kurbo/

cubicbez.rs

// Copyright 2018 the Kurbo Authors
// SPDX-License-Identifier: Apache-2.0 OR MIT

//! Cubic Bézier segments.

use alloc::vec;
use alloc::vec::Vec;
use core::ops::{Mul, Range};

use crate::MAX_EXTREMA;
use crate::{Line, QuadSpline, Vec2};
use arrayvec::ArrayVec;

use crate::common::{
    GAUSS_LEGENDRE_COEFFS_8, GAUSS_LEGENDRE_COEFFS_8_HALF, GAUSS_LEGENDRE_COEFFS_16_HALF,
    GAUSS_LEGENDRE_COEFFS_24_HALF, solve_cubic, solve_quadratic, solve_quartic,
};
use crate::{
    Affine, Nearest, ParamCurve, ParamCurveArclen, ParamCurveArea, ParamCurveCurvature,
    ParamCurveDeriv, ParamCurveExtrema, ParamCurveNearest, PathEl, Point, QuadBez, Rect, Shape,
};

#[cfg(not(feature = "std"))]
use crate::common::FloatFuncs;

const MAX_SPLINE_SPLIT: usize = 100;

/// A single cubic Bézier segment.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "schemars", derive(schemars::JsonSchema))]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[allow(missing_docs)]
pub struct CubicBez {
    pub p0: Point,
    pub p1: Point,
    pub p2: Point,
    pub p3: Point,
}

/// An iterator which produces quadratic Bézier segments.
struct ToQuads {
    c: CubicBez,
    i: usize,
    n: usize,
}

/// Classification result for cusp detection.
#[derive(Debug)]
pub enum CuspType {
    /// Cusp is a loop.
    Loop,
    /// Cusp has two closely spaced inflection points.
    DoubleInflection,
}

impl CubicBez {
    /// Create a new cubic Bézier segment.
    #[inline(always)]
    pub fn new<P: Into<Point>>(p0: P, p1: P, p2: P, p3: P) -> CubicBez {
        CubicBez {
            p0: p0.into(),
            p1: p1.into(),
            p2: p2.into(),
            p3: p3.into(),
        }
    }

    /// Convert to quadratic Béziers.
    ///
    /// The iterator returns the start and end parameter in the cubic of each quadratic
    /// segment, along with the quadratic.
    ///
    /// Note that the resulting quadratic Béziers are not in general G1 continuous;
    /// they are optimized for minimizing distance error.
    ///
    /// This iterator will always produce at least one `QuadBez`.
    #[inline]
    pub fn to_quads(&self, accuracy: f64) -> impl Iterator<Item = (f64, f64, QuadBez)> {
        // The maximum error, as a vector from the cubic to the best approximating
        // quadratic, is proportional to the third derivative, which is constant
        // across the segment. Thus, the error scales down as the third power of
        // the number of subdivisions. Our strategy then is to subdivide `t` evenly.
        //
        // This is an overestimate of the error because only the component
        // perpendicular to the first derivative is important. But the simplicity is
        // appealing.

        // This magic number is the square of 36 / sqrt(3).
        // See: http://caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
        let max_hypot2 = 432.0 * accuracy * accuracy;
        let p1x2 = 3.0 * self.p1.to_vec2() - self.p0.to_vec2();
        let p2x2 = 3.0 * self.p2.to_vec2() - self.p3.to_vec2();
        let err = (p2x2 - p1x2).hypot2();
        let n = ((err / max_hypot2).powf(1. / 6.0).ceil() as usize).max(1);

        ToQuads { c: *self, n, i: 0 }
    }

    /// Return a [`QuadSpline`] approximating this cubic Bézier.
    ///
    /// Returns `None` if no suitable approximation is found within the given
    /// tolerance.
    pub fn approx_spline(&self, accuracy: f64) -> Option<QuadSpline> {
        (1..=MAX_SPLINE_SPLIT).find_map(|n| self.approx_spline_n(n, accuracy))
    }

    // Approximate a cubic curve with a quadratic spline of `n` curves
    fn approx_spline_n(&self, n: usize, accuracy: f64) -> Option<QuadSpline> {
        if n == 1 {
            return self
                .try_approx_quadratic(accuracy)
                .map(|quad| QuadSpline::new(vec![quad.p0, quad.p1, quad.p2]));
        }
        let mut cubics = self.split_into_n(n);

        // The above function guarantees that the iterator returns n items,
        // which is why we're unwrapping things with wild abandon.
        let mut next_cubic = cubics.next().unwrap();
        let mut next_q1: Point = next_cubic.approx_quad_control(0.0);
        let mut q2 = self.p0;
        let mut d1 = Vec2::ZERO;
        let mut spline = vec![self.p0, next_q1];
        for i in 1..=n {
            let current_cubic: CubicBez = next_cubic;
            let q0 = q2;
            let q1 = next_q1;
            q2 = if i < n {
                next_cubic = cubics.next().unwrap();
                next_q1 = next_cubic.approx_quad_control(i as f64 / (n - 1) as f64);

                spline.push(next_q1);
                q1.midpoint(next_q1)
            } else {
                current_cubic.p3
            };
            let d0 = d1;
            d1 = q2.to_vec2() - current_cubic.p3.to_vec2();

            if d1.hypot() > accuracy
                || !CubicBez::new(
                    d0.to_point(),
                    q0.lerp(q1, 2.0 / 3.0) - current_cubic.p1.to_vec2(),
                    q2.lerp(q1, 2.0 / 3.0) - current_cubic.p2.to_vec2(),
                    d1.to_point(),
                )
                .fit_inside(accuracy)
            {
                return None;
            }
        }
        spline.push(self.p3);
        Some(QuadSpline::new(spline))
    }

    fn approx_quad_control(&self, t: f64) -> Point {
        let p1 = self.p0 + (self.p1 - self.p0) * 1.5;
        let p2 = self.p3 + (self.p2 - self.p3) * 1.5;
        p1.lerp(p2, t)
    }

    /// Approximate a cubic with a single quadratic
    ///
    /// Returns a quadratic approximating the given cubic that maintains
    /// endpoint tangents if that is within tolerance, or `None` otherwise.
    fn try_approx_quadratic(&self, accuracy: f64) -> Option<QuadBez> {
        if let Some(q1) = Line::new(self.p0, self.p1)
            .crossing_point(Line::new(self.p2, self.p3))
            .or_else(|| {
                // with 3 to 4 consecutive equal points (with no valid crossing_point),
                // we can try to use the common point as the quadratic control point.
                // This matches fonttools' cu2qu: https://github.com/fonttools/fonttools/pull/3904
                (self.p1 == self.p2 && (self.p0 == self.p1 || self.p2 == self.p3))
                    .then_some(self.p1)
            })
        {
            let c1 = self.p0.lerp(q1, 2.0 / 3.0);
            let c2 = self.p3.lerp(q1, 2.0 / 3.0);
            if !CubicBez::new(
                Point::ZERO,
                c1 - self.p1.to_vec2(),
                c2 - self.p2.to_vec2(),
                Point::ZERO,
            )
            .fit_inside(accuracy)
            {
                return None;
            }
            return Some(QuadBez::new(self.p0, q1, self.p3));
        }
        None
    }

    fn split_into_n(&self, n: usize) -> impl Iterator<Item = CubicBez> {
        // for certain small values of `n` we precompute all our values.
        // if we have six or fewer items we precompute them.
        let mut storage = ArrayVec::<_, 6>::new();

        match n {
            1 => storage.push(*self),
            2 => {
                let (l, r) = self.subdivide();
                storage.try_extend_from_slice(&[r, l]).unwrap();
            }
            3 => {
                let (left, mid, right) = self.subdivide_3();
                storage.try_extend_from_slice(&[right, mid, left]).unwrap();
            }
            4 => {
                let (l, r) = self.subdivide();
                let (ll, lr) = l.subdivide();
                let (rl, rr) = r.subdivide();
                storage.try_extend_from_slice(&[rr, rl, lr, ll]).unwrap();
            }
            6 => {
                let (l, r) = self.subdivide();
                let (l1, l2, l3) = l.subdivide_3();
                let (r1, r2, r3) = r.subdivide_3();
                storage
                    .try_extend_from_slice(&[r3, r2, r1, l3, l2, l1])
                    .unwrap();
            }
            _ => (),
        }

        // a limitation of returning 'impl Trait' is that the implementation
        // can only return a single concrete type; that is you cannot return
        // Vec::into_iter() from one branch, and then HashSet::into_iter from
        // another branch.
        //
        // This means we have to get a bit fancy, and have a single concrete
        // type that represents both of our possible cases.

        let mut storage = if storage.is_empty() {
            None
        } else {
            Some(storage)
        };

        // used in the fallback case
        let mut i = 0;
        let (a, b, c, d) = self.parameters();
        let dt = 1.0 / n as f64;
        let delta_2 = dt * dt;
        let delta_3 = dt * delta_2;

        core::iter::from_fn(move || {
            // if storage exists, we use it exclusively
            if let Some(storage) = storage.as_mut() {
                return storage.pop();
            }

            // if storage does not exist, we are exclusively working down here.
            if i >= n {
                return None;
            }

            let t1 = i as f64 * dt;
            let t1_2 = t1 * t1;
            let a1 = a * delta_3;
            let b1 = (3.0 * a * t1 + b) * delta_2;
            let c1 = (2.0 * b * t1 + c + 3.0 * a * t1_2) * dt;
            let d1 = a * t1 * t1_2 + b * t1_2 + c * t1 + d;
            let result = CubicBez::from_parameters(a1, b1, c1, d1);
            i += 1;
            Some(result)
        })
    }

    fn parameters(&self) -> (Vec2, Vec2, Vec2, Vec2) {
        let c = (self.p1 - self.p0) * 3.0;
        let b = (self.p2 - self.p1) * 3.0 - c;
        let d = self.p0.to_vec2();
        let a = self.p3.to_vec2() - d - c - b;
        (a, b, c, d)
    }

    /// Rust port of cu2qu [calc_cubic_points](https://github.com/fonttools/fonttools/blob/3b9a73ff8379ab49d3ce35aaaaf04b3a7d9d1655/Lib/fontTools/cu2qu/cu2qu.py#L63-L68)
    fn from_parameters(a: Vec2, b: Vec2, c: Vec2, d: Vec2) -> Self {
        let p0 = d.to_point();
        let p1 = c.div_exact(3.0).to_point() + d;
        let p2 = (b + c).div_exact(3.0).to_point() + p1.to_vec2();
        let p3 = (a + d + c + b).to_point();
        CubicBez::new(p0, p1, p2, p3)
    }

    fn subdivide_3(&self) -> (CubicBez, CubicBez, CubicBez) {
        let (p0, p1, p2, p3) = (
            self.p0.to_vec2(),
            self.p1.to_vec2(),
            self.p2.to_vec2(),
            self.p3.to_vec2(),
        );
        // The original Python cu2qu code here does not use division operator to divide by 27 but
        // instead uses multiplication by the reciprocal 1 / 27. We want to match it exactly
        // to avoid any floating point differences, hence in this particular case we do not use div_exact.
        // I could directly use the Vec2 Div trait (also implemented as multiplication by reciprocal)
        // but I prefer to be explicit here.
        // Source: https://github.com/fonttools/fonttools/blob/85c80be/Lib/fontTools/cu2qu/cu2qu.py#L215-L218
        // See also: https://github.com/linebender/kurbo/issues/272
        let one_27th = 27.0_f64.recip();
        let mid1 = ((8.0 * p0 + 12.0 * p1 + 6.0 * p2 + p3) * one_27th).to_point();
        let deriv1 = (p3 + 3.0 * p2 - 4.0 * p0) * one_27th;
        let mid2 = ((p0 + 6.0 * p1 + 12.0 * p2 + 8.0 * p3) * one_27th).to_point();
        let deriv2 = (4.0 * p3 - 3.0 * p1 - p0) * one_27th;

        let left = CubicBez::new(
            self.p0,
            (2.0 * p0 + p1).div_exact(3.0).to_point(),
            mid1 - deriv1,
            mid1,
        );
        let mid = CubicBez::new(mid1, mid1 + deriv1, mid2 - deriv2, mid2);
        let right = CubicBez::new(
            mid2,
            mid2 + deriv2,
            (p2 + 2.0 * p3).div_exact(3.0).to_point(),
            self.p3,
        );
        (left, mid, right)
    }

    /// Does this curve fit inside the given distance from the origin?
    ///
    /// Rust port of cu2qu [cubic_farthest_fit_inside](https://github.com/fonttools/fonttools/blob/3b9a73ff8379ab49d3ce35aaaaf04b3a7d9d1655/Lib/fontTools/cu2qu/cu2qu.py#L281)
    fn fit_inside(&self, distance: f64) -> bool {
        if self.p2.to_vec2().hypot() <= distance && self.p1.to_vec2().hypot() <= distance {
            return true;
        }
        let mid =
            (self.p0.to_vec2() + 3.0 * (self.p1.to_vec2() + self.p2.to_vec2()) + self.p3.to_vec2())
                * 0.125;
        if mid.hypot() > distance {
            return false;
        }
        // Split in two. Note that cu2qu here uses a 3/8 subdivision. I don't know why.
        let (left, right) = self.subdivide();
        left.fit_inside(distance) && right.fit_inside(distance)
    }

    /// Is this cubic Bezier curve finite?
    #[inline]
    pub const fn is_finite(&self) -> bool {
        self.p0.is_finite() && self.p1.is_finite() && self.p2.is_finite() && self.p3.is_finite()
    }

    /// Is this cubic Bezier curve NaN?
    #[inline]
    pub const fn is_nan(&self) -> bool {
        self.p0.is_nan() || self.p1.is_nan() || self.p2.is_nan() || self.p3.is_nan()
    }

    /// Determine the inflection points.
    ///
    /// Return value is t parameter for the inflection points of the curve segment.
    /// There are a maximum of two for a cubic Bézier.
    ///
    /// See <https://www.caffeineowl.com/graphics/2d/vectorial/cubic-inflexion.html>
    /// for the theory.
    pub fn inflections(&self) -> ArrayVec<f64, 2> {
        let a = self.p1 - self.p0;
        let b = (self.p2 - self.p1) - a;
        let c = (self.p3 - self.p0) - 3. * (self.p2 - self.p1);
        solve_quadratic(a.cross(b), a.cross(c), b.cross(c))
            .iter()
            .copied()
            .filter(|t| *t >= 0.0 && *t <= 1.0)
            .collect()
    }

    /// Find points on the curve where the tangent line passes through the
    /// given point.
    ///
    /// Result is array of t values such that the tangent line from the curve
    /// evaluated at that point goes through the argument point.
    pub fn tangents_to_point(&self, p: Point) -> ArrayVec<f64, 4> {
        let (a, b, c, d_orig) = self.parameters();
        let d = d_orig - p.to_vec2();
        // coefficients of x(t) \cross x'(t)
        let c4 = b.cross(a);
        let c3 = 2.0 * c.cross(a);
        let c2 = c.cross(b) + 3.0 * d.cross(a);
        let c1 = 2.0 * d.cross(b);
        let c0 = d.cross(c);
        solve_quartic(c0, c1, c2, c3, c4)
            .iter()
            .copied()
            .filter(|t| *t >= 0.0 && *t <= 1.0)
            .collect()
    }

    /// Preprocess a cubic Bézier to ease numerical robustness.
    ///
    /// If the cubic Bézier segment has zero or near-zero derivatives as an interior
    /// cusp, perturb the control points to make curvature finite, avoiding
    /// numerical robustness problems in offset and stroke.
    pub(crate) fn regularize_cusp(&self, dimension: f64) -> CubicBez {
        let mut c = *self;
        // First step: if control point is too near the endpoint, nudge it away
        // along the tangent.
        if let Some(cusp_type) = self.detect_cusp(dimension) {
            let d01 = c.p1 - c.p0;
            let d01h = d01.hypot();
            let d23 = c.p3 - c.p2;
            let d23h = d23.hypot();
            match cusp_type {
                CuspType::Loop => {
                    c.p1 += (dimension / d01h) * d01;
                    c.p2 -= (dimension / d23h) * d23;
                }
                CuspType::DoubleInflection => {
                    // Avoid making control distance smaller than dimension
                    if d01h > 2.0 * dimension {
                        c.p1 -= (dimension / d01h) * d01;
                    }
                    if d23h > 2.0 * dimension {
                        c.p2 += (dimension / d23h) * d23;
                    }
                }
            }
        }
        c
    }

    /// Detect whether there is a cusp.
    ///
    /// Return a cusp classification if there is a cusp with curvature greater than
    /// the reciprocal of the given dimension.
    fn detect_cusp(&self, dimension: f64) -> Option<CuspType> {
        let d01 = self.p1 - self.p0;
        let d02 = self.p2 - self.p0;
        let d03 = self.p3 - self.p0;
        let d12 = self.p2 - self.p1;
        let d23 = self.p3 - self.p2;
        let det_012 = d01.cross(d02);
        let det_123 = d12.cross(d23);
        let det_013 = d01.cross(d03);
        let det_023 = d02.cross(d03);
        if det_012 * det_123 > 0.0 && det_012 * det_013 < 0.0 && det_012 * det_023 < 0.0 {
            let q = self.deriv();
            // accuracy isn't used for quadratic nearest
            let nearest = q.nearest(Point::ORIGIN, 1e-9);
            // detect whether curvature at minimum derivative exceeds 1/dimension,
            // without division.
            let d = q.eval(nearest.t);
            let d2 = q.deriv().eval(nearest.t);
            let cross = d.to_vec2().cross(d2.to_vec2());
            if nearest.distance_sq.powi(3) <= (cross * dimension).powi(2) {
                let a = 3. * det_012 + det_023 - 2. * det_013;
                let b = -3. * det_012 + det_013;
                let c = det_012;
                let d = b * b - 4. * a * c;
                if d > 0.0 {
                    return Some(CuspType::DoubleInflection);
                } else {
                    return Some(CuspType::Loop);
                }
            }
        }
        None
    }

    /// Finds the value of `t` for which `self.eval(t)` is about `y`.
    ///
    /// Assumes that this segment is monotonic in `y` and that it crosses
    /// the height `y`. (Under these assumptions, there is a unique answer.)
    pub(crate) fn solve_monotonic_for_y(&self, y: f64) -> f64 {
        let start = self.start();
        let end = self.end();

        debug_assert!(start.y.min(end.y) <= y && y <= start.y.max(end.y));

        let p1 = self.p1;
        let p2 = self.p2;
        let a = end.y - 3.0 * p2.y + 3.0 * p1.y - start.y;
        let b = 3.0 * (p2.y - 2.0 * p1.y + start.y);
        let c = 3.0 * (p1.y - start.y);
        let d = start.y - y;
        for t in solve_cubic(d, c, b, a) {
            if (0.0..=1.0).contains(&t) {
                return t;
            }
        }

        // Even though we asserted that our y range contains `y`, it's possible
        // that we failed to find a solution numerically. (For example, rounding
        // of a, b, or c might have pushed the root outside of [0.0, 1.0].)
        // If we failed to find a solution, the real solution should be close
        // to one of the endpoints. So just find whichever endpoint was closer.
        if (start.y - y).abs() <= (end.y - y).abs() {
            0.0
        } else {
            1.0
        }
    }
}

/// An iterator for cubic beziers.
pub struct CubicBezIter {
    cubic: CubicBez,
    ix: usize,
}

impl Shape for CubicBez {
    type PathElementsIter<'iter> = CubicBezIter;

    #[inline]
    fn path_elements(&self, _tolerance: f64) -> CubicBezIter {
        CubicBezIter {
            cubic: *self,
            ix: 0,
        }
    }

    #[inline(always)]
    fn area(&self) -> f64 {
        0.0
    }

    #[inline]
    fn perimeter(&self, accuracy: f64) -> f64 {
        self.arclen(accuracy)
    }

    #[inline(always)]
    fn winding(&self, _pt: Point) -> i32 {
        0
    }

    #[inline]
    fn bounding_box(&self) -> Rect {
        ParamCurveExtrema::bounding_box(self)
    }
}

impl Iterator for CubicBezIter {
    type Item = PathEl;

    fn next(&mut self) -> Option<PathEl> {
        self.ix += 1;
        match self.ix {
            1 => Some(PathEl::MoveTo(self.cubic.p0)),
            2 => Some(PathEl::CurveTo(self.cubic.p1, self.cubic.p2, self.cubic.p3)),
            _ => None,
        }
    }
}

impl ParamCurve for CubicBez {
    #[inline]
    fn eval(&self, t: f64) -> Point {
        let mt = 1.0 - t;
        let v = self.p0.to_vec2() * (mt * mt * mt)
            + (self.p1.to_vec2() * (mt * mt * 3.0)
                + (self.p2.to_vec2() * (mt * 3.0) + self.p3.to_vec2() * t) * t)
                * t;
        v.to_point()
    }

    fn subsegment(&self, range: Range<f64>) -> CubicBez {
        let (t0, t1) = (range.start, range.end);
        let p0 = self.eval(t0);
        let p3 = self.eval(t1);
        let d = self.deriv();
        let scale = (t1 - t0) * (1.0 / 3.0);
        let p1 = p0 + scale * d.eval(t0).to_vec2();
        let p2 = p3 - scale * d.eval(t1).to_vec2();
        CubicBez { p0, p1, p2, p3 }
    }

    /// Subdivide into halves, using de Casteljau.
    #[inline]
    fn subdivide(&self) -> (CubicBez, CubicBez) {
        let pm = self.eval(0.5);
        (
            CubicBez::new(
                self.p0,
                self.p0.midpoint(self.p1),
                ((self.p0.to_vec2() + self.p1.to_vec2() * 2.0 + self.p2.to_vec2()) * 0.25)
                    .to_point(),
                pm,
            ),
            CubicBez::new(
                pm,
                ((self.p1.to_vec2() + self.p2.to_vec2() * 2.0 + self.p3.to_vec2()) * 0.25)
                    .to_point(),
                self.p2.midpoint(self.p3),
                self.p3,
            ),
        )
    }

    #[inline(always)]
    fn start(&self) -> Point {
        self.p0
    }

    #[inline(always)]
    fn end(&self) -> Point {
        self.p3
    }
}

impl ParamCurveDeriv for CubicBez {
    type DerivResult = QuadBez;

    #[inline]
    fn deriv(&self) -> QuadBez {
        QuadBez::new(
            (3.0 * (self.p1 - self.p0)).to_point(),
            (3.0 * (self.p2 - self.p1)).to_point(),
            (3.0 * (self.p3 - self.p2)).to_point(),
        )
    }
}

fn arclen_quadrature_core(coeffs: &[(f64, f64)], dm: Vec2, dm1: Vec2, dm2: Vec2) -> f64 {
    coeffs
        .iter()
        .map(|&(wi, xi)| {
            let d = dm + dm2 * (xi * xi);
            let dpx = (d + dm1 * xi).hypot();
            let dmx = (d - dm1 * xi).hypot();
            (2.25f64.sqrt() * wi) * (dpx + dmx)
        })
        .sum::<f64>()
}

fn arclen_rec(c: &CubicBez, accuracy: f64, depth: usize) -> f64 {
    let d03 = c.p3 - c.p0;
    let d01 = c.p1 - c.p0;
    let d12 = c.p2 - c.p1;
    let d23 = c.p3 - c.p2;
    let lp_lc = d01.hypot() + d12.hypot() + d23.hypot() - d03.hypot();
    let dd1 = d12 - d01;
    let dd2 = d23 - d12;
    // It might be faster to do direct multiplies, the data dependencies would be shorter.
    // The following values don't have the factor of 3 for first deriv
    let dm = 0.25 * (d01 + d23) + 0.5 * d12; // first derivative at midpoint
    let dm1 = 0.5 * (dd2 + dd1); // second derivative at midpoint
    let dm2 = 0.25 * (dd2 - dd1); // 0.5 * (third derivative at midpoint)

    let est = GAUSS_LEGENDRE_COEFFS_8
        .iter()
        .map(|&(wi, xi)| {
            wi * {
                let d_norm2 = (dm + dm1 * xi + dm2 * (xi * xi)).hypot2();
                let dd_norm2 = (dm1 + dm2 * (2.0 * xi)).hypot2();
                dd_norm2 / d_norm2
            }
        })
        .sum::<f64>();
    let est_gauss8_error = (est.powi(3) * 2.5e-6).min(3e-2) * lp_lc;
    if est_gauss8_error < accuracy {
        return arclen_quadrature_core(GAUSS_LEGENDRE_COEFFS_8_HALF, dm, dm1, dm2);
    }
    let est_gauss16_error = (est.powi(6) * 1.5e-11).min(9e-3) * lp_lc;
    if est_gauss16_error < accuracy {
        return arclen_quadrature_core(GAUSS_LEGENDRE_COEFFS_16_HALF, dm, dm1, dm2);
    }
    let est_gauss24_error = (est.powi(9) * 3.5e-16).min(3.5e-3) * lp_lc;
    if est_gauss24_error < accuracy || depth >= 20 {
        return arclen_quadrature_core(GAUSS_LEGENDRE_COEFFS_24_HALF, dm, dm1, dm2);
    }
    let (c0, c1) = c.subdivide();
    arclen_rec(&c0, accuracy * 0.5, depth + 1) + arclen_rec(&c1, accuracy * 0.5, depth + 1)
}

impl ParamCurveArclen for CubicBez {
    /// Arclength of a cubic Bézier segment.
    ///
    /// This is an adaptive subdivision approach using Legendre-Gauss quadrature
    /// in the base case, and an error estimate to decide when to subdivide.
    fn arclen(&self, accuracy: f64) -> f64 {
        arclen_rec(self, accuracy, 0)
    }
}

impl ParamCurveArea for CubicBez {
    #[inline]
    fn signed_area(&self) -> f64 {
        (self.p0.x * (6.0 * self.p1.y + 3.0 * self.p2.y + self.p3.y)
            + 3.0
                * (self.p1.x * (-2.0 * self.p0.y + self.p2.y + self.p3.y)
                    - self.p2.x * (self.p0.y + self.p1.y - 2.0 * self.p3.y))
            - self.p3.x * (self.p0.y + 3.0 * self.p1.y + 6.0 * self.p2.y))
            * (1.0 / 20.0)
    }
}

impl ParamCurveNearest for CubicBez {
    /// Find the nearest point using a quintic solver.
    ///
    /// The polynomial `|self - p|^2` has degree 6, so we find its critical
    /// points and evaluate them all to find the best one.
    fn nearest(&self, p: Point, accuracy: f64) -> Nearest {
        fn eval_t(p: Point, t_best: &mut f64, r_best: &mut Option<f64>, t: f64, p0: Point) {
            let r = (p0 - p).hypot2();
            if r_best.map(|r_best| r < r_best).unwrap_or(true) {
                *r_best = Some(r);
                *t_best = t;
            }
        }

        let mut r_best = None;
        let mut t_best = 0.0;

        // Reparameterize `self - p` as q0 + q1 t + q2 t^2 + q3 t^3.
        let q0 = self.p0 - p;
        let q1 = 3.0 * (self.p1 - self.p0);
        let q2 = 3.0 * (self.p0.to_vec2() - 2.0 * self.p1.to_vec2() + self.p2.to_vec2());
        let q3 = -self.p0.to_vec2() + 3.0 * self.p1.to_vec2() - 3.0 * self.p2.to_vec2()
            + self.p3.to_vec2();

        // Coefficients of the degree-5 polynomial (self - p) \cdot tangent.
        let c0 = q0.dot(q1);
        let c1 = q1.hypot2() + 2.0 * q2.dot(q0);
        let c2 = 3.0 * (q2.dot(q1) + q3.dot(q0));
        let c3 = 4.0 * q3.dot(q1) + 2.0 * q2.hypot2();
        let c4 = 5.0 * q3.dot(q2);
        let c5 = 3.0 * q3.hypot2();

        let roots = polycool::Poly::new([c0, c1, c2, c3, c4, c5]).roots_between(0.0, 1.0, accuracy);

        for &t in &roots {
            eval_t(p, &mut t_best, &mut r_best, t, self.eval(t));
        }

        // If we found all 5 critical points, we can skip evaluating the endpoints.
        if roots.len() != 5 {
            eval_t(p, &mut t_best, &mut r_best, 0.0, self.p0);
            eval_t(p, &mut t_best, &mut r_best, 1.0, self.p3);
        }

        Nearest {
            t: t_best,
            distance_sq: r_best.unwrap(),
        }
    }
}

impl ParamCurveCurvature for CubicBez {}

impl ParamCurveExtrema for CubicBez {
    fn extrema(&self) -> ArrayVec<f64, MAX_EXTREMA> {
        fn one_coord(result: &mut ArrayVec<f64, MAX_EXTREMA>, d0: f64, d1: f64, d2: f64) {
            let a = d0 - 2.0 * d1 + d2;
            let b = 2.0 * (d1 - d0);
            let c = d0;
            let roots = solve_quadratic(c, b, a);
            for &t in &roots {
                if t > 0.0 && t < 1.0 {
                    result.push(t);
                }
            }
        }
        let mut result = ArrayVec::new();
        let d0 = self.p1 - self.p0;
        let d1 = self.p2 - self.p1;
        let d2 = self.p3 - self.p2;
        one_coord(&mut result, d0.x, d1.x, d2.x);
        one_coord(&mut result, d0.y, d1.y, d2.y);
        result.sort_by(|a, b| a.partial_cmp(b).unwrap());
        result
    }
}

impl Mul<CubicBez> for Affine {
    type Output = CubicBez;

    #[inline]
    fn mul(self, c: CubicBez) -> CubicBez {
        CubicBez {
            p0: self * c.p0,
            p1: self * c.p1,
            p2: self * c.p2,
            p3: self * c.p3,
        }
    }
}

impl Iterator for ToQuads {
    type Item = (f64, f64, QuadBez);

    #[inline]
    fn next(&mut self) -> Option<(f64, f64, QuadBez)> {
        if self.i == self.n {
            return None;
        }
        let t0 = self.i as f64 / self.n as f64;
        let t1 = (self.i + 1) as f64 / self.n as f64;
        let seg = self.c.subsegment(t0..t1);
        let p1x2 = 3.0 * seg.p1.to_vec2() - seg.p0.to_vec2();
        let p2x2 = 3.0 * seg.p2.to_vec2() - seg.p3.to_vec2();
        let result = QuadBez::new(seg.p0, ((p1x2 + p2x2) / 4.0).to_point(), seg.p3);
        self.i += 1;
        Some((t0, t1, result))
    }

    fn size_hint(&self) -> (usize, Option<usize>) {
        let remaining = self.n - self.i;
        (remaining, Some(remaining))
    }
}

/// Convert multiple cubic Bézier curves to quadratic splines.
///
/// Ensures that the resulting splines have the same number of control points.
///
/// Rust port of cu2qu [cubic_approx_quadratic](https://github.com/fonttools/fonttools/blob/3b9a73ff8379ab49d3ce35aaaaf04b3a7d9d1655/Lib/fontTools/cu2qu/cu2qu.py#L322)
pub fn cubics_to_quadratic_splines(curves: &[CubicBez], accuracy: f64) -> Option<Vec<QuadSpline>> {
    let mut result = Vec::new();
    let mut split_order = 0;

    while split_order <= MAX_SPLINE_SPLIT {
        split_order += 1;
        result.clear();

        for curve in curves {
            match curve.approx_spline_n(split_order, accuracy) {
                Some(spline) => result.push(spline),
                None => break,
            }
        }

        if result.len() == curves.len() {
            return Some(result);
        }
    }
    None
}
#[cfg(test)]
mod tests {
    use crate::{
        Affine, CubicBez, Nearest, ParamCurve, ParamCurveArclen, ParamCurveArea, ParamCurveDeriv,
        ParamCurveExtrema, ParamCurveNearest, Point, QuadBez, QuadSpline,
        cubics_to_quadratic_splines,
    };

    #[test]
    fn cubicbez_deriv() {
        // y = x^2
        let c = CubicBez::new(
            (0.0, 0.0),
            (1.0 / 3.0, 0.0),
            (2.0 / 3.0, 1.0 / 3.0),
            (1.0, 1.0),
        );
        let deriv = c.deriv();

        let n = 10;
        for i in 0..=n {
            let t = (i as f64) * (n as f64).recip();
            let delta = 1e-6;
            let p = c.eval(t);
            let p1 = c.eval(t + delta);
            let d_approx = (p1 - p) * delta.recip();
            let d = deriv.eval(t).to_vec2();
            assert!((d - d_approx).hypot() < delta * 2.0);
        }
    }

    #[test]
    fn cubicbez_arclen() {
        // y = x^2
        let c = CubicBez::new(
            (0.0, 0.0),
            (1.0 / 3.0, 0.0),
            (2.0 / 3.0, 1.0 / 3.0),
            (1.0, 1.0),
        );
        let true_arclen = 0.5 * 5.0f64.sqrt() + 0.25 * (2.0 + 5.0f64.sqrt()).ln();
        for i in 0..12 {
            let accuracy = 0.1f64.powi(i);
            let error = c.arclen(accuracy) - true_arclen;
            assert!(error.abs() < accuracy);
        }
    }

    #[test]
    fn cubicbez_inv_arclen() {
        // y = x^2 / 100
        let c = CubicBez::new(
            (0.0, 0.0),
            (100.0 / 3.0, 0.0),
            (200.0 / 3.0, 100.0 / 3.0),
            (100.0, 100.0),
        );
        let true_arclen = 100.0 * (0.5 * 5.0f64.sqrt() + 0.25 * (2.0 + 5.0f64.sqrt()).ln());
        for i in 0..12 {
            let accuracy = 0.1f64.powi(i);
            let n = 10;
            for j in 0..=n {
                let arc = (j as f64) * ((n as f64).recip() * true_arclen);
                let t = c.inv_arclen(arc, accuracy * 0.5);
                let actual_arc = c.subsegment(0.0..t).arclen(accuracy * 0.5);
                assert!(
                    (arc - actual_arc).abs() < accuracy,
                    "at accuracy {accuracy:e}, wanted {actual_arc} got {arc}"
                );
            }
        }
        // corner case: user passes accuracy larger than total arc length
        let accuracy = true_arclen * 1.1;
        let arc = true_arclen * 0.5;
        let t = c.inv_arclen(arc, accuracy);
        let actual_arc = c.subsegment(0.0..t).arclen(accuracy);
        assert!(
            (arc - actual_arc).abs() < 2.0 * accuracy,
            "at accuracy {accuracy:e}, want {actual_arc} got {arc}"
        );
    }

    #[test]
    fn cubicbez_inv_arclen_accuracy() {
        let c = CubicBez::new((0.2, 0.73), (0.35, 1.08), (0.85, 1.08), (1.0, 0.73));
        let true_t = c.inv_arclen(0.5, 1e-12);
        for i in 1..12 {
            let accuracy = (0.1f64).powi(i);
            let approx_t = c.inv_arclen(0.5, accuracy);
            assert!((approx_t - true_t).abs() <= accuracy);
        }
    }

    #[test]
    #[allow(clippy::float_cmp)]
    fn cubicbez_signed_area_linear() {
        // y = 1 - x
        let c = CubicBez::new(
            (1.0, 0.0),
            (2.0 / 3.0, 1.0 / 3.0),
            (1.0 / 3.0, 2.0 / 3.0),
            (0.0, 1.0),
        );
        let epsilon = 1e-12;
        assert_eq!((Affine::rotate(0.5) * c).signed_area(), 0.5);
        assert!(((Affine::rotate(0.5) * c).signed_area() - 0.5).abs() < epsilon);
        assert!(((Affine::translate((0.0, 1.0)) * c).signed_area() - 1.0).abs() < epsilon);
        assert!(((Affine::translate((1.0, 0.0)) * c).signed_area() - 1.0).abs() < epsilon);
    }

    #[test]
    fn cubicbez_signed_area() {
        // y = 1 - x^3
        let c = CubicBez::new((1.0, 0.0), (2.0 / 3.0, 1.0), (1.0 / 3.0, 1.0), (0.0, 1.0));
        let epsilon = 1e-12;
        assert!((c.signed_area() - 0.75).abs() < epsilon);
        assert!(((Affine::rotate(0.5) * c).signed_area() - 0.75).abs() < epsilon);
        assert!(((Affine::translate((0.0, 1.0)) * c).signed_area() - 1.25).abs() < epsilon);
        assert!(((Affine::translate((1.0, 0.0)) * c).signed_area() - 1.25).abs() < epsilon);
    }

    #[test]
    fn cubicbez_nearest() {
        fn verify(result: Nearest, expected: f64) {
            assert!(
                (result.t - expected).abs() < 1e-6,
                "got {result:?} expected {expected}"
            );
        }
        // y = x^3
        let c = CubicBez::new((0.0, 0.0), (1.0 / 3.0, 0.0), (2.0 / 3.0, 0.0), (1.0, 1.0));
        verify(c.nearest((0.1, 0.001).into(), 1e-6), 0.1);
        verify(c.nearest((0.2, 0.008).into(), 1e-6), 0.2);
        verify(c.nearest((0.3, 0.027).into(), 1e-6), 0.3);
        verify(c.nearest((0.4, 0.064).into(), 1e-6), 0.4);
        verify(c.nearest((0.5, 0.125).into(), 1e-6), 0.5);
        verify(c.nearest((0.6, 0.216).into(), 1e-6), 0.6);
        verify(c.nearest((0.7, 0.343).into(), 1e-6), 0.7);
        verify(c.nearest((0.8, 0.512).into(), 1e-6), 0.8);
        verify(c.nearest((0.9, 0.729).into(), 1e-6), 0.9);
        verify(c.nearest((1.0, 1.0).into(), 1e-6), 1.0);
        verify(c.nearest((1.1, 1.1).into(), 1e-6), 1.0);
        verify(c.nearest((-0.1, 0.0).into(), 1e-6), 0.0);
        let a = Affine::rotate(0.5);
        verify((a * c).nearest(a * Point::new(0.1, 0.001), 1e-6), 0.1);

        // Here's a case that tripped up the old solver because the start is close to
        // degenerate and the end is actually degenerate; see #446.
        let curve = CubicBez::new(
            (461.0, 123.0),
            (460.99999999999994, 123.00000000000004),
            (111.0, 319.0),
            (111.0, 319.0),
        );
        let p = Point::new(282.0379003395483, 223.21877580985594);
        let eps = 0.0005;
        let nearest = curve.nearest(p, eps);
        let q = curve.eval(nearest.t);
        let r = curve.eval(0.5075474297354187);

        assert!((q - p).hypot() <= (r - p).hypot() + eps);
    }

    // ensure to_quads returns something given collinear points
    #[test]
    fn degenerate_to_quads() {
        let c = CubicBez::new((0., 9.), (6., 6.), (12., 3.0), (18., 0.0));
        let quads = c.to_quads(1e-6).collect::<Vec<_>>();
        assert_eq!(quads.len(), 1, "{:?}", &quads);
    }

    #[test]
    fn cubicbez_extrema() {
        // y = x^2
        let q = CubicBez::new((0.0, 0.0), (0.0, 1.0), (1.0, 1.0), (1.0, 0.0));
        let extrema = q.extrema();
        assert_eq!(extrema.len(), 1);
        assert!((extrema[0] - 0.5).abs() < 1e-6);

        let q = CubicBez::new((0.4, 0.5), (0.0, 1.0), (1.0, 0.0), (0.5, 0.4));
        let extrema = q.extrema();
        assert_eq!(extrema.len(), 4);
    }

    #[test]
    fn cubicbez_toquads() {
        // y = x^3
        let c = CubicBez::new((0.0, 0.0), (1.0 / 3.0, 0.0), (2.0 / 3.0, 0.0), (1.0, 1.0));
        for i in 0..10 {
            let accuracy = 0.1f64.powi(i);
            let mut worst: f64 = 0.0;
            for (t0, t1, q) in c.to_quads(accuracy) {
                let epsilon = 1e-12;
                assert!((q.start() - c.eval(t0)).hypot() < epsilon);
                assert!((q.end() - c.eval(t1)).hypot() < epsilon);
                let n = 4;
                for j in 0..=n {
                    let t = (j as f64) * (n as f64).recip();
                    let p = q.eval(t);
                    let err = (p.y - p.x.powi(3)).abs();
                    worst = worst.max(err);
                    assert!(err < accuracy, "got {err} wanted {accuracy}");
                }
            }
        }
    }

    #[test]
    fn cubicbez_approx_spline() {
        let c1 = CubicBez::new(
            (550.0, 258.0),
            (1044.0, 482.0),
            (2029.0, 1841.0),
            (1934.0, 1554.0),
        );

        let quad = c1.try_approx_quadratic(344.0);
        let expected = QuadBez::new(
            Point::new(550.0, 258.0),
            Point::new(1673.665720592873, 767.5164401068898),
            Point::new(1934.0, 1554.0),
        );
        assert!(quad.is_some());
        assert_eq!(quad.unwrap(), expected);

        let quad = c1.try_approx_quadratic(343.0);
        assert!(quad.is_none());

        let spline = c1.approx_spline_n(2, 343.0);
        assert!(spline.is_some());
        let spline = spline.unwrap();
        let expected = [
            Point::new(550.0, 258.0),
            Point::new(920.5, 426.0),
            Point::new(2005.25, 1769.25),
            Point::new(1934.0, 1554.0),
        ];
        assert_eq!(spline.points().len(), expected.len());
        for (got, &wanted) in spline.points().iter().zip(expected.iter()) {
            assert!(got.distance(wanted) < 5.0);
        }

        let spline = c1.approx_spline(5.0);
        let expected = [
            Point::new(550.0, 258.0),
            Point::new(673.5, 314.0),
            Point::new(984.8777777777776, 584.2666666666667),
            Point::new(1312.6305555555557, 927.825),
            Point::new(1613.1194444444443, 1267.425),
            Point::new(1842.7055555555555, 1525.8166666666666),
            Point::new(1957.75, 1625.75),
            Point::new(1934.0, 1554.0),
        ];
        assert!(spline.is_some());
        let spline = spline.unwrap();
        assert_eq!(spline.points().len(), expected.len());
        for (got, &wanted) in spline.points().iter().zip(expected.iter()) {
            assert!(got.distance(wanted) < 5.0);
        }
    }

    #[test]
    fn cubicbez_cubics_to_quadratic_splines() {
        let curves = vec![
            CubicBez::new(
                (550.0, 258.0),
                (1044.0, 482.0),
                (2029.0, 1841.0),
                (1934.0, 1554.0),
            ),
            CubicBez::new(
                (859.0, 384.0),
                (1998.0, 116.0),
                (1596.0, 1772.0),
                (8.0, 1824.0),
            ),
            CubicBez::new(
                (1090.0, 937.0),
                (418.0, 1300.0),
                (125.0, 91.0),
                (104.0, 37.0),
            ),
        ];
        let converted = cubics_to_quadratic_splines(&curves, 5.0);
        assert!(converted.is_some());
        let converted = converted.unwrap();
        assert_eq!(converted[0].points().len(), 8);
        assert_eq!(converted[1].points().len(), 8);
        assert_eq!(converted[2].points().len(), 8);
        assert!(converted[0].points()[1].distance(Point::new(673.5, 314.0)) < 0.0001);
        assert!(
            converted[0].points()[2].distance(Point::new(88639.0 / 90.0, 52584.0 / 90.0)) < 0.0001
        );
    }

    #[test]
    fn cubicbez_approx_spline_div_exact() {
        // Ensure rounding behavior for division matches fonttools
        // cu2qu.
        // See <https://github.com/linebender/kurbo/issues/272>
        let cubic = CubicBez::new(
            Point::new(408.0, 321.0),
            Point::new(408.0, 452.0),
            Point::new(342.0, 560.0),
            Point::new(260.0, 560.0),
        );
        let spline = cubic.approx_spline(1.0).unwrap();
        assert_eq!(
            spline.points(),
            &[
                Point::new(408.0, 321.0),
                // Previous behavior produced 386.49999999999994 for the
                // y coordinate leading to inconsistent rounding.
                Point::new(408.0, 386.5),
                Point::new(368.16666666666663, 495.0833333333333),
                Point::new(301.0, 560.0),
                Point::new(260.0, 560.0)
            ]
        );
    }

    #[test]
    fn cubicbez_inflections() {
        let c = CubicBez::new((0., 0.), (0.8, 1.), (0.2, 1.), (1., 0.));
        let inflections = c.inflections();
        assert_eq!(inflections.len(), 2);
        assert!((inflections[0] - 0.311018).abs() < 1e-6);
        assert!((inflections[1] - 0.688982).abs() < 1e-6);
        let c = CubicBez::new((0., 0.), (1., 1.), (2., -1.), (3., 0.));
        let inflections = c.inflections();
        assert_eq!(inflections.len(), 1);
        assert!((inflections[0] - 0.5).abs() < 1e-6);
        let c = CubicBez::new((0., 0.), (1., 1.), (2., 1.), (3., 0.));
        let inflections = c.inflections();
        assert_eq!(inflections.len(), 0);
    }

    #[test]
    fn cubic_to_quadratic_matches_python() {
        // from https://github.com/googlefonts/fontmake-rs/issues/217
        let cubic = CubicBez {
            p0: (796.0, 319.0).into(),
            p1: (727.0, 314.0).into(),
            p2: (242.0, 303.0).into(),
            p3: (106.0, 303.0).into(),
        };

        // FontTools can approximate this curve successfully in 7 splits, we can too
        assert!(cubic.approx_spline_n(7, 1.0).is_some());

        // FontTools can solve this with accuracy 0.001, we can too
        assert!(cubics_to_quadratic_splines(&[cubic], 0.001).is_some());
    }

    #[test]
    fn cubic_to_quadratic_all_points_equal() {
        let pt = Point::new(5.0, 5.0);
        let cubic = CubicBez::new(pt, pt, pt, pt);
        let quads = cubics_to_quadratic_splines(&[cubic], 0.1).unwrap();
        assert_eq!(quads, [QuadSpline::new(vec![pt, pt, pt])]);
    }

    #[test]
    fn cubic_to_quadratic_3_points_equal_single_quad_within_tolerance() {
        let p0 = Point::new(5.0, 5.0);
        let p1 = Point::new(5.0, 5.1);
        let cubic = CubicBez::new(p0, p0, p0, p1);
        let quads = cubics_to_quadratic_splines(&[cubic], 0.1).unwrap();
        // a single quadratic bezier approximates this cubic for the given tolerance
        assert_eq!(quads, [QuadSpline::new(vec![p0, p0, p1])]);
    }

    #[test]
    fn cubic_to_quadratic_3_points_equal_exceeding_tolerance() {
        let p0 = Point::new(5.0, 5.0);
        let p1 = Point::new(5.0, 5.1);
        let cubic = CubicBez::new(p0, p0, p0, p1);
        let quads = cubics_to_quadratic_splines(&[cubic], 0.01).unwrap();
        // 2 quadratic off-curves are required to approximate the same cubic
        // given the smaller tolerance
        assert_eq!(
            quads,
            [QuadSpline::new(vec![p0, p0, (5.0, 5.025).into(), p1])]
        );
    }

    #[test]
    fn cubics_to_quadratic_splines_matches_python() {
        // https://github.com/linebender/kurbo/pull/273
        let light = CubicBez::new((378., 608.), (378., 524.), (355., 455.), (266., 455.));
        let regular = CubicBez::new((367., 607.), (367., 511.), (338., 472.), (243., 472.));
        let bold = CubicBez::new(
            (372.425, 593.05),
            (372.425, 524.95),
            (355.05, 485.95),
            (274., 485.95),
        );
        let qsplines = cubics_to_quadratic_splines(&[light, regular, bold], 1.0).unwrap();
        assert_eq!(
            qsplines,
            [
                QuadSpline::new(vec![
                    (378.0, 608.0).into(),
                    (378.0, 566.0).into(),
                    (359.0833333333333, 496.5).into(),
                    (310.5, 455.0).into(),
                    (266.0, 455.0).into(),
                ]),
                QuadSpline::new(vec![
                    (367.0, 607.0).into(),
                    (367.0, 559.0).into(),
                    // Previous behavior produced 496.5 for the y coordinate
                    (344.5833333333333, 499.49999999999994).into(),
                    (290.5, 472.0).into(),
                    (243.0, 472.0).into(),
                ]),
                QuadSpline::new(vec![
                    (372.425, 593.05).into(),
                    (372.425, 559.0).into(),
                    (356.98333333333335, 511.125).into(),
                    (314.525, 485.95).into(),
                    (274.0, 485.95).into(),
                ]),
            ]
        );
    }
}