoints. The intermediate pairs
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/// each control the length of one segment of the curve. Multiple repeated knot values are called
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/// "knot multiplicity". Knot multiplicity in the intermediate knots causes a "zero-length" segment,
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/// and can create sharp corners.
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///
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/// ### Rationality
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/// The 'R' part of NURBS stands for "Rational". This has to do with NURBS allowing each control point to
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/// be assigned a weighting, which controls how much it affects the curve compared to the other points.
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///
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/// ### Interpolation
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/// The curve will not pass through the control points except where a knot has multiplicity four.
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///
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/// ### Tangency
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/// Tangents are automatically computed based on the position of control points.
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///
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/// ### Continuity
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/// When there is no knot multiplicity, the curve is C2 continuous, meaning it has no holes or jumps and the
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/// tangent vector changes smoothly along the entire curve length. Like the [`CubicBSpline`], the acceleration
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/// continuity makes it useful for camera paths. Knot multiplicity of 2 in intermediate knots reduces the
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/// continuity to C2, and knot multiplicity of 3 reduces the continuity to C0. The curve is always at least
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/// C0, meaning it has no jumps or holes.
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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 weights = [1.0, 1.0, 2.0, 1.0];
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/// let knots = [0.0, 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 5.0];
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/// let nurbs = CubicNurbs::new(points, Some(weights), Some(knots))
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/// .expect("NURBS construction failed!")
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/// .to_curve();
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/// let positions: Vec<_> = nurbs.iter_positions(100).collect();
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/// ```
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#[derive(Clone, Debug)]
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#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
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pub struct CubicNurbs<P: VectorSpace> {
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/// The control points of the NURBS
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pub control_points: Vec<P>,
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/// Weights
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pub weights: Vec<f32>,
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/// Knots
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pub knots: Vec<f32>,
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}
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impl<P: VectorSpace> CubicNurbs<P> {
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/// Build a Non-Uniform Rational B-Spline.
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///
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/// If provided, weights must be the same length as the control points. Defaults to equal weights.
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///
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/// If provided, the number of knots must be n + 4 elements, where n is the amount of control
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/// points. Defaults to open uniform knots: [`Self::open_uniform_knots`]. Knots cannot
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/// all be equal.
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///
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/// At least 4 points must be provided, otherwise an error will be returned.
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pub fn new(
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control_points: impl Into<Vec<P>>,
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weights: Option<impl Into<Vec<f32>>>,
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knots: Option<impl Into<Vec<f32>>>,
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) -> Result<Self, CubicNurbsError> {
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let mut control_points: Vec<P> = control_points.into();
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let control_points_len = control_points.len();
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if control_points_len < 4 {
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return Err(CubicNurbsError::NotEnoughControlPoints {
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provided: control_points_len,
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});
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}
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let weights = weights
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.map(Into::into)
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.unwrap_or_else(|| vec![1.0; control_points_len]);
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let mut knots: Vec<f32> = knots.map(Into::into).unwrap_or_else(|| {
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Self::open_uniform_knots(control_points_len)
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.expect("The amount of control points was checked")
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});
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let expected_knots_len = Self::knots_len(control_points_len);
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// Check the number of knots is correct
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if knots.len() != expected_knots_len {
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return Err(CubicNurbsError::KnotsNumberMismatch {
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expected: expected_knots_len,
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provided: knots.len(),
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});
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}
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// Ensure the knots are non-descending (previous element is less than or equal
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// to the next)
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if knots.windows(2).any(|win| win[0] > win[1]) {
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return Err(CubicNurbsError::DescendingKnots);
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}
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// Ensure the knots are non-constant
|
|
if knots.windows(2).all(|win| win[0] == win[1]) {
|
|
return Err(CubicNurbsError::ConstantKnots);
|
|
}
|
|
|
|
// Check that the number of weights equals the number of control points
|
|
if weights.len() != control_points_len {
|
|
return Err(CubicNurbsError::WeightsNumberMismatch {
|
|
expected: control_points_len,
|
|
provided: weights.len(),
|
|
});
|
|
}
|
|
|
|
// To align the evaluation behavior of nurbs with the other splines,
|
|
// make the intervals between knots form an exact cover of [0, N], where N is
|
|
// the number of segments of the final curve.
|
|
let curve_length = (control_points.len() - 3) as f32;
|
|
let min = *knots.first().unwrap();
|
|
let max = *knots.last().unwrap();
|
|
let knot_delta = max - min;
|
|
knots = knots
|
|
.into_iter()
|
|
.map(|k| k - min)
|
|
.map(|k| k * curve_length / knot_delta)
|
|
.collect();
|
|
|
|
control_points
|
|
.iter_mut()
|
|
.zip(weights.iter())
|
|
.for_each(|(p, w)| *p = *p * *w);
|
|
|
|
Ok(Self {
|
|
control_points,
|
|
weights,
|
|
knots,
|
|
})
|
|
}
|
|
|
|
/// Generates uniform knots that will generate the same curve as [`CubicBSpline`].
|
|
///
|
|
/// "Uniform" means that the difference between two subsequent knots is the same.
|
|
///
|
|
/// Will return `None` if there are less than 4 control points.
|
|
pub fn uniform_knots(control_points: usize) -> Option<Vec<f32>> {
|
|
if control_points < 4 {
|
|
return None;
|
|
}
|
|
Some(
|
|
(0..Self::knots_len(control_points))
|
|
.map(|v| v as f32)
|
|
.collect(),
|
|
)
|
|
}
|
|
|
|
/// Generates open uniform knots, which makes the ends of the curve pass through the
|
|
/// start and end points.
|
|
///
|
|
/// The start and end knots have multiplicity 4, and intermediate knots have multiplicity 0 and
|
|
/// difference of 1.
|
|
///
|
|
/// Will return `None` if there are less than 4 control points.
|
|
pub fn open_uniform_knots(control_points: usize) -> Option<Vec<f32>> {
|
|
if control_points < 4 {
|
|
return None;
|
|
}
|
|
let last_knots_value = control_points - 3;
|
|
Some(
|
|
std::iter::repeat(0.0)
|
|
.take(4)
|
|
.chain((1..last_knots_value).map(|v| v as f32))
|
|
.chain(std::iter::repeat(last_knots_value as f32).take(4))
|
|
.collect(),
|
|
)
|
|
}
|
|
|
|
#[inline(always)]
|
|
const fn knots_len(control_points_len: usize) -> usize {
|
|
control_points_len + 4
|
|
}
|
|
|
|
/// Generates a non-uniform B-spline characteristic matrix from a sequence of six knots. Each six
|
|
/// knots describe the relationship between four successive control points. For padding reasons,
|
|
/// this takes a vector of 8 knots, but only six are actually used.
|
|
fn generate_matrix(knots: &[f32; 8]) -> [[f32; 4]; 4] {
|
|
// 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 3.1.
|
|
|
|
let t = knots;
|
|
// In the notation of the paper:
|
|
// t[1] := t_i-2
|
|
// t[2] := t_i-1
|
|
// t[3] := t_i (the lower extent of the current knot span)
|
|
// t[4] := t_i+1 (the upper extent of the current knot span)
|
|
// t[5] := t_i+2
|
|
// t[6] := t_i+3
|
|
|
|
let m00 = (t[4] - t[3]).powi(2) / ((t[4] - t[2]) * (t[4] - t[1]));
|
|
let m02 = (t[3] - t[2]).powi(2) / ((t[5] - t[2]) * (t[4] - t[2]));
|
|
let m12 = (3.0 * (t[4] - t[3]) * (t[3] - t[2])) / ((t[5] - t[2]) * (t[4] - t[2]));
|
|
let m22 = 3.0 * (t[4] - t[3]).powi(2) / ((t[5] - t[2]) * (t[4] - t[2]));
|
|
let m33 = (t[4] - t[3]).powi(2) / ((t[6] - t[3]) * (t[5] - t[3]));
|
|
let m32 = -m22 / 3.0 - m33 - (t[4] - t[3]).powi(2) / ((t[5] - t[3]) * (t[5] - t[2]));
|
|
[
|
|
[m00, 1.0 - m00 - m02, m02, 0.0],
|
|
[-3.0 * m00, 3.0 * m00 - m12, m12, 0.0],
|
|
[3.0 * m00, -3.0 * m00 - m22, m22, 0.0],
|
|
[-m00, m00 - m32 - m33, m32, m33],
|
|
]
|
|
}
|
|
}
|
|
impl<P: VectorSpace> RationalGenerator<P> for CubicNurbs<P> {
|
|
#[inline]
|
|
fn to_curve(&self) -> RationalCurve<P> {
|
|
let segments = self
|
|
.control_points
|
|
.windows(4)
|
|
.zip(self.weights.windows(4))
|
|
.zip(self.knots.windows(8))
|
|
.filter(|(_, knots)| knots[4] - knots[3] > 0.0)
|
|
.map(|((points, weights), knots)| {
|
|
// This is curve segment i. It uses control points P_i, P_i+2, P_i+2 and P_i+3,
|
|
// It is associated with knot span i+3 (which is the interval between knots i+3
|
|
// and i+4) and its characteristic matrix uses knots i+1 through i+6 (because
|
|
// those define the two knot spans on either side).
|
|
let span = knots[4] - knots[3];
|
|
let coefficient_knots = knots.try_into().expect("Knot windows are of length 6");
|
|
let matrix = Self::generate_matrix(coefficient_knots);
|
|
RationalSegment::coefficients(
|
|
points.try_into().expect("Point windows are of length 4"),
|
|
weights.try_into().expect("Weight windows are of length 4"),
|
|
span,
|
|
matrix,
|
|
)
|
|
})
|
|
.collect();
|
|
RationalCurve { segments }
|
|
}
|
|
}
|
|
|
|
/// A spline interpolated linearly between the nearest 2 points.
|
|
///
|
|
/// ### Interpolation
|
|
/// The curve passes through every control point.
|
|
///
|
|
/// ### Tangency
|
|
/// The curve is not generally differentiable at control points.
|
|
///
|
|
/// ### Continuity
|
|
/// The curve is C0 continuous, meaning it has no holes or jumps.
|
|
#[derive(Clone, Debug)]
|
|
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
|
|
pub struct LinearSpline<P: VectorSpace> {
|
|
/// The control points of the NURBS
|
|
pub points: Vec<P>,
|
|
}
|
|
impl<P: VectorSpace> LinearSpline<P> {
|
|
/// Create a new linear spline
|
|
pub fn new(points: impl Into<Vec<P>>) -> Self {
|
|
Self {
|
|
points: points.into(),
|
|
}
|
|
}
|
|
}
|
|
impl<P: VectorSpace> CubicGenerator<P> for LinearSpline<P> {
|
|
#[inline]
|
|
fn to_curve(&self) -> CubicCurve<P> {
|
|
let segments = self
|
|
.points
|
|
.windows(2)
|
|
.map(|points| {
|
|
let a = points[0];
|
|
let b = points[1];
|
|
CubicSegment {
|
|
coeff: [a, b - a, P::default(), P::default()],
|
|
}
|
|
})
|
|
.collect();
|
|
CubicCurve { segments }
|
|
}
|
|
}
|
|
|
|
/// Implement this on cubic splines that can generate a cubic curve from their spline parameters.
|
|
pub trait CubicGenerator<P: VectorSpace> {
|
|
/// 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.
|
|
/// Can be evaluated as a parametric curve over the domain `[0, 1)`.
|
|
///
|
|
/// Segments can be chained together to form a longer compound curve.
|
|
#[derive(Copy, Clone, Debug, Default, PartialEq)]
|
|
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug, Default))]
|
|
pub struct CubicSegment<P: VectorSpace> {
|
|
/// Coefficients of the segment
|
|
pub coeff: [P; 4],
|
|
}
|
|
|
|
impl<P: VectorSpace> 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;
|
|
// Evaluate `a + bt + ct^2 + dt^3`, avoiding exponentiation
|
|
a + (b + (c + d * t) * t) * t
|
|
}
|
|
|
|
/// Instantaneous velocity of a point at parametric value `t`.
|
|
#[inline]
|
|
pub fn velocity(&self, t: f32) -> P {
|
|
let [_, b, c, d] = self.coeff;
|
|
// Evaluate the derivative, which is `b + 2ct + 3dt^2`, avoiding exponentiation
|
|
b + (c * 2.0 + d * 3.0 * t) * t
|
|
}
|
|
|
|
/// Instantaneous acceleration of a point at parametric value `t`.
|
|
#[inline]
|
|
pub fn acceleration(&self, t: f32) -> P {
|
|
let [_, _, c, d] = self.coeff;
|
|
// Evaluate the second derivative, which is `2c + 6dt`
|
|
c * 2.0 + d * 6.0 * t
|
|
}
|
|
|
|
/// Calculate polynomial coefficients for the cubic curve using a characteristic matrix.
|
|
#[inline]
|
|
fn coefficients(p: [P; 4], char_matrix: [[f32; 4]; 4]) -> Self {
|
|
let [c0, c1, c2, c3] = char_matrix;
|
|
// These are the polynomial coefficients, computed by multiplying the characteristic
|
|
// matrix by the point matrix.
|
|
let 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],
|
|
];
|
|
Self { coeff }
|
|
}
|
|
}
|
|
|
|
/// 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 = CubicBezier::new([[p0, p1.into(), p2.into(), p3]]).to_curve();
|
|
bezier.segments[0]
|
|
}
|
|
|
|
/// 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 single parametric curve. Has domain `[0, N)`
|
|
/// where `N` is the number of attached segments.
|
|
///
|
|
/// Use any struct that implements the [`CubicGenerator`] trait to create a new curve, such as
|
|
/// [`CubicBezier`].
|
|
#[derive(Clone, Debug, PartialEq)]
|
|
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
|
|
pub struct CubicCurve<P: VectorSpace> {
|
|
/// Segments of the curve
|
|
pub segments: Vec<CubicSegment<P>>,
|
|
}
|
|
|
|
impl<P: VectorSpace> 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 = 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 segments = self.segments.len() as f32;
|
|
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)
|
|
}
|
|
|
|
#[inline]
|
|
/// Adds a segment to the curve
|
|
pub fn push_segment(&mut self, segment: CubicSegment<P>) {
|
|
self.segments.push(segment);
|
|
}
|
|
|
|
/// 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)
|
|
}
|
|
}
|
|
}
|
|
|
|
impl<P: VectorSpace> Extend<CubicSegment<P>> for CubicCurve<P> {
|
|
fn extend<T: IntoIterator<Item = CubicSegment<P>>>(&mut self, iter: T) {
|
|
self.segments.extend(iter);
|
|
}
|
|
}
|
|
|
|
impl<P: VectorSpace> IntoIterator for CubicCurve<P> {
|
|
type IntoIter = <Vec<CubicSegment<P>> as IntoIterator>::IntoIter;
|
|
|
|
type Item = CubicSegment<P>;
|
|
|
|
fn into_iter(self) -> Self::IntoIter {
|
|
self.segments.into_iter()
|
|
}
|
|
}
|
|
|
|
/// Implement this on cubic splines that can generate a rational cubic curve from their spline parameters.
|
|
pub trait RationalGenerator<P: VectorSpace> {
|
|
/// Build a [`RationalCurve`] by computing the interpolation coefficients for each curve segment.
|
|
fn to_curve(&self) -> RationalCurve<P>;
|
|
}
|
|
|
|
/// A segment of a rational cubic curve, used to hold precomputed coefficients for fast interpolation.
|
|
/// Can be evaluated as a parametric curve over the domain `[0, knot_span)`.
|
|
///
|
|
/// Segments can be chained together to form a longer compound curve.
|
|
#[derive(Copy, Clone, Debug, Default, PartialEq)]
|
|
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug, Default))]
|
|
pub struct RationalSegment<P: VectorSpace> {
|
|
/// The coefficients matrix of the cubic curve.
|
|
pub coeff: [P; 4],
|
|
/// The homogeneous weight coefficients.
|
|
pub weight_coeff: [f32; 4],
|
|
/// The width of the domain of this segment.
|
|
pub knot_span: f32,
|
|
}
|
|
|
|
impl<P: VectorSpace> RationalSegment<P> {
|
|
/// Instantaneous position of a point at parametric value `t` in `[0, knot_span)`.
|
|
#[inline]
|
|
pub fn position(&self, t: f32) -> P {
|
|
let [a, b, c, d] = self.coeff;
|
|
let [x, y, z, w] = self.weight_coeff;
|
|
// Compute a cubic polynomial for the control points
|
|
let numerator = a + (b + (c + d * t) * t) * t;
|
|
// Compute a cubic polynomial for the weights
|
|
let denominator = x + (y + (z + w * t) * t) * t;
|
|
numerator / denominator
|
|
}
|
|
|
|
/// Instantaneous velocity of a point at parametric value `t` in `[0, knot_span)`.
|
|
#[inline]
|
|
pub fn velocity(&self, t: f32) -> P {
|
|
// A derivation for the following equations can be found in "Matrix representation for NURBS
|
|
// curves and surfaces" by Choi et al. See equation 19.
|
|
|
|
let [a, b, c, d] = self.coeff;
|
|
let [x, y, z, w] = self.weight_coeff;
|
|
// Compute a cubic polynomial for the control points
|
|
let numerator = a + (b + (c + d * t) * t) * t;
|
|
// Compute a cubic polynomial for the weights
|
|
let denominator = x + (y + (z + w * t) * t) * t;
|
|
|
|
// Compute the derivative of the control point polynomial
|
|
let numerator_derivative = b + (c * 2.0 + d * 3.0 * t) * t;
|
|
// Compute the derivative of the weight polynomial
|
|
let denominator_derivative = y + (z * 2.0 + w * 3.0 * t) * t;
|
|
|
|
// Velocity is the first derivative (wrt to the parameter `t`)
|
|
// Position = N/D therefore
|
|
// Velocity = (N/D)' = N'/D - N * D'/D^2 = (N' * D - N * D')/D^2
|
|
numerator_derivative / denominator
|
|
- numerator * (denominator_derivative / denominator.powi(2))
|
|
}
|
|
|
|
/// Instantaneous acceleration of a point at parametric value `t` in `[0, knot_span)`.
|
|
#[inline]
|
|
pub fn acceleration(&self, t: f32) -> P {
|
|
// A derivation for the following equations can be found in "Matrix representation for NURBS
|
|
// curves and surfaces" by Choi et al. See equation 20. Note: In come copies of this paper, equation 20
|
|
// is printed with the following two errors:
|
|
// + The first term has incorrect sign.
|
|
// + The second term uses R when it should use the first derivative.
|
|
|
|
let [a, b, c, d] = self.coeff;
|
|
let [x, y, z, w] = self.weight_coeff;
|
|
// Compute a cubic polynomial for the control points
|
|
let numerator = a + (b + (c + d * t) * t) * t;
|
|
// Compute a cubic polynomial for the weights
|
|
let denominator = x + (y + (z + w * t) * t) * t;
|
|
|
|
// Compute the derivative of the control point polynomial
|
|
let numerator_derivative = b + (c * 2.0 + d * 3.0 * t) * t;
|
|
// Compute the derivative of the weight polynomial
|
|
let denominator_derivative = y + (z * 2.0 + w * 3.0 * t) * t;
|
|
|
|
// Compute the second derivative of the control point polynomial
|
|
let numerator_second_derivative = c * 2.0 + d * 6.0 * t;
|
|
// Compute the second derivative of the weight polynomial
|
|
let denominator_second_derivative = z * 2.0 + w * 6.0 * t;
|
|
|
|
// Velocity is the first derivative (wrt to the parameter `t`)
|
|
// Position = N/D therefore
|
|
// Velocity = (N/D)' = N'/D - N * D'/D^2 = (N' * D - N * D')/D^2
|
|
// Acceleration = (N/D)'' = ((N' * D - N * D')/D^2)' = N''/D + N' * (-2D'/D^2) + N * (-D''/D^2 + 2D'^2/D^3)
|
|
numerator_second_derivative / denominator
|
|
+ numerator_derivative * (-2.0 * denominator_derivative / denominator.powi(2))
|
|
+ numerator
|
|
* (-denominator_second_derivative / denominator.powi(2)
|
|
+ 2.0 * denominator_derivative.powi(2) / denominator.powi(3))
|
|
}
|
|
|
|
/// Calculate polynomial coefficients for the cubic polynomials using a characteristic matrix.
|
|
#[inline]
|
|
fn coefficients(
|
|
control_points: [P; 4],
|
|
weights: [f32; 4],
|
|
knot_span: f32,
|
|
char_matrix: [[f32; 4]; 4],
|
|
) -> Self {
|
|
// An explanation of this use can be found in "Matrix representation for NURBS curves and surfaces"
|
|
// by Choi et al. See section "Evaluation of NURB Curves and Surfaces", and equation 16.
|
|
|
|
let [c0, c1, c2, c3] = char_matrix;
|
|
let p = control_points;
|
|
let w = weights;
|
|
// These are the control point polynomial coefficients, computed by multiplying the characteristic
|
|
// matrix by the point matrix.
|
|
let 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],
|
|
];
|
|
// These are the weight polynomial coefficients, computed by multiplying the characteristic
|
|
// matrix by the weight matrix.
|
|
let weight_coeff = [
|
|
w[0] * c0[0] + w[1] * c0[1] + w[2] * c0[2] + w[3] * c0[3],
|
|
w[0] * c1[0] + w[1] * c1[1] + w[2] * c1[2] + w[3] * c1[3],
|
|
w[0] * c2[0] + w[1] * c2[1] + w[2] * c2[2] + w[3] * c2[3],
|
|
w[0] * c3[0] + w[1] * c3[1] + w[2] * c3[2] + w[3] * c3[3],
|
|
];
|
|
Self {
|
|
coeff,
|
|
weight_coeff,
|
|
knot_span,
|
|
}
|
|
}
|
|
}
|
|
|
|
/// A collection of [`RationalSegment`]s chained into a single parametric curve.
|
|
///
|
|
/// Use any struct that implements the [`RationalGenerator`] trait to create a new curve, such as
|
|
/// [`CubicNurbs`], or convert [`CubicCurve`] using `into/from`.
|
|
#[derive(Clone, Debug, PartialEq)]
|
|
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug))]
|
|
pub struct RationalCurve<P: VectorSpace> {
|
|
/// The segments in the curve
|
|
pub segments: Vec<RationalSegment<P>>,
|
|
}
|
|
|
|
impl<P: VectorSpace> RationalCurve<P> {
|
|
/// Compute the position of a point on the 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 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 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 = 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 domain = self.domain();
|
|
let step = domain / subdivisions as f32;
|
|
(0..=subdivisions).map(move |i| i as f32 * step)
|
|
}
|
|
|
|
/// The list of segments contained in this `RationalCurve`.
|
|
///
|
|
/// This spline's global `t` value is equal to how many segments it has.
|
|
///
|
|
/// All method accepting `t` on `RationalCurve` 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) -> &[RationalSegment<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)
|
|
}
|
|
|
|
/// Adds a segment to the curve.
|
|
#[inline]
|
|
pub fn push_segment(&mut self, segment: RationalSegment<P>) {
|
|
self.segments.push(segment);
|
|
}
|
|
|
|
/// Returns the [`RationalSegment`] and local `t` value given a spline's global `t` value.
|
|
/// Input `t` will be clamped to the domain of the curve. Returned value will be in `[0, 1]`.
|
|
#[inline]
|
|
fn segment(&self, mut t: f32) -> (&RationalSegment<P>, f32) {
|
|
if t <= 0.0 {
|
|
(&self.segments[0], 0.0)
|
|
} else if self.segments.len() == 1 {
|
|
(&self.segments[0], t / self.segments[0].knot_span)
|
|
} else {
|
|
// Try to fit t into each segment domain
|
|
for segment in self.segments.iter() {
|
|
if t < segment.knot_span {
|
|
// The division here makes t a normalized parameter in [0, 1] that can be properly
|
|
// evaluated against a cubic curve segment. See equations 6 & 16 from "Matrix representation
|
|
// of NURBS curves and surfaces" by Choi et al. or equation 3 from "General Matrix
|
|
// Representations for B-Splines" by Qin.
|
|
return (segment, t / segment.knot_span);
|
|
}
|
|
t -= segment.knot_span;
|
|
}
|
|
return (self.segments.last().unwrap(), 1.0);
|
|
}
|
|
}
|
|
|
|
/// Returns the length of the domain of the parametric curve.
|
|
#[inline]
|
|
pub fn domain(&self) -> f32 {
|
|
self.segments.iter().map(|segment| segment.knot_span).sum()
|
|
}
|
|
}
|
|
|
|
impl<P: VectorSpace> Extend<RationalSegment<P>> for RationalCurve<P> {
|
|
fn extend<T: IntoIterator<Item = RationalSegment<P>>>(&mut self, iter: T) {
|
|
self.segments.extend(iter);
|
|
}
|
|
}
|
|
|
|
impl<P: VectorSpace> IntoIterator for RationalCurve<P> {
|
|
type IntoIter = <Vec<RationalSegment<P>> as IntoIterator>::IntoIter;
|
|
|
|
type Item = RationalSegment<P>;
|
|
|
|
fn into_iter(self) -> Self::IntoIter {
|
|
self.segments.into_iter()
|
|
}
|
|
}
|
|
|
|
impl<P: VectorSpace> From<CubicSegment<P>> for RationalSegment<P> {
|
|
fn from(value: CubicSegment<P>) -> Self {
|
|
Self {
|
|
coeff: value.coeff,
|
|
weight_coeff: [1.0, 0.0, 0.0, 0.0],
|
|
knot_span: 1.0, // Cubic curves are uniform, so every segment has domain [0, 1).
|
|
}
|
|
}
|
|
}
|
|
|
|
impl<P: VectorSpace> From<CubicCurve<P>> for RationalCurve<P> {
|
|
fn from(value: CubicCurve<P>) -> Self {
|
|
Self {
|
|
segments: value.segments.into_iter().map(Into::into).collect(),
|
|
}
|
|
}
|
|
}
|
|
|
|
#[cfg(test)]
|
|
mod tests {
|
|
use glam::{vec2, Vec2};
|
|
|
|
use crate::cubic_splines::{
|
|
CubicBSpline, CubicBezier, CubicGenerator, CubicNurbs, CubicSegment, RationalCurve,
|
|
RationalGenerator,
|
|
};
|
|
|
|
/// 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 = CubicBezier::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);
|
|
}
|
|
|
|
/// Test that a simple cardinal spline passes through all of its control points with
|
|
/// the correct tangents.
|
|
#[test]
|
|
fn cardinal_control_pts() {
|
|
use super::CubicCardinalSpline;
|
|
|
|
let tension = 0.2;
|
|
let [p0, p1, p2, p3] = [vec2(-1., -2.), vec2(0., 1.), vec2(1., 2.), vec2(-2., 1.)];
|
|
let curve = CubicCardinalSpline::new(tension, [p0, p1, p2, p3]).to_curve();
|
|
|
|
// Positions at segment endpoints
|
|
assert!(curve.position(0.).abs_diff_eq(p0, FLOAT_EQ));
|
|
assert!(curve.position(1.).abs_diff_eq(p1, FLOAT_EQ));
|
|
assert!(curve.position(2.).abs_diff_eq(p2, FLOAT_EQ));
|
|
assert!(curve.position(3.).abs_diff_eq(p3, FLOAT_EQ));
|
|
|
|
// Tangents at segment endpoints
|
|
assert!(curve
|
|
.velocity(0.)
|
|
.abs_diff_eq((p1 - p0) * tension * 2., FLOAT_EQ));
|
|
assert!(curve
|
|
.velocity(1.)
|
|
.abs_diff_eq((p2 - p0) * tension, FLOAT_EQ));
|
|
assert!(curve
|
|
.velocity(2.)
|
|
.abs_diff_eq((p3 - p1) * tension, FLOAT_EQ));
|
|
assert!(curve
|
|
.velocity(3.)
|
|
.abs_diff_eq((p3 - p2) * tension * 2., FLOAT_EQ));
|
|
}
|
|
|
|
/// Test that [`RationalCurve`] properly generalizes [`CubicCurve`]. A Cubic upgraded to a rational
|
|
/// should produce pretty much the same output.
|
|
#[test]
|
|
fn cubic_to_rational() {
|
|
const EPSILON: f32 = 0.00001;
|
|
|
|
let points = [
|
|
vec2(0.0, 0.0),
|
|
vec2(1.0, 1.0),
|
|
vec2(1.0, 1.0),
|
|
vec2(2.0, -1.0),
|
|
vec2(3.0, 1.0),
|
|
vec2(0.0, 0.0),
|
|
];
|
|
|
|
let b_spline = CubicBSpline::new(points).to_curve();
|
|
let rational_b_spline = RationalCurve::from(b_spline.clone());
|
|
|
|
/// Tests if two vectors of points are approximately the same
|
|
fn compare_vectors(cubic_curve: Vec<Vec2>, rational_curve: Vec<Vec2>, name: &str) {
|
|
assert_eq!(
|
|
cubic_curve.len(),
|
|
rational_curve.len(),
|
|
"{name} vector lengths mismatch"
|
|
);
|
|
for (i, (a, b)) in cubic_curve.iter().zip(rational_curve.iter()).enumerate() {
|
|
assert!(
|
|
a.distance(*b) < EPSILON,
|
|
"Mismatch at {name} value {i}. CubicCurve: {} Converted RationalCurve: {}",
|
|
a,
|
|
b
|
|
);
|
|
}
|
|
}
|
|
|
|
// Both curves should yield the same values
|
|
let cubic_positions: Vec<_> = b_spline.iter_positions(10).collect();
|
|
let rational_positions: Vec<_> = rational_b_spline.iter_positions(10).collect();
|
|
compare_vectors(cubic_positions, rational_positions, "position");
|
|
|
|
let cubic_velocities: Vec<_> = b_spline.iter_velocities(10).collect();
|
|
let rational_velocities: Vec<_> = rational_b_spline.iter_velocities(10).collect();
|
|
compare_vectors(cubic_velocities, rational_velocities, "velocity");
|
|
|
|
let cubic_accelerations: Vec<_> = b_spline.iter_accelerations(10).collect();
|
|
let rational_accelerations: Vec<_> = rational_b_spline.iter_accelerations(10).collect();
|
|
compare_vectors(cubic_accelerations, rational_accelerations, "acceleration");
|
|
}
|
|
|
|
/// Test that a nurbs curve can approximate a portion of a circle.
|
|
#[test]
|
|
fn nurbs_circular_arc() {
|
|
use std::f32::consts::FRAC_PI_2;
|
|
const EPSILON: f32 = 0.0000001;
|
|
|
|
// The following NURBS parameters were determined by constraining the first two
|
|
// points to the line y=1, the second two points to the line x=1, and the distance
|
|
// between each pair of points to be equal. One can solve the weights by assuming the
|
|
// first and last weights to be one, the intermediate weights to be equal, and
|
|
// subjecting ones self to a lot of tedious matrix algebra.
|
|
|
|
let alpha = FRAC_PI_2;
|
|
let leg = 2.0 * f32::sin(alpha / 2.0) / (1.0 + 2.0 * f32::cos(alpha / 2.0));
|
|
let weight = (1.0 + 2.0 * f32::cos(alpha / 2.0)) / 3.0;
|
|
let points = [
|
|
vec2(1.0, 0.0),
|
|
vec2(1.0, leg),
|
|
vec2(leg, 1.0),
|
|
vec2(0.0, 1.0),
|
|
];
|
|
let weights = [1.0, weight, weight, 1.0];
|
|
let knots = [0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0];
|
|
let spline = CubicNurbs::new(points, Some(weights), Some(knots)).unwrap();
|
|
let curve = spline.to_curve();
|
|
for (i, point) in curve.iter_positions(10).enumerate() {
|
|
assert!(
|
|
f32::abs(point.length() - 1.0) < EPSILON,
|
|
"Point {i} is not on the unit circle: {point:?} has length {}",
|
|
point.length()
|
|
);
|
|
}
|
|
}
|
|
}
|