phanoob
1f7f37d1ef
When an extrude cuts with profiles whose outlines stay more than 1 µm apart (the letters of a text), their tools are apart too, so they are joined and cut from the body in one boolean instead of one per profile. A ten-letter text cut drops from about 13 s to 1.5 s with the same result. Sheet metal targets and touching profiles keep cutting tool by tool. Tapered extrudes are unaffected.
851 lines
37 KiB
Rust
851 lines
37 KiB
Rust
//! Planar profiles: closed loops of lines, circular arcs, cubic Béziers and conics (rational
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//! quadratic Béziers), and regions (an outer loop minus holes). Sketches produce them; the
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//! kernel turns them into faces.
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use std::f64::consts::TAU;
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use serde::{Deserialize, Serialize};
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use crate::Vec2;
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/// One boundary segment in sketch coordinates.
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#[derive(Clone, Copy, Debug, PartialEq, Serialize, Deserialize)]
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#[serde(tag = "type", rename_all = "snake_case")]
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pub enum Seg2 {
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Line {
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a: Vec2,
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b: Vec2,
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},
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/// Arc from `start` angle sweeping `sweep` radians (positive = counter-clockwise).
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Arc {
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center: Vec2,
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radius: f64,
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start: f64,
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sweep: f64,
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},
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/// Cubic Bézier.
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Cubic {
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p0: Vec2,
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p1: Vec2,
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p2: Vec2,
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p3: Vec2,
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},
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/// Rational quadratic Bézier from `a` to `b` with control `apex` of weight `w` (end
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/// weights 1): exact conic arcs (ellipses, parabolas, hyperbolas).
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Conic {
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a: Vec2,
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apex: Vec2,
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b: Vec2,
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w: f64,
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},
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/// Bézier of degree `n` (4…7) over `p[0..=n]` (the rest unused): pieces of higher-degree
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/// splines.
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Bezier {
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n: u8,
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p: [Vec2; 8],
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},
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}
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/// Highest degree of [`Seg2::Bezier`].
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pub const MAX_BEZIER_DEGREE: usize = 7;
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/// A Bézier segment over `pts` (2…8 points): a line, a cubic or a higher-degree piece. A
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/// quadratic becomes a conic of weight 1.
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pub fn bezier(pts: &[Vec2]) -> Option<Seg2> {
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Some(match *pts {
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[a, b] => Seg2::Line { a, b },
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[a, apex, b] => Seg2::Conic { a, apex, b, w: 1.0 },
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[p0, p1, p2, p3] => Seg2::Cubic { p0, p1, p2, p3 },
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_ if pts.len() <= MAX_BEZIER_DEGREE + 1 => {
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let mut p = [Vec2::ZERO; 8];
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for (d, s) in p.iter_mut().zip(pts) {
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*d = *s;
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}
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Seg2::Bezier { n: (pts.len() - 1) as u8, p }
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}
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_ => return None,
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})
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}
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/// de Casteljau: the point at `t` and the two halves' control points.
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fn casteljau(c: &[Vec2], t: f64) -> (Vec2, Vec<Vec2>, Vec<Vec2>) {
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let mut w = c.to_vec();
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let mut left = Vec::with_capacity(c.len());
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let mut right = Vec::with_capacity(c.len());
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while let Some(first) = w.first().copied() {
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left.push(first);
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if let Some(last) = w.last().copied() {
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right.push(last);
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}
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if w.len() == 1 {
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break;
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}
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w = w.windows(2).map(|x| x[0].lerp(x[1], t)).collect();
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}
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right.reverse();
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(left.last().copied().unwrap_or(Vec2::ZERO), left, right)
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}
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/// Gauss–Legendre nodes and weights on [0, 1] (5 points).
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const GL5: [(f64, f64); 5] = [
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(0.046_910_077_030_668, 0.118_463_442_528_095),
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(0.230_765_344_947_158, 0.239_314_335_249_683),
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(0.5, 0.284_444_444_444_444),
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(0.769_234_655_052_842, 0.239_314_335_249_683),
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(0.953_089_922_969_332, 0.118_463_442_528_095),
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];
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impl Seg2 {
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fn bez(n: u8, p: &[Vec2; 8]) -> &[Vec2] {
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&p[..(n as usize).clamp(1, MAX_BEZIER_DEGREE) + 1]
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}
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/// Derivative with respect to the parameter at `t` (Béziers and conics; lines and arcs too).
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pub fn derivative(&self, t: f64) -> Vec2 {
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match *self {
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Seg2::Line { a, b } => b - a,
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Seg2::Arc { radius, start, sweep, .. } => Vec2::from_angle(start + sweep * t).perp() * (radius * sweep),
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Seg2::Cubic { p0, p1, p2, p3 } => {
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let u = 1.0 - t;
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(p1 - p0) * (3.0 * u * u) + (p2 - p1) * (6.0 * u * t) + (p3 - p2) * (3.0 * t * t)
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}
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Seg2::Conic { a, apex, b, w } => {
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let u = 1.0 - t;
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let (n, d) = (a * (u * u) + apex * (2.0 * w * u * t) + b * (t * t), u * u + 2.0 * w * u * t + t * t);
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let (dn, dd) = (a * (-2.0 * u) + apex * (2.0 * w * (1.0 - 2.0 * t)) + b * (2.0 * t), -2.0 * u + 2.0 * w * (1.0 - 2.0 * t) + 2.0 * t);
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(dn * d - n * dd) / (d * d).max(1e-300)
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}
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Seg2::Bezier { n, ref p } => {
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let c = Self::bez(n, p);
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let diffs: Vec<Vec2> = c.windows(2).map(|w| (w[1] - w[0]) * (c.len() - 1) as f64).collect();
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casteljau(&diffs, t).0
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}
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}
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}
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pub fn start(&self) -> Vec2 {
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match *self {
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Seg2::Line { a, .. } | Seg2::Conic { a, .. } => a,
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Seg2::Cubic { p0, .. } => p0,
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Seg2::Bezier { p, .. } => p[0],
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Seg2::Arc { center, radius, start, .. } => center + Vec2::from_angle(start) * radius,
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}
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}
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pub fn end(&self) -> Vec2 {
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match *self {
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Seg2::Line { b, .. } | Seg2::Conic { b, .. } => b,
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Seg2::Cubic { p3, .. } => p3,
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Seg2::Bezier { n, ref p } => Self::bez(n, p).last().copied().unwrap_or(p[0]),
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Seg2::Arc { center, radius, start, sweep } => center + Vec2::from_angle(start + sweep) * radius,
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}
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}
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pub fn point_at(&self, t: f64) -> Vec2 {
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match *self {
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Seg2::Line { a, b } => a.lerp(b, t),
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Seg2::Arc { center, radius, start, sweep } => center + Vec2::from_angle(start + sweep * t) * radius,
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Seg2::Cubic { p0, p1, p2, p3 } => {
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let u = 1.0 - t;
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p0 * (u * u * u) + p1 * (3.0 * u * u * t) + p2 * (3.0 * u * t * t) + p3 * (t * t * t)
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}
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Seg2::Conic { a, apex, b, w } => {
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let u = 1.0 - t;
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let d = u * u + 2.0 * w * u * t + t * t;
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(a * (u * u) + apex * (2.0 * w * u * t) + b * (t * t)) / d.max(1e-300)
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}
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Seg2::Bezier { n, ref p } => casteljau(Self::bez(n, p), t).0,
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}
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}
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pub fn mid(&self) -> Vec2 {
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self.point_at(0.5)
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}
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/// Unit tangent at the start (direction of travel).
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pub fn start_tangent(&self) -> Vec2 {
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match *self {
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Seg2::Line { a, b } => (b - a).normalized().unwrap_or(Vec2::X),
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Seg2::Arc { start, sweep, .. } => Vec2::from_angle(start).perp() * sweep.signum(),
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Seg2::Cubic { p0, p1, p2, .. } => (p1 - p0).normalized().or_else(|| (p2 - p0).normalized()).unwrap_or(Vec2::X),
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Seg2::Conic { a, apex, b, .. } => (apex - a).normalized().or_else(|| (b - a).normalized()).unwrap_or(Vec2::X),
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Seg2::Bezier { n, ref p } => {
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let c = Self::bez(n, p);
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c.iter().skip(1).find_map(|q| (*q - c[0]).normalized()).unwrap_or(Vec2::X)
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}
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}
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}
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/// Unit tangent at the end (direction of travel).
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pub fn end_tangent(&self) -> Vec2 {
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match *self {
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Seg2::Line { a, b } => (b - a).normalized().unwrap_or(Vec2::X),
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Seg2::Arc { start, sweep, .. } => Vec2::from_angle(start + sweep).perp() * sweep.signum(),
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Seg2::Cubic { p1, p2, p3, .. } => (p3 - p2).normalized().or_else(|| (p3 - p1).normalized()).unwrap_or(Vec2::X),
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Seg2::Conic { a, apex, b, .. } => (b - apex).normalized().or_else(|| (b - a).normalized()).unwrap_or(Vec2::X),
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Seg2::Bezier { n, ref p } => {
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let c = Self::bez(n, p);
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let last = c.last().copied().unwrap_or(p[0]);
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c.iter().rev().skip(1).find_map(|q| (last - *q).normalized()).unwrap_or(Vec2::X)
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}
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}
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}
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pub fn reversed(&self) -> Seg2 {
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match *self {
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Seg2::Line { a, b } => Seg2::Line { a: b, b: a },
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Seg2::Arc { center, radius, start, sweep } => Seg2::Arc { center, radius, start: start + sweep, sweep: -sweep },
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Seg2::Cubic { p0, p1, p2, p3 } => Seg2::Cubic { p0: p3, p1: p2, p2: p1, p3: p0 },
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Seg2::Conic { a, apex, b, w } => Seg2::Conic { a: b, apex, b: a, w },
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Seg2::Bezier { n, ref p } => {
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let mut q = *p;
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let k = (n as usize).clamp(1, MAX_BEZIER_DEGREE);
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q[..=k].reverse();
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Seg2::Bezier { n, p: q }
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}
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}
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}
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pub fn length(&self) -> f64 {
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match *self {
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Seg2::Line { a, b } => a.dist(b),
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Seg2::Arc { radius, sweep, .. } => (radius * sweep).abs(),
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Seg2::Cubic { .. } | Seg2::Conic { .. } | Seg2::Bezier { .. } => self.integrate(|s, t| s.derivative(t).len()),
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}
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}
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/// The segment split at parameter `t` in (0, 1) (same kind, exact; conic halves are
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/// re-normalised so their end weights stay 1).
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pub fn split_at(&self, t: f64) -> (Seg2, Seg2) {
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let t = t.clamp(1e-9, 1.0 - 1e-9);
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match *self {
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Seg2::Line { a, b } => {
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let m = a.lerp(b, t);
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(Seg2::Line { a, b: m }, Seg2::Line { a: m, b })
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}
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Seg2::Arc { center, radius, start, sweep } => (
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Seg2::Arc { center, radius, start, sweep: sweep * t },
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Seg2::Arc { center, radius, start: start + sweep * t, sweep: sweep * (1.0 - t) },
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),
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Seg2::Cubic { p0, p1, p2, p3 } => {
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let (a, b, c) = (p0.lerp(p1, t), p1.lerp(p2, t), p2.lerp(p3, t));
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let (d, e) = (a.lerp(b, t), b.lerp(c, t));
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let m = d.lerp(e, t);
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(Seg2::Cubic { p0, p1: a, p2: d, p3: m }, Seg2::Cubic { p0: m, p1: e, p2: c, p3 })
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}
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Seg2::Conic { a, apex, b, w } => {
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// Homogeneous de Casteljau: (a, 1), (w·apex, w), (b, 1).
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let h = |p: Vec2, wt: f64| (p * wt, wt);
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let lerp = |x: (Vec2, f64), y: (Vec2, f64)| (x.0 * (1.0 - t) + y.0 * t, x.1 * (1.0 - t) + y.1 * t);
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let (p0, p1, p2) = (h(a, 1.0), h(apex, w), h(b, 1.0));
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let (q0, q1) = (lerp(p0, p1), lerp(p1, p2));
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let r = lerp(q0, q1);
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let pt = |x: (Vec2, f64)| x.0 / x.1.max(1e-300);
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let m = pt(r);
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(
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Seg2::Conic { a, apex: pt(q0), b: m, w: q0.1 / r.1.max(1e-300).sqrt() },
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Seg2::Conic { a: m, apex: pt(q1), b, w: q1.1 / r.1.max(1e-300).sqrt() },
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)
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}
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Seg2::Bezier { n, ref p } => {
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let (_, l, r) = casteljau(Self::bez(n, p), t);
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let mk = |c: &[Vec2]| bezier(c).unwrap_or(*self);
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(mk(&l), mk(&r))
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}
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}
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}
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/// Parameter of the closest point: exact projection for lines,
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/// sampling followed by Newton for the remaining curves.
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pub fn closest_param(&self, p: Vec2) -> f64 {
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let finite = |p: Vec2| p.x.is_finite() && p.y.is_finite();
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match *self {
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Seg2::Line { a, b } if finite(a) && finite(b) && finite(p) => {
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let d = b - a;
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if a == b {
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return 0.0;
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}
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let denominator = d.len2();
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let t = (p - a).dot(d) / denominator;
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if denominator > 0.0 && denominator.is_finite() && t.is_finite() {
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return t.clamp(0.0, 1.0);
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}
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}
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_ => {}
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}
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let n = 64;
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let mut t =
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(0..=n).map(|i| i as f64 / n as f64).min_by(|x, y| self.point_at(*x).dist(p).total_cmp(&self.point_at(*y).dist(p))).unwrap_or(0.0);
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for _ in 0..30 {
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let (q, d) = (self.point_at(t), self.derivative(t));
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let (d2, h) = ((self.derivative((t + 1e-6).min(1.0)) - self.derivative((t - 1e-6).max(0.0))) / 2e-6, q - p);
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let f = h.dot(d);
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let df = d.dot(d) + h.dot(d2);
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if df.abs() < 1e-300 {
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break;
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}
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let nt = (t - f / df).clamp(0.0, 1.0);
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if (nt - t).abs() < 1e-14 {
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t = nt;
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break;
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}
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t = nt;
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}
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t
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}
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/// The two halves of the segment (same kind, exact).
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pub fn split_half(&self) -> (Seg2, Seg2) {
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match *self {
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Seg2::Line { a, b } => {
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let m = a.lerp(b, 0.5);
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(Seg2::Line { a, b: m }, Seg2::Line { a: m, b })
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}
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Seg2::Arc { center, radius, start, sweep } => (
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Seg2::Arc { center, radius, start, sweep: sweep / 2.0 },
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Seg2::Arc { center, radius, start: start + sweep / 2.0, sweep: sweep / 2.0 },
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),
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Seg2::Cubic { p0, p1, p2, p3 } => {
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let (a, b, c) = (p0.lerp(p1, 0.5), p1.lerp(p2, 0.5), p2.lerp(p3, 0.5));
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let (d, e) = (a.lerp(b, 0.5), b.lerp(c, 0.5));
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let m = d.lerp(e, 0.5);
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(Seg2::Cubic { p0, p1: a, p2: d, p3: m }, Seg2::Cubic { p0: m, p1: e, p2: c, p3 })
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}
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Seg2::Conic { a, apex, b, w } => {
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// Rational de Casteljau at the middle, weights normalised to 1 at the ends.
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let m = self.point_at(0.5);
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let w2 = ((1.0 + w) / 2.0).max(0.0).sqrt();
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let (l, r) = ((a + apex * w) / (1.0 + w), (apex * w + b) / (1.0 + w));
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(Seg2::Conic { a, apex: l, b: m, w: w2 }, Seg2::Conic { a: m, apex: r, b, w: w2 })
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}
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Seg2::Bezier { .. } => self.split_at(0.5),
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}
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}
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/// ∫₀¹ f(t) dt by composite Gauss–Legendre (16 pieces of 5 points).
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fn integrate(&self, f: impl Fn(&Seg2, f64) -> f64) -> f64 {
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let n = 16;
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let mut sum = 0.0;
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for k in 0..n {
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let (t0, h) = (k as f64 / n as f64, 1.0 / n as f64);
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for (x, w) in GL5 {
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sum += w * h * f(self, t0 + x * h);
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}
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}
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sum
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}
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/// Contribution to the loop's signed area (shoelace integral ½∮(x dy − y dx)).
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pub fn area_term(&self) -> f64 {
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match *self {
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Seg2::Line { a, b } => 0.5 * a.cross(b),
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Seg2::Arc { radius, sweep, .. } => {
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let (a, b) = (self.start(), self.end());
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// Chord term plus the circular segment between chord and arc.
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0.5 * a.cross(b) + 0.5 * radius * radius * (sweep - sweep.sin())
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}
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Seg2::Cubic { .. } | Seg2::Conic { .. } | Seg2::Bezier { .. } => self.integrate(|s, t| 0.5 * s.point_at(t).cross(s.derivative(t))),
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}
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}
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/// Polyline approximation (including both end points) with at most `tol` chord error.
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pub fn polyline(&self, tol: f64) -> Vec<Vec2> {
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match *self {
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Seg2::Line { a, b } => vec![a, b],
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Seg2::Arc { radius, sweep, .. } => {
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let tol = tol.max(radius.abs() * 1e-6).max(1e-9);
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let step = if tol < radius { 2.0 * (1.0 - tol / radius).clamp(-1.0, 1.0).acos() } else { TAU / 8.0 };
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let n = ((sweep.abs() / step.max(1e-6)).ceil() as usize).clamp(2, 4096);
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(0..=n).map(|i| self.point_at(i as f64 / n as f64)).collect()
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}
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Seg2::Cubic { p0, p1, p2, p3 } => {
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// Chord error of n pieces ≈ max second difference / (8 n²).
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let bow = (p0 - p1 * 2.0 + p2).len().max((p1 - p2 * 2.0 + p3).len()) * 6.0;
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let n = ((bow / (8.0 * tol.max(1e-9))).sqrt().ceil() as usize).clamp(2, 4096);
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(0..=n).map(|i| self.point_at(i as f64 / n as f64)).collect()
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}
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Seg2::Conic { a, apex, b, w } => {
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let bow = (a - apex * 2.0 + b).len() * 2.0 * w.max(1.0);
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let n = ((bow / (8.0 * tol.max(1e-9))).sqrt().ceil() as usize).clamp(2, 4096);
|
||
(0..=n).map(|i| self.point_at(i as f64 / n as f64)).collect()
|
||
}
|
||
Seg2::Bezier { n, ref p } => {
|
||
// Second derivative ≤ n(n−1) × the largest second difference of the controls.
|
||
let c = Self::bez(n, p);
|
||
let d2 = c.windows(3).map(|w| (w[0] - w[1] * 2.0 + w[2]).len()).fold(0.0, f64::max) * (n as f64) * (n as f64 - 1.0);
|
||
let k = ((d2 / (8.0 * tol.max(1e-9))).sqrt().ceil() as usize).clamp(2, 4096);
|
||
(0..=k).map(|i| self.point_at(i as f64 / k as f64)).collect()
|
||
}
|
||
}
|
||
}
|
||
}
|
||
|
||
/// A closed loop of segments, each starting where the previous one ends.
|
||
#[derive(Clone, Debug, Default, PartialEq, Serialize, Deserialize)]
|
||
pub struct Loop2 {
|
||
pub segs: Vec<Seg2>,
|
||
}
|
||
|
||
impl Loop2 {
|
||
/// Full circle as two half arcs (counter-clockwise).
|
||
pub fn circle(center: Vec2, radius: f64) -> Loop2 {
|
||
Loop2 {
|
||
segs: vec![
|
||
Seg2::Arc { center, radius, start: 0.0, sweep: std::f64::consts::PI },
|
||
Seg2::Arc { center, radius, start: std::f64::consts::PI, sweep: std::f64::consts::PI },
|
||
],
|
||
}
|
||
}
|
||
/// Full circle starting (and split) at angle `start`.
|
||
pub fn circle_from(center: Vec2, radius: f64, start: f64) -> Loop2 {
|
||
use std::f64::consts::PI;
|
||
Loop2 { segs: vec![Seg2::Arc { center, radius, start, sweep: PI }, Seg2::Arc { center, radius, start: start + PI, sweep: PI }] }
|
||
}
|
||
pub fn polygon(pts: &[Vec2]) -> Loop2 {
|
||
let n = pts.len();
|
||
Loop2 { segs: (0..n).filter_map(|i| Some(Seg2::Line { a: *pts.get(i)?, b: *pts.get((i + 1) % n)? })).collect() }
|
||
}
|
||
/// Signed area: positive when counter-clockwise.
|
||
pub fn signed_area(&self) -> f64 {
|
||
self.segs.iter().map(Seg2::area_term).sum()
|
||
}
|
||
pub fn reversed(&self) -> Loop2 {
|
||
Loop2 { segs: self.segs.iter().rev().map(Seg2::reversed).collect() }
|
||
}
|
||
/// Whether the loop runs counter-clockwise ([`Loop2::ccw`] keeps it as it is).
|
||
pub fn is_ccw(&self) -> bool {
|
||
self.signed_area() >= 0.0
|
||
}
|
||
/// Counter-clockwise copy.
|
||
pub fn ccw(&self) -> Loop2 {
|
||
if self.is_ccw() { self.clone() } else { self.reversed() }
|
||
}
|
||
pub fn polyline(&self, tol: f64) -> Vec<Vec2> {
|
||
let mut out = Vec::new();
|
||
for s in &self.segs {
|
||
let p = s.polyline(tol);
|
||
out.extend(p.iter().take(p.len().saturating_sub(1)));
|
||
}
|
||
out
|
||
}
|
||
/// Even-odd point containment against the polyline approximation.
|
||
pub fn contains(&self, p: Vec2) -> bool {
|
||
let poly = self.polyline(1e-3);
|
||
let n = poly.len();
|
||
let mut inside = false;
|
||
for i in 0..n {
|
||
let (Some(a), Some(b)) = (poly.get(i), poly.get((i + 1) % n)) else { continue };
|
||
if (a.y > p.y) != (b.y > p.y) {
|
||
let x = a.x + (p.y - a.y) / (b.y - a.y) * (b.x - a.x);
|
||
if p.x < x {
|
||
inside = !inside;
|
||
}
|
||
}
|
||
}
|
||
inside
|
||
}
|
||
/// Area-weighted centroid (of the polyline approximation).
|
||
pub fn centroid(&self) -> Vec2 {
|
||
let poly = self.polyline(1e-3);
|
||
let n = poly.len();
|
||
let (mut a2, mut c) = (0.0, Vec2::ZERO);
|
||
for i in 0..n {
|
||
let (Some(p), Some(q)) = (poly.get(i), poly.get((i + 1) % n)) else { continue };
|
||
let k = p.cross(*q);
|
||
a2 += k;
|
||
c += (*p + *q) * k;
|
||
}
|
||
if a2.abs() < 1e-15 { poly.first().copied().unwrap_or_default() } else { c / (3.0 * a2) }
|
||
}
|
||
/// Closed within `tol`?
|
||
pub fn is_closed(&self, tol: f64) -> bool {
|
||
let n = self.segs.len();
|
||
n > 0
|
||
&& (0..n).all(|i| match (self.segs.get(i), self.segs.get((i + 1) % n)) {
|
||
(Some(a), Some(b)) => a.end().dist(b.start()) <= tol,
|
||
_ => false,
|
||
})
|
||
}
|
||
}
|
||
|
||
/// A planar region: outer loop (counter-clockwise) minus holes (clockwise).
|
||
#[derive(Clone, Debug, Default, PartialEq, Serialize, Deserialize)]
|
||
pub struct Region2 {
|
||
pub outer: Loop2,
|
||
pub holes: Vec<Loop2>,
|
||
}
|
||
|
||
impl Region2 {
|
||
pub fn area(&self) -> f64 {
|
||
self.outer.signed_area().abs() - self.holes.iter().map(|h| h.signed_area().abs()).sum::<f64>()
|
||
}
|
||
/// A point strictly inside the region (for picking and labels).
|
||
pub fn interior_point(&self) -> Vec2 {
|
||
let c = self.centroid();
|
||
if self.contains(c) {
|
||
return c;
|
||
}
|
||
// Scan a few horizontal lines for the midpoint of the widest inside span.
|
||
let poly = self.outer.polyline(1e-2);
|
||
let (mut y0, mut y1) = (f64::INFINITY, f64::NEG_INFINITY);
|
||
for p in &poly {
|
||
y0 = y0.min(p.y);
|
||
y1 = y1.max(p.y);
|
||
}
|
||
for k in 1..16 {
|
||
let y = y0 + (y1 - y0) * k as f64 / 16.0;
|
||
let mut xs: Vec<f64> = Vec::new();
|
||
for l in std::iter::once(&self.outer).chain(&self.holes) {
|
||
let pl = l.polyline(1e-2);
|
||
let n = pl.len();
|
||
for i in 0..n {
|
||
let (Some(a), Some(b)) = (pl.get(i), pl.get((i + 1) % n)) else { continue };
|
||
if (a.y > y) != (b.y > y) {
|
||
xs.push(a.x + (y - a.y) / (b.y - a.y) * (b.x - a.x));
|
||
}
|
||
}
|
||
}
|
||
xs.sort_by(f64::total_cmp);
|
||
for w in xs.chunks(2) {
|
||
if let [a, b] = w {
|
||
let p = Vec2::new((a + b) * 0.5, y);
|
||
if self.contains(p) {
|
||
return p;
|
||
}
|
||
}
|
||
}
|
||
}
|
||
c
|
||
}
|
||
pub fn contains(&self, p: Vec2) -> bool {
|
||
self.outer.contains(p) && !self.holes.iter().any(|h| h.contains(p))
|
||
}
|
||
/// Triangles covering the region (for filled display), by slabs between the vertex
|
||
/// heights: inside each slab the boundary edges don't cross, so consecutive crossings pair
|
||
/// up into trapezoids.
|
||
pub fn triangulate(&self, tol: f64) -> Vec<[Vec2; 3]> {
|
||
let mut edges: Vec<(Vec2, Vec2)> = Vec::new();
|
||
for l in std::iter::once(&self.outer).chain(&self.holes) {
|
||
let pl = l.polyline(tol);
|
||
let n = pl.len();
|
||
for i in 0..n {
|
||
let (Some(a), Some(b)) = (pl.get(i), pl.get((i + 1) % n)) else { continue };
|
||
if (a.y - b.y).abs() > 1e-12 {
|
||
edges.push(if a.y < b.y { (*a, *b) } else { (*b, *a) });
|
||
}
|
||
}
|
||
}
|
||
if edges.len() > 20_000 {
|
||
return Vec::new();
|
||
}
|
||
let mut ys: Vec<f64> = edges.iter().flat_map(|(a, b)| [a.y, b.y]).collect();
|
||
ys.sort_by(f64::total_cmp);
|
||
ys.dedup_by(|a, b| (*a - *b).abs() < 1e-12);
|
||
let x_at = |e: &(Vec2, Vec2), y: f64| e.0.x + (y - e.0.y) / (e.1.y - e.0.y) * (e.1.x - e.0.x);
|
||
// Sweep upward: an edge joins the slabs once its lower end is passed and leaves after
|
||
// its upper end, so each slab sorts only the edges crossing it (a finely sampled ring
|
||
// has hundreds of slabs and edges, but four edges in each slab).
|
||
edges.sort_by(|a, b| a.0.y.total_cmp(&b.0.y));
|
||
let mut next = 0;
|
||
let mut span: Vec<&(Vec2, Vec2)> = Vec::new();
|
||
let mut out = Vec::new();
|
||
for w in ys.windows(2) {
|
||
let (y0, y1) = (w[0], w[1]);
|
||
let ym = (y0 + y1) * 0.5;
|
||
while let Some(e) = edges.get(next)
|
||
&& e.0.y <= ym
|
||
{
|
||
span.push(e);
|
||
next += 1;
|
||
}
|
||
span.retain(|e| e.1.y >= ym);
|
||
span.sort_by(|a, b| x_at(a, ym).total_cmp(&x_at(b, ym)));
|
||
for pair in span.chunks(2) {
|
||
let [l, r] = pair else { continue };
|
||
let (a, b) = (Vec2::new(x_at(l, y0), y0), Vec2::new(x_at(r, y0), y0));
|
||
let (c, d) = (Vec2::new(x_at(r, y1), y1), Vec2::new(x_at(l, y1), y1));
|
||
out.push([a, b, c]);
|
||
out.push([a, c, d]);
|
||
}
|
||
}
|
||
out
|
||
}
|
||
pub fn centroid(&self) -> Vec2 {
|
||
let ao = self.outer.signed_area().abs();
|
||
let mut c = self.outer.centroid() * ao;
|
||
let mut a = ao;
|
||
for h in &self.holes {
|
||
let ah = h.signed_area().abs();
|
||
c -= h.centroid() * ah;
|
||
a -= ah;
|
||
}
|
||
if a.abs() < 1e-15 { self.outer.centroid() } else { c / a }
|
||
}
|
||
}
|
||
|
||
/// Most boundary segment pairs [`regions_apart`] compares before it gives up (and says no).
|
||
const APART_BUDGET: usize = 50_000_000;
|
||
|
||
/// Whether no two of the regions meet or come within `gap` of each other (their boundaries
|
||
/// are farther apart than `gap` and none lies inside another; one inside another's hole is
|
||
/// apart). Tools made from them alike (extruded over one interval) are then apart too, and
|
||
/// can be cut as one. False when unsure (a degenerate region, too much to compare).
|
||
pub fn regions_apart(regions: &[Region2], gap: f64) -> bool {
|
||
if !(gap.is_finite() && gap > 0.0) || regions.len() > 10_000 {
|
||
return false;
|
||
}
|
||
// The polylines lie within `tol` of the curves: boundaries this much farther than the gap
|
||
// apart are apart.
|
||
let tol = gap / 4.0;
|
||
let reach = gap + 2.0 * tol;
|
||
struct Outline {
|
||
loops: Vec<Vec<Vec2>>,
|
||
lo: Vec2,
|
||
hi: Vec2,
|
||
inside: Vec2,
|
||
}
|
||
let mut outlines = Vec::with_capacity(regions.len());
|
||
for r in regions {
|
||
let loops: Vec<Vec<Vec2>> = std::iter::once(&r.outer).chain(&r.holes).map(|l| l.polyline(tol)).collect();
|
||
let Some(outer) = loops.first().filter(|l| l.len() >= 2) else { return false };
|
||
if outer.iter().any(|p| !p.is_finite()) {
|
||
return false;
|
||
}
|
||
let lo = outer.iter().fold(Vec2::new(f64::MAX, f64::MAX), |m, p| Vec2::new(m.x.min(p.x), m.y.min(p.y)));
|
||
let hi = outer.iter().fold(Vec2::new(f64::MIN, f64::MIN), |m, p| Vec2::new(m.x.max(p.x), m.y.max(p.y)));
|
||
let Some(inside) = outer.first().copied() else { return false };
|
||
outlines.push(Outline { loops, lo, hi, inside });
|
||
}
|
||
let mut order: Vec<usize> = (0..outlines.len()).collect();
|
||
order.sort_by(|&i, &j| {
|
||
let x = |k: usize| outlines.get(k).map(|o| o.lo.x).unwrap_or(0.0);
|
||
x(i).total_cmp(&x(j))
|
||
});
|
||
let mut work = 0usize;
|
||
for (k, &i) in order.iter().enumerate() {
|
||
let Some(a) = outlines.get(i) else { return false };
|
||
for &j in order.iter().skip(k + 1) {
|
||
let Some(b) = outlines.get(j) else { return false };
|
||
if b.lo.x > a.hi.x + reach {
|
||
break;
|
||
}
|
||
if b.lo.y > a.hi.y + reach || a.lo.y > b.hi.y + reach {
|
||
continue;
|
||
}
|
||
// A boundary point of one inside the other: one holds the other (boundaries that do
|
||
// not come near each other do not cross).
|
||
let (Some(ra), Some(rb)) = (regions.get(i), regions.get(j)) else { return false };
|
||
if ra.contains(b.inside) || rb.contains(a.inside) {
|
||
return false;
|
||
}
|
||
for la in &a.loops {
|
||
for lb in &b.loops {
|
||
work = work.saturating_add(la.len().saturating_mul(lb.len()));
|
||
if work > APART_BUDGET {
|
||
return false;
|
||
}
|
||
// (Each loop closed: its last point back to its first.)
|
||
for (p, q) in la.iter().zip(la.iter().skip(1).chain(la.first())) {
|
||
for (r, s) in lb.iter().zip(lb.iter().skip(1).chain(lb.first())) {
|
||
if segment_distance(*p, *q, *r, *s) <= reach {
|
||
return false;
|
||
}
|
||
}
|
||
}
|
||
}
|
||
}
|
||
}
|
||
}
|
||
true
|
||
}
|
||
|
||
/// The distance between segments `ab` and `cd` (0 where they cross).
|
||
fn segment_distance(a: Vec2, b: Vec2, c: Vec2, d: Vec2) -> f64 {
|
||
let side = |p: Vec2, q: Vec2, r: Vec2| (q - p).cross(r - p);
|
||
let (d1, d2, d3, d4) = (side(c, d, a), side(c, d, b), side(a, b, c), side(a, b, d));
|
||
if ((d1 > 0.0 && d2 < 0.0) || (d1 < 0.0 && d2 > 0.0)) && ((d3 > 0.0 && d4 < 0.0) || (d3 < 0.0 && d4 > 0.0)) {
|
||
return 0.0;
|
||
}
|
||
let to = |p: Vec2, s: Vec2, e: Vec2| {
|
||
let v = e - s;
|
||
let len2 = v.dot(v);
|
||
let t = if len2 > 0.0 { ((p - s).dot(v) / len2).clamp(0.0, 1.0) } else { 0.0 };
|
||
p.dist(s + v * t)
|
||
};
|
||
to(a, c, d).min(to(b, c, d)).min(to(c, a, b)).min(to(d, a, b))
|
||
}
|
||
|
||
#[cfg(test)]
|
||
mod tests {
|
||
use super::*;
|
||
|
||
fn square(x: f64, y: f64, size: f64) -> Loop2 {
|
||
Loop2::polygon(&[Vec2::new(x, y), Vec2::new(x + size, y), Vec2::new(x + size, y + size), Vec2::new(x, y + size)])
|
||
}
|
||
|
||
fn solid(l: Loop2) -> Region2 {
|
||
Region2 { outer: l, holes: Vec::new() }
|
||
}
|
||
|
||
#[test]
|
||
fn regions_are_apart_only_when_nothing_meets_or_holds_another() {
|
||
let gap = 1e-3;
|
||
// Side by side, a millimetre apart.
|
||
assert!(regions_apart(&[solid(square(0.0, 0.0, 5.0)), solid(square(6.0, 0.0, 5.0))], gap));
|
||
// Touching along a side, or at a corner, or closer than the gap: not apart.
|
||
assert!(!regions_apart(&[solid(square(0.0, 0.0, 5.0)), solid(square(5.0, 0.0, 5.0))], gap));
|
||
assert!(!regions_apart(&[solid(square(0.0, 0.0, 5.0)), solid(square(5.0, 5.0, 5.0))], gap));
|
||
assert!(!regions_apart(&[solid(square(0.0, 0.0, 5.0)), solid(square(5.0005, 0.0, 5.0))], gap));
|
||
// Overlapping, or one inside another: not apart.
|
||
assert!(!regions_apart(&[solid(square(0.0, 0.0, 5.0)), solid(square(3.0, 3.0, 5.0))], gap));
|
||
assert!(!regions_apart(&[solid(square(0.0, 0.0, 10.0)), solid(square(2.0, 2.0, 2.0))], gap));
|
||
// Inside another's hole, clear of its edge: apart (a letter in the middle of an O).
|
||
let ring = Region2 { outer: square(0.0, 0.0, 10.0), holes: vec![square(2.0, 2.0, 6.0).reversed()] };
|
||
assert!(regions_apart(&[ring.clone(), solid(square(4.0, 4.0, 2.0))], gap));
|
||
// Filling the hole: its edge is the hole's.
|
||
assert!(!regions_apart(&[ring, solid(square(2.0, 2.0, 6.0))], gap));
|
||
// Curved boundaries: two circles a little apart, then touching.
|
||
assert!(regions_apart(&[solid(Loop2::circle(Vec2::ZERO, 1.0)), solid(Loop2::circle(Vec2::new(2.01, 0.0), 1.0))], gap));
|
||
assert!(!regions_apart(&[solid(Loop2::circle(Vec2::ZERO, 1.0)), solid(Loop2::circle(Vec2::new(2.0, 0.0), 1.0))], gap));
|
||
// Nothing to compare, and nonsense gaps.
|
||
assert!(regions_apart(&[], gap) && regions_apart(&[solid(square(0.0, 0.0, 1.0))], gap));
|
||
assert!(!regions_apart(&[solid(square(0.0, 0.0, 1.0)), solid(square(5.0, 0.0, 1.0))], f64::NAN));
|
||
}
|
||
|
||
#[test]
|
||
fn closest_line_projection_and_degeneracy() {
|
||
let line = Seg2::Line { a: Vec2::new(-3.0, 4.0), b: Vec2::new(7.0, 4.0) };
|
||
for (p, t) in [(Vec2::new(-9.0, 20.0), 0.0), (Vec2::new(2.0, -5.0), 0.5), (Vec2::new(15.0, 4.0), 1.0)] {
|
||
assert_eq!(line.closest_param(p), t);
|
||
assert_eq!(line.reversed().closest_param(p), 1.0 - t);
|
||
}
|
||
let a = Vec2::new(1e100, -1e100);
|
||
let b = a + Vec2::new(1e90, 2e90);
|
||
let large = Seg2::Line { a, b };
|
||
assert!((large.closest_param(a.lerp(b, 0.5)) - 0.5).abs() < 1e-6);
|
||
assert_eq!(Seg2::Line { a, b: a }.closest_param(Vec2::ZERO), 0.0);
|
||
let tiny = Seg2::Line { a: Vec2::ZERO, b: Vec2::new(1e-12, 0.0) };
|
||
assert_eq!(tiny.closest_param(Vec2::new(0.5e-12, 1.0)), 0.5);
|
||
for line in [line, line.reversed(), tiny] {
|
||
for k in 0..53 {
|
||
let p = Vec2::new((k * 17 % 37) as f64 - 18.0, (k * 31 % 29) as f64 - 14.0);
|
||
let closest = line.point_at(line.closest_param(p)).dist(p);
|
||
let dense = (0..=8192).map(|i| line.point_at(i as f64 / 8192.0).dist(p)).fold(f64::INFINITY, f64::min);
|
||
assert!(closest <= dense + 1e-12);
|
||
}
|
||
}
|
||
}
|
||
|
||
fn sampled_closest(s: &Seg2, p: Vec2) -> f64 {
|
||
let mut t = (0..=64).map(|i| i as f64 / 64.0).min_by(|x, y| s.point_at(*x).dist(p).total_cmp(&s.point_at(*y).dist(p))).unwrap_or(0.0);
|
||
for _ in 0..30 {
|
||
let (q, d) = (s.point_at(t), s.derivative(t));
|
||
let (d2, h) = ((s.derivative((t + 1e-6).min(1.0)) - s.derivative((t - 1e-6).max(0.0))) / 2e-6, q - p);
|
||
let df = d.dot(d) + h.dot(d2);
|
||
if df.abs() < 1e-300 {
|
||
break;
|
||
}
|
||
let nt = (t - h.dot(d) / df).clamp(0.0, 1.0);
|
||
if (nt - t).abs() < 1e-14 {
|
||
t = nt;
|
||
break;
|
||
}
|
||
t = nt;
|
||
}
|
||
t
|
||
}
|
||
|
||
#[test]
|
||
fn closest_projection_preserves_clearance_tolerance_and_generic_fallback() {
|
||
use std::f64::consts::TAU;
|
||
let curves = [
|
||
Seg2::Line { a: Vec2::new(-4.0, 7.0), b: Vec2::new(3.0, -2.0) },
|
||
Seg2::Arc { center: Vec2::new(3.0, -2.0), radius: 6.0, start: 5.9, sweep: 1.3 },
|
||
Seg2::Arc { center: Vec2::ZERO, radius: 2.0, start: 0.2, sweep: -TAU },
|
||
Seg2::Arc { center: Vec2::ZERO, radius: 2.0, start: 0.0, sweep: TAU },
|
||
Seg2::Arc { center: Vec2::ZERO, radius: 2.0, start: 0.0, sweep: 0.0 },
|
||
Seg2::Arc { center: Vec2::ZERO, radius: 2.0, start: 0.0, sweep: 1e-12 },
|
||
Seg2::Cubic { p0: Vec2::ZERO, p1: Vec2::new(2.0, 7.0), p2: Vec2::new(6.0, -3.0), p3: Vec2::new(8.0, 4.0) },
|
||
];
|
||
for (i, s) in curves.iter().enumerate() {
|
||
for k in 0..101 {
|
||
let p = Vec2::new((k * 17 % 37) as f64 - 18.0, (k * 31 % 29) as f64 - 14.0);
|
||
let old = sampled_closest(s, p);
|
||
let new = s.closest_param(p);
|
||
if i > 0 {
|
||
assert_eq!(new.to_bits(), old.to_bits());
|
||
}
|
||
assert!(s.point_at(new).dist(p) <= s.point_at(old).dist(p) + 1e-10);
|
||
let on_curve = s.point_at(k as f64 / 100.0);
|
||
assert!(s.point_at(s.closest_param(on_curve)).dist(on_curve) < 1e-9);
|
||
}
|
||
}
|
||
}
|
||
|
||
/// The sweep visits only the edges crossing each slab; it must give the triangles that
|
||
/// testing every edge against every slab gives, for a finely sampled ring too (#164).
|
||
#[test]
|
||
fn triangulation_sweep_matches_every_edge_against_every_slab() {
|
||
fn every_edge(r: &Region2, tol: f64) -> Vec<[Vec2; 3]> {
|
||
let mut edges = Vec::new();
|
||
for l in std::iter::once(&r.outer).chain(&r.holes) {
|
||
let pl = l.polyline(tol);
|
||
for i in 0..pl.len() {
|
||
let (a, b) = (pl[i], pl[(i + 1) % pl.len()]);
|
||
if (a.y - b.y).abs() > 1e-12 {
|
||
edges.push(if a.y < b.y { (a, b) } else { (b, a) });
|
||
}
|
||
}
|
||
}
|
||
let mut ys: Vec<f64> = edges.iter().flat_map(|(a, b): &(Vec2, Vec2)| [a.y, b.y]).collect();
|
||
ys.sort_by(f64::total_cmp);
|
||
ys.dedup_by(|a, b| (*a - *b).abs() < 1e-12);
|
||
let x_at = |e: &(Vec2, Vec2), y: f64| e.0.x + (y - e.0.y) / (e.1.y - e.0.y) * (e.1.x - e.0.x);
|
||
let mut out = Vec::new();
|
||
for w in ys.windows(2) {
|
||
let ym = (w[0] + w[1]) * 0.5;
|
||
let mut span: Vec<_> = edges.iter().filter(|e| e.0.y <= ym && e.1.y >= ym).collect();
|
||
span.sort_by(|a, b| x_at(a, ym).total_cmp(&x_at(b, ym)));
|
||
for pair in span.chunks(2) {
|
||
let [l, r] = pair else { continue };
|
||
let (a, b) = (Vec2::new(x_at(l, w[0]), w[0]), Vec2::new(x_at(r, w[0]), w[0]));
|
||
let (c, d) = (Vec2::new(x_at(r, w[1]), w[1]), Vec2::new(x_at(l, w[1]), w[1]));
|
||
out.extend([[a, b, c], [a, c, d]]);
|
||
}
|
||
}
|
||
out
|
||
}
|
||
let comb = Loop2::polygon(&[
|
||
Vec2::new(0.0, 0.0),
|
||
Vec2::new(9.0, 0.0),
|
||
Vec2::new(9.0, 6.0),
|
||
Vec2::new(7.0, 2.0),
|
||
Vec2::new(5.0, 7.0),
|
||
Vec2::new(3.0, 2.5),
|
||
Vec2::new(1.0, 6.0),
|
||
]);
|
||
let ring = Region2 { outer: Loop2::circle(Vec2::ZERO, 45.0), holes: vec![Loop2::circle(Vec2::ZERO, 40.0).reversed()] };
|
||
let regions = [Region2 { outer: comb, holes: vec![Loop2::circle(Vec2::new(4.5, 1.2), 0.6).reversed()] }, ring];
|
||
for (r, tol) in regions.iter().zip([1e-3, 1e-4]) {
|
||
let tris = r.triangulate(tol);
|
||
assert!(tris.len() > 100);
|
||
assert_eq!(tris, every_edge(r, tol));
|
||
}
|
||
}
|
||
|
||
#[test]
|
||
fn triangulated_area() {
|
||
let sq = Loop2::polygon(&[Vec2::new(0.0, 0.0), Vec2::new(10.0, 0.0), Vec2::new(10.0, 8.0), Vec2::new(5.0, 3.0), Vec2::new(0.0, 8.0)]);
|
||
let r = Region2 { outer: sq, holes: vec![Loop2::circle(Vec2::new(3.0, 1.5), 1.0).reversed()] };
|
||
let tris = r.triangulate(1e-3);
|
||
let a: f64 = tris.iter().map(|[a, b, c]| (*b - *a).cross(*c - *a).abs() * 0.5).sum();
|
||
// The circle is a polyline within the tolerance, so the areas agree to about that.
|
||
assert!((a - r.area()).abs() < 1e-3 * r.area(), "{a} vs {}", r.area());
|
||
for [a, b, c] in &tris {
|
||
let m = (*a + *b + *c) / 3.0;
|
||
assert!((*b - *a).cross(*c - *a).abs() < 1e-12 || r.contains(m), "{m:?}");
|
||
}
|
||
}
|
||
|
||
#[test]
|
||
fn areas() {
|
||
let sq = Loop2::polygon(&[Vec2::new(0.0, 0.0), Vec2::new(4.0, 0.0), Vec2::new(4.0, 3.0), Vec2::new(0.0, 3.0)]);
|
||
assert!((sq.signed_area() - 12.0).abs() < 1e-12);
|
||
assert!((sq.reversed().signed_area() + 12.0).abs() < 1e-12);
|
||
let c = Loop2::circle(Vec2::new(5.0, 5.0), 2.0);
|
||
assert!((c.signed_area() - std::f64::consts::PI * 4.0).abs() < 1e-9);
|
||
assert!(c.is_closed(1e-9));
|
||
let r = Region2 { outer: Loop2::circle(Vec2::ZERO, 10.0), holes: vec![Loop2::circle(Vec2::ZERO, 5.0).reversed()] };
|
||
assert!((r.area() - std::f64::consts::PI * 75.0).abs() < 1e-9);
|
||
let ip = r.interior_point();
|
||
assert!(r.contains(ip), "{ip:?}");
|
||
assert!(sq.contains(Vec2::new(1.0, 1.0)) && !sq.contains(Vec2::new(5.0, 1.0)));
|
||
assert!(sq.centroid().dist(Vec2::new(2.0, 1.5)) < 1e-9);
|
||
}
|
||
}
|