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/* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
* file, You can obtain one at https://mozilla.org/MPL/2.0/. */
//! Parametric Bézier curves.
//!
//! This is based on `WebCore/platform/graphics/UnitBezier.h` in WebKit.
#![deny(missing_docs)]
use crate::values::CSSFloat;
const NEWTON_METHOD_ITERATIONS: u8 = 8;
/// A unit cubic Bézier curve, used for timing functions in CSS transitions and animations.
pub struct Bezier {
ax: f64,
bx: f64,
cx: f64,
ay: f64,
by: f64,
cy: f64,
}
impl Bezier {
/// Calculate the output of a unit cubic Bézier curve from the two middle control points.
///
/// X coordinate is time, Y coordinate is function advancement.
/// The nominal range for both is 0 to 1.
///
/// The start and end points are always (0, 0) and (1, 1) so that a transition or animation
/// starts at 0% and ends at 100%.
pub fn calculate_bezier_output(
progress: f64,
epsilon: f64,
x1: f32,
y1: f32,
x2: f32,
y2: f32,
) -> f64 {
// Check for a linear curve.
if x1 == y1 && x2 == y2 {
return progress;
}
// Ensure that we return 0 or 1 on both edges.
if progress == 0.0 {
return 0.0;
}
if progress == 1.0 {
return 1.0;
}
// For negative values, try to extrapolate with tangent (p1 - p0) or,
// if p1 is coincident with p0, with (p2 - p0).
if progress < 0.0 {
if x1 > 0.0 {
return progress * y1 as f64 / x1 as f64;
}
if y1 == 0.0 && x2 > 0.0 {
return progress * y2 as f64 / x2 as f64;
}
// If we can't calculate a sensible tangent, don't extrapolate at all.
return 0.0;
}
// For values greater than 1, try to extrapolate with tangent (p2 - p3) or,
// if p2 is coincident with p3, with (p1 - p3).
if progress > 1.0 {
if x2 < 1.0 {
return 1.0 + (progress - 1.0) * (y2 as f64 - 1.0) / (x2 as f64 - 1.0);
}
if y2 == 1.0 && x1 < 1.0 {
return 1.0 + (progress - 1.0) * (y1 as f64 - 1.0) / (x1 as f64 - 1.0);
}
// If we can't calculate a sensible tangent, don't extrapolate at all.
return 1.0;
}
Bezier::new(x1, y1, x2, y2).solve(progress, epsilon)
}
#[inline]
fn new(x1: CSSFloat, y1: CSSFloat, x2: CSSFloat, y2: CSSFloat) -> Bezier {
let cx = 3. * x1 as f64;
let bx = 3. * (x2 as f64 - x1 as f64) - cx;
let cy = 3. * y1 as f64;
let by = 3. * (y2 as f64 - y1 as f64) - cy;
Bezier {
ax: 1.0 - cx - bx,
bx: bx,
cx: cx,
ay: 1.0 - cy - by,
by: by,
cy: cy,
}
}
#[inline]
fn sample_curve_x(&self, t: f64) -> f64 {
// ax * t^3 + bx * t^2 + cx * t
((self.ax * t + self.bx) * t + self.cx) * t
}
#[inline]
fn sample_curve_y(&self, t: f64) -> f64 {
((self.ay * t + self.by) * t + self.cy) * t
}
#[inline]
fn sample_curve_derivative_x(&self, t: f64) -> f64 {
(3.0 * self.ax * t + 2.0 * self.bx) * t + self.cx
}
#[inline]
fn solve_curve_x(&self, x: f64, epsilon: f64) -> f64 {
// Fast path: Use Newton's method.
let mut t = x;
for _ in 0..NEWTON_METHOD_ITERATIONS {
let x2 = self.sample_curve_x(t);
if x2.approx_eq(x, epsilon) {
return t;
}
let dx = self.sample_curve_derivative_x(t);
if dx.approx_eq(0.0, 1e-6) {
break;
}
t -= (x2 - x) / dx;
}
// Slow path: Use bisection.
let (mut lo, mut hi, mut t) = (0.0, 1.0, x);
if t < lo {
return lo;
}
if t > hi {
return hi;
}
while lo < hi {
let x2 = self.sample_curve_x(t);
if x2.approx_eq(x, epsilon) {
return t;
}
if x > x2 {
lo = t
} else {
hi = t
}
t = (hi - lo) / 2.0 + lo
}
t
}
/// Solve the bezier curve for a given `x` and an `epsilon`, that should be
/// between zero and one.
#[inline]
fn solve(&self, x: f64, epsilon: f64) -> f64 {
self.sample_curve_y(self.solve_curve_x(x, epsilon))
}
}
trait ApproxEq {
fn approx_eq(self, value: Self, epsilon: Self) -> bool;
}
impl ApproxEq for f64 {
#[inline]
fn approx_eq(self, value: f64, epsilon: f64) -> bool {
(self - value).abs() < epsilon
}
}