webrender/
ellipse.rs

1/* This Source Code Form is subject to the terms of the Mozilla Public
2 * License, v. 2.0. If a copy of the MPL was not distributed with this
3 * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
4
5use api::units::*;
6use euclid::Size2D;
7use std::f32::consts::FRAC_PI_2;
8
9
10/// Number of steps to integrate arc length over.
11const STEP_COUNT: usize = 20;
12
13/// Represents an ellipse centred at a local space origin.
14#[derive(Debug, Clone)]
15pub struct Ellipse<U> {
16    pub radius: Size2D<f32, U>,
17    pub total_arc_length: f32,
18}
19
20impl<U> Ellipse<U> {
21    pub fn new(radius: Size2D<f32, U>) -> Ellipse<U> {
22        // Approximate the total length of the first quadrant of this ellipse.
23        let total_arc_length = get_simpson_length(FRAC_PI_2, radius.width, radius.height);
24
25        Ellipse {
26            radius,
27            total_arc_length,
28        }
29    }
30
31    /// Binary search to estimate the angle of an ellipse
32    /// for a given arc length. This only searches over the
33    /// first quadrant of an ellipse.
34    pub fn find_angle_for_arc_length(&self, arc_length: f32) -> f32 {
35        // Clamp arc length to [0, pi].
36        let arc_length = arc_length.max(0.0).min(self.total_arc_length);
37
38        let epsilon = 0.01;
39        let mut low = 0.0;
40        let mut high = FRAC_PI_2;
41        let mut theta = 0.0;
42        let mut new_low = 0.0;
43        let mut new_high = FRAC_PI_2;
44
45        while low <= high {
46            theta = 0.5 * (low + high);
47            let length = get_simpson_length(theta, self.radius.width, self.radius.height);
48
49            if (length - arc_length).abs() < epsilon {
50                break;
51            } else if length < arc_length {
52                new_low = theta;
53            } else {
54                new_high = theta;
55            }
56
57            // If we have stopped moving down the arc, the answer that we have is as good as
58            // it is going to get. We break to avoid going into an infinite loop.
59            if new_low == low && new_high == high {
60                break;
61            }
62
63            high = new_high;
64            low = new_low;
65        }
66
67        theta
68    }
69
70    /// Get a point and tangent on this ellipse from a given angle.
71    /// This only works for the first quadrant of the ellipse.
72    pub fn get_point_and_tangent(&self, theta: f32) -> (LayoutPoint, LayoutPoint) {
73        let (sin_theta, cos_theta) = theta.sin_cos();
74        let point = LayoutPoint::new(
75            self.radius.width * cos_theta,
76            self.radius.height * sin_theta,
77        );
78        let tangent = LayoutPoint::new(
79            -self.radius.width * sin_theta,
80            self.radius.height * cos_theta,
81        );
82        (point, tangent)
83    }
84
85    pub fn contains(&self, point: LayoutPoint) -> bool {
86        self.signed_distance(point.to_vector()) <= 0.0
87    }
88
89    /// Find the signed distance from this ellipse given a point.
90    /// Taken from http://www.iquilezles.org/www/articles/ellipsedist/ellipsedist.htm
91    fn signed_distance(&self, point: LayoutVector2D) -> f32 {
92        // This algorithm fails for circles, so we handle them here.
93        if self.radius.width == self.radius.height {
94            return point.length() - self.radius.width;
95        }
96
97        let mut p = LayoutVector2D::new(point.x.abs(), point.y.abs());
98        let mut ab = self.radius.to_vector();
99        if p.x > p.y {
100            p = p.yx();
101            ab = ab.yx();
102        }
103
104        let l = ab.y * ab.y - ab.x * ab.x;
105
106        let m = ab.x * p.x / l;
107        let n = ab.y * p.y / l;
108        let m2 = m * m;
109        let n2 = n * n;
110
111        let c = (m2 + n2 - 1.0) / 3.0;
112        let c3 = c * c * c;
113
114        let q = c3 + m2 * n2 * 2.0;
115        let d = c3 + m2 * n2;
116        let g = m + m * n2;
117
118        let co = if d < 0.0 {
119            let p = (q / c3).acos() / 3.0;
120            let s = p.cos();
121            let t = p.sin() * (3.0_f32).sqrt();
122            let rx = (-c * (s + t + 2.0) + m2).sqrt();
123            let ry = (-c * (s - t + 2.0) + m2).sqrt();
124            (ry + l.signum() * rx + g.abs() / (rx * ry) - m) / 2.0
125        } else {
126            let h = 2.0 * m * n * d.sqrt();
127            let s = (q + h).signum() * (q + h).abs().powf(1.0 / 3.0);
128            let u = (q - h).signum() * (q - h).abs().powf(1.0 / 3.0);
129            let rx = -s - u - c * 4.0 + 2.0 * m2;
130            let ry = (s - u) * (3.0_f32).sqrt();
131            let rm = (rx * rx + ry * ry).sqrt();
132            let p = ry / (rm - rx).sqrt();
133            (p + 2.0 * g / rm - m) / 2.0
134        };
135
136        let si = (1.0 - co * co).sqrt();
137        let r = LayoutVector2D::new(ab.x * co, ab.y * si);
138        (r - p).length() * (p.y - r.y).signum()
139    }
140}
141
142/// Use Simpsons rule to approximate the arc length of
143/// part of an ellipse. Note that this only works over
144/// the range of [0, pi/2].
145// TODO(gw): This is a simplistic way to estimate the
146// arc length of an ellipse segment. We can probably use
147// a faster / more accurate method!
148fn get_simpson_length(theta: f32, rx: f32, ry: f32) -> f32 {
149    let df = theta / STEP_COUNT as f32;
150    let mut sum = 0.0;
151
152    for i in 0 .. (STEP_COUNT + 1) {
153        let (sin_theta, cos_theta) = (i as f32 * df).sin_cos();
154        let a = rx * sin_theta;
155        let b = ry * cos_theta;
156        let y = (a * a + b * b).sqrt();
157        let q = if i == 0 || i == STEP_COUNT {
158            1.0
159        } else if i % 2 == 0 {
160            2.0
161        } else {
162            4.0
163        };
164
165        sum += q * y;
166    }
167
168    (df / 3.0) * sum
169}
170
171#[cfg(test)]
172pub mod test {
173    use super::*;
174
175    #[test]
176    fn find_angle_for_arc_length_for_long_eclipse() {
177        // Ensure that finding the angle on giant ellipses produces and answer and
178        // doesn't send us into an infinite loop.
179        let ellipse = Ellipse::new(LayoutSize::new(57500.0, 25.0));
180        let _ = ellipse.find_angle_for_arc_length(55674.53);
181        assert!(true);
182
183        let ellipse = Ellipse::new(LayoutSize::new(25.0, 57500.0));
184        let _ = ellipse.find_angle_for_arc_length(55674.53);
185        assert!(true);
186    }
187}
webrender/ellipse.rs

webrender/
ellipse.rs
