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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
use api::units::*;
use euclid::Size2D;
use std::f32::consts::FRAC_PI_2;
/// Number of steps to integrate arc length over.
const STEP_COUNT: usize = 20;
/// Represents an ellipse centred at a local space origin.
#[derive(Debug, Clone)]
pub struct Ellipse<U> {
pub radius: Size2D<f32, U>,
pub total_arc_length: f32,
}
impl<U> Ellipse<U> {
pub fn new(radius: Size2D<f32, U>) -> Ellipse<U> {
// Approximate the total length of the first quadrant of this ellipse.
let total_arc_length = get_simpson_length(FRAC_PI_2, radius.width, radius.height);
Ellipse {
radius,
total_arc_length,
}
}
/// Binary search to estimate the angle of an ellipse
/// for a given arc length. This only searches over the
/// first quadrant of an ellipse.
pub fn find_angle_for_arc_length(&self, arc_length: f32) -> f32 {
// Clamp arc length to [0, pi].
let arc_length = arc_length.max(0.0).min(self.total_arc_length);
let epsilon = 0.01;
let mut low = 0.0;
let mut high = FRAC_PI_2;
let mut theta = 0.0;
let mut new_low = 0.0;
let mut new_high = FRAC_PI_2;
while low <= high {
theta = 0.5 * (low + high);
let length = get_simpson_length(theta, self.radius.width, self.radius.height);
if (length - arc_length).abs() < epsilon {
break;
} else if length < arc_length {
new_low = theta;
} else {
new_high = theta;
}
// If we have stopped moving down the arc, the answer that we have is as good as
// it is going to get. We break to avoid going into an infinite loop.
if new_low == low && new_high == high {
break;
}
high = new_high;
low = new_low;
}
theta
}
/// Get a point and tangent on this ellipse from a given angle.
/// This only works for the first quadrant of the ellipse.
pub fn get_point_and_tangent(&self, theta: f32) -> (LayoutPoint, LayoutPoint) {
let (sin_theta, cos_theta) = theta.sin_cos();
let point = LayoutPoint::new(
self.radius.width * cos_theta,
self.radius.height * sin_theta,
);
let tangent = LayoutPoint::new(
-self.radius.width * sin_theta,
self.radius.height * cos_theta,
);
(point, tangent)
}
pub fn contains(&self, point: LayoutPoint) -> bool {
self.signed_distance(point.to_vector()) <= 0.0
}
/// Find the signed distance from this ellipse given a point.
fn signed_distance(&self, point: LayoutVector2D) -> f32 {
// This algorithm fails for circles, so we handle them here.
if self.radius.width == self.radius.height {
return point.length() - self.radius.width;
}
let mut p = LayoutVector2D::new(point.x.abs(), point.y.abs());
let mut ab = self.radius.to_vector();
if p.x > p.y {
p = p.yx();
ab = ab.yx();
}
let l = ab.y * ab.y - ab.x * ab.x;
let m = ab.x * p.x / l;
let n = ab.y * p.y / l;
let m2 = m * m;
let n2 = n * n;
let c = (m2 + n2 - 1.0) / 3.0;
let c3 = c * c * c;
let q = c3 + m2 * n2 * 2.0;
let d = c3 + m2 * n2;
let g = m + m * n2;
let co = if d < 0.0 {
let p = (q / c3).acos() / 3.0;
let s = p.cos();
let t = p.sin() * (3.0_f32).sqrt();
let rx = (-c * (s + t + 2.0) + m2).sqrt();
let ry = (-c * (s - t + 2.0) + m2).sqrt();
(ry + l.signum() * rx + g.abs() / (rx * ry) - m) / 2.0
} else {
let h = 2.0 * m * n * d.sqrt();
let s = (q + h).signum() * (q + h).abs().powf(1.0 / 3.0);
let u = (q - h).signum() * (q - h).abs().powf(1.0 / 3.0);
let rx = -s - u - c * 4.0 + 2.0 * m2;
let ry = (s - u) * (3.0_f32).sqrt();
let rm = (rx * rx + ry * ry).sqrt();
let p = ry / (rm - rx).sqrt();
(p + 2.0 * g / rm - m) / 2.0
};
let si = (1.0 - co * co).sqrt();
let r = LayoutVector2D::new(ab.x * co, ab.y * si);
(r - p).length() * (p.y - r.y).signum()
}
}
/// Use Simpsons rule to approximate the arc length of
/// part of an ellipse. Note that this only works over
/// the range of [0, pi/2].
// TODO(gw): This is a simplistic way to estimate the
// arc length of an ellipse segment. We can probably use
// a faster / more accurate method!
fn get_simpson_length(theta: f32, rx: f32, ry: f32) -> f32 {
let df = theta / STEP_COUNT as f32;
let mut sum = 0.0;
for i in 0 .. (STEP_COUNT + 1) {
let (sin_theta, cos_theta) = (i as f32 * df).sin_cos();
let a = rx * sin_theta;
let b = ry * cos_theta;
let y = (a * a + b * b).sqrt();
let q = if i == 0 || i == STEP_COUNT {
1.0
} else if i % 2 == 0 {
2.0
} else {
4.0
};
sum += q * y;
}
(df / 3.0) * sum
}
#[cfg(test)]
pub mod test {
use super::*;
#[test]
fn find_angle_for_arc_length_for_long_eclipse() {
// Ensure that finding the angle on giant ellipses produces and answer and
// doesn't send us into an infinite loop.
let ellipse = Ellipse::new(LayoutSize::new(57500.0, 25.0));
let _ = ellipse.find_angle_for_arc_length(55674.53);
assert!(true);
let ellipse = Ellipse::new(LayoutSize::new(25.0, 57500.0));
let _ = ellipse.find_angle_for_arc_length(55674.53);
assert!(true);
}
}