lyon_geom/

arc.rs

//! Elliptic arc related maths and tools.

use core::mem::swap;
use core::ops::Range;

use num_traits::NumCast;

use crate::scalar::{cast, Float, Scalar};
use crate::segment::{BoundingBox, Segment};
use crate::{point, vector, Angle, Box2D, Point, Rotation, Transform, Vector};
use crate::{CubicBezierSegment, Line, LineSegment, QuadraticBezierSegment};

/// An elliptic arc curve segment.
#[derive(Copy, Clone, Debug, PartialEq)]
#[cfg_attr(feature = "serialization", derive(Serialize, Deserialize))]
pub struct Arc<S> {
    pub center: Point<S>,
    pub radii: Vector<S>,
    pub start_angle: Angle<S>,
    pub sweep_angle: Angle<S>,
    pub x_rotation: Angle<S>,
}

/// An elliptic arc curve segment using the SVG's end-point notation.
#[derive(Copy, Clone, Debug, PartialEq)]
#[cfg_attr(feature = "serialization", derive(Serialize, Deserialize))]
pub struct SvgArc<S> {
    pub from: Point<S>,
    pub to: Point<S>,
    pub radii: Vector<S>,
    pub x_rotation: Angle<S>,
    pub flags: ArcFlags,
}

impl<S: Scalar> Arc<S> {
    pub fn cast<NewS: NumCast>(self) -> Arc<NewS> {
        Arc {
            center: self.center.cast(),
            radii: self.radii.cast(),
            start_angle: self.start_angle.cast(),
            sweep_angle: self.sweep_angle.cast(),
            x_rotation: self.x_rotation.cast(),
        }
    }

    /// Create simple circle.
    pub fn circle(center: Point<S>, radius: S) -> Self {
        Arc {
            center,
            radii: vector(radius, radius),
            start_angle: Angle::zero(),
            sweep_angle: Angle::two_pi(),
            x_rotation: Angle::zero(),
        }
    }

    /// Convert from the SVG arc notation.
    pub fn from_svg_arc(arc: &SvgArc<S>) -> Arc<S> {
        debug_assert!(!arc.from.x.is_nan());
        debug_assert!(!arc.from.y.is_nan());
        debug_assert!(!arc.to.x.is_nan());
        debug_assert!(!arc.to.y.is_nan());
        debug_assert!(!arc.radii.x.is_nan());
        debug_assert!(!arc.radii.y.is_nan());
        debug_assert!(!arc.x_rotation.get().is_nan());
        // The SVG spec specifies what we should do if one of the two
        // radii is zero and not the other, but it's better to handle
        // this out of arc code and generate a line_to instead of an arc.
        assert!(!arc.is_straight_line());

        let mut rx = S::abs(arc.radii.x);
        let mut ry = S::abs(arc.radii.y);

        let xr = arc.x_rotation.get() % (S::TWO * S::PI());
        let cos_phi = Float::cos(xr);
        let sin_phi = Float::sin(xr);
        let hd_x = (arc.from.x - arc.to.x) / S::TWO;
        let hd_y = (arc.from.y - arc.to.y) / S::TWO;
        let hs_x = (arc.from.x + arc.to.x) / S::TWO;
        let hs_y = (arc.from.y + arc.to.y) / S::TWO;

        // F6.5.1
        let p = Point::new(
            cos_phi * hd_x + sin_phi * hd_y,
            -sin_phi * hd_x + cos_phi * hd_y,
        );

        // Sanitize the radii.
        // If rf > 1 it means the radii are too small for the arc to
        // possibly connect the end points. In this situation we scale
        // them up according to the formula provided by the SVG spec.

        // F6.6.2
        let rf = p.x * p.x / (rx * rx) + p.y * p.y / (ry * ry);
        if rf > S::ONE {
            let scale = S::sqrt(rf);
            rx *= scale;
            ry *= scale;
        }

        let rxry = rx * ry;
        let rxpy = rx * p.y;
        let rypx = ry * p.x;
        let sum_of_sq = rxpy * rxpy + rypx * rypx;

        debug_assert_ne!(sum_of_sq, S::ZERO);

        // F6.5.2
        let sign_coe = if arc.flags.large_arc == arc.flags.sweep {
            -S::ONE
        } else {
            S::ONE
        };
        let coe = sign_coe * S::sqrt(S::abs((rxry * rxry - sum_of_sq) / sum_of_sq));
        let transformed_cx = coe * rxpy / ry;
        let transformed_cy = -coe * rypx / rx;

        // F6.5.3
        let center = point(
            cos_phi * transformed_cx - sin_phi * transformed_cy + hs_x,
            sin_phi * transformed_cx + cos_phi * transformed_cy + hs_y,
        );

        let start_v: Vector<S> = vector((p.x - transformed_cx) / rx, (p.y - transformed_cy) / ry);
        let end_v: Vector<S> = vector((-p.x - transformed_cx) / rx, (-p.y - transformed_cy) / ry);

        let two_pi = S::TWO * S::PI();

        let start_angle = start_v.angle_from_x_axis();

        let mut sweep_angle = (end_v.angle_from_x_axis() - start_angle).radians % two_pi;

        if arc.flags.sweep && sweep_angle < S::ZERO {
            sweep_angle += two_pi;
        } else if !arc.flags.sweep && sweep_angle > S::ZERO {
            sweep_angle -= two_pi;
        }

        Arc {
            center,
            radii: vector(rx, ry),
            start_angle,
            sweep_angle: Angle::radians(sweep_angle),
            x_rotation: arc.x_rotation,
        }
    }

    /// Convert to the SVG arc notation.
    pub fn to_svg_arc(&self) -> SvgArc<S> {
        let from = self.sample(S::ZERO);
        let to = self.sample(S::ONE);
        let flags = ArcFlags {
            sweep: self.sweep_angle.get() >= S::ZERO,
            large_arc: S::abs(self.sweep_angle.get()) >= S::PI(),
        };
        SvgArc {
            from,
            to,
            radii: self.radii,
            x_rotation: self.x_rotation,
            flags,
        }
    }

    /// Approximate the arc with a sequence of quadratic bézier curves.
    #[inline]
    pub fn for_each_quadratic_bezier<F>(&self, cb: &mut F)
    where
        F: FnMut(&QuadraticBezierSegment<S>),
    {
        arc_to_quadratic_beziers_with_t(self, &mut |curve, _| cb(curve));
    }

    /// Approximate the arc with a sequence of quadratic bézier curves.
    #[inline]
    pub fn for_each_quadratic_bezier_with_t<F>(&self, cb: &mut F)
    where
        F: FnMut(&QuadraticBezierSegment<S>, Range<S>),
    {
        arc_to_quadratic_beziers_with_t(self, cb);
    }

    /// Approximate the arc with a sequence of cubic bézier curves.
    #[inline]
    pub fn for_each_cubic_bezier<F>(&self, cb: &mut F)
    where
        F: FnMut(&CubicBezierSegment<S>),
    {
        arc_to_cubic_beziers(self, cb);
    }

    /// Sample the curve at t (expecting t between 0 and 1).
    #[inline]
    pub fn sample(&self, t: S) -> Point<S> {
        let angle = self.get_angle(t);
        self.center + sample_ellipse(self.radii, self.x_rotation, angle).to_vector()
    }

    #[inline]
    pub fn x(&self, t: S) -> S {
        self.sample(t).x
    }

    #[inline]
    pub fn y(&self, t: S) -> S {
        self.sample(t).y
    }

    /// Sample the curve's tangent at t (expecting t between 0 and 1).
    #[inline]
    pub fn sample_tangent(&self, t: S) -> Vector<S> {
        self.tangent_at_angle(self.get_angle(t))
    }

    /// Sample the curve's angle at t (expecting t between 0 and 1).
    #[inline]
    pub fn get_angle(&self, t: S) -> Angle<S> {
        self.start_angle + Angle::radians(self.sweep_angle.get() * t)
    }

    #[inline]
    pub fn end_angle(&self) -> Angle<S> {
        self.start_angle + self.sweep_angle
    }

    #[inline]
    pub fn from(&self) -> Point<S> {
        self.sample(S::ZERO)
    }

    #[inline]
    pub fn to(&self) -> Point<S> {
        self.sample(S::ONE)
    }

    /// Return the sub-curve inside a given range of t.
    ///
    /// This is equivalent splitting at the range's end points.
    pub fn split_range(&self, t_range: Range<S>) -> Self {
        let angle_1 = Angle::radians(self.sweep_angle.get() * t_range.start);
        let angle_2 = Angle::radians(self.sweep_angle.get() * t_range.end);

        Arc {
            center: self.center,
            radii: self.radii,
            start_angle: self.start_angle + angle_1,
            sweep_angle: angle_2 - angle_1,
            x_rotation: self.x_rotation,
        }
    }

    /// Split this curve into two sub-curves.
    pub fn split(&self, t: S) -> (Arc<S>, Arc<S>) {
        let split_angle = Angle::radians(self.sweep_angle.get() * t);
        (
            Arc {
                center: self.center,
                radii: self.radii,
                start_angle: self.start_angle,
                sweep_angle: split_angle,
                x_rotation: self.x_rotation,
            },
            Arc {
                center: self.center,
                radii: self.radii,
                start_angle: self.start_angle + split_angle,
                sweep_angle: self.sweep_angle - split_angle,
                x_rotation: self.x_rotation,
            },
        )
    }

    /// Return the curve before the split point.
    pub fn before_split(&self, t: S) -> Arc<S> {
        let split_angle = Angle::radians(self.sweep_angle.get() * t);
        Arc {
            center: self.center,
            radii: self.radii,
            start_angle: self.start_angle,
            sweep_angle: split_angle,
            x_rotation: self.x_rotation,
        }
    }

    /// Return the curve after the split point.
    pub fn after_split(&self, t: S) -> Arc<S> {
        let split_angle = Angle::radians(self.sweep_angle.get() * t);
        Arc {
            center: self.center,
            radii: self.radii,
            start_angle: self.start_angle + split_angle,
            sweep_angle: self.sweep_angle - split_angle,
            x_rotation: self.x_rotation,
        }
    }

    /// Swap the direction of the segment.
    pub fn flip(&self) -> Self {
        let mut arc = *self;
        arc.start_angle += self.sweep_angle;
        arc.sweep_angle = -self.sweep_angle;

        arc
    }

    /// Approximates the curve with sequence of line segments.
    ///
    /// The `tolerance` parameter defines the maximum distance between the curve and
    /// its approximation.
    pub fn for_each_flattened<F>(&self, tolerance: S, callback: &mut F)
    where
        F: FnMut(&LineSegment<S>),
    {
        let mut from = self.from();
        let mut iter = *self;
        loop {
            let t = iter.flattening_step(tolerance);
            if t >= S::ONE {
                break;
            }
            iter = iter.after_split(t);
            let to = iter.from();
            callback(&LineSegment { from, to });
            from = to;
        }

        callback(&LineSegment {
            from,
            to: self.to(),
        });
    }

    /// Approximates the curve with sequence of line segments.
    ///
    /// The `tolerance` parameter defines the maximum distance between the curve and
    /// its approximation.
    ///
    /// The end of the t parameter range at the final segment is guaranteed to be equal to `1.0`.
    pub fn for_each_flattened_with_t<F>(&self, tolerance: S, callback: &mut F)
    where
        F: FnMut(&LineSegment<S>, Range<S>),
    {
        let mut iter = *self;
        let mut t0 = S::ZERO;
        let mut from = self.from();
        loop {
            let step = iter.flattening_step(tolerance);

            if step >= S::ONE {
                break;
            }

            iter = iter.after_split(step);
            let t1 = t0 + step * (S::ONE - t0);
            let to = iter.from();
            callback(&LineSegment { from, to }, t0..t1);
            from = to;
            t0 = t1;
        }

        callback(
            &LineSegment {
                from,
                to: self.to(),
            },
            t0..S::ONE,
        );
    }

    /// Finds the interval of the beginning of the curve that can be approximated with a
    /// line segment.
    fn flattening_step(&self, tolerance: S) -> S {
        // cos(theta) = (r - tolerance) / r
        // angle = 2 * theta
        // s = angle / sweep

        // Here we make the approximation that for small tolerance values we consider
        // the radius to be constant over each approximated segment.
        let r = (self.from() - self.center).length();
        let a = S::TWO * S::acos((r - tolerance) / r);
        let result = S::min(a / self.sweep_angle.radians.abs(), S::ONE);

        if result < S::EPSILON {
            return S::ONE;
        }

        result
    }

    /// Returns the flattened representation of the curve as an iterator, starting *after* the
    /// current point.
    pub fn flattened(&self, tolerance: S) -> Flattened<S> {
        Flattened::new(*self, tolerance)
    }

    /// Returns a conservative rectangle that contains the curve.
    pub fn fast_bounding_box(&self) -> Box2D<S> {
        Transform::rotation(self.x_rotation).outer_transformed_box(&Box2D {
            min: self.center - self.radii,
            max: self.center + self.radii,
        })
    }

    /// Returns a conservative rectangle that contains the curve.
    pub fn bounding_box(&self) -> Box2D<S> {
        let from = self.from();
        let to = self.to();
        let mut min = Point::min(from, to);
        let mut max = Point::max(from, to);
        self.for_each_local_x_extremum_t(&mut |t| {
            let p = self.sample(t);
            min.x = S::min(min.x, p.x);
            max.x = S::max(max.x, p.x);
        });
        self.for_each_local_y_extremum_t(&mut |t| {
            let p = self.sample(t);
            min.y = S::min(min.y, p.y);
            max.y = S::max(max.y, p.y);
        });

        Box2D { min, max }
    }

    pub fn for_each_local_x_extremum_t<F>(&self, cb: &mut F)
    where
        F: FnMut(S),
    {
        let rx = self.radii.x;
        let ry = self.radii.y;
        let a1 = Angle::radians(-S::atan(ry * Float::tan(self.x_rotation.radians) / rx));
        let a2 = Angle::pi() + a1;

        self.for_each_extremum_inner(a1, a2, cb);
    }

    pub fn for_each_local_y_extremum_t<F>(&self, cb: &mut F)
    where
        F: FnMut(S),
    {
        let rx = self.radii.x;
        let ry = self.radii.y;
        let a1 = Angle::radians(S::atan(ry / (Float::tan(self.x_rotation.radians) * rx)));
        let a2 = Angle::pi() + a1;

        self.for_each_extremum_inner(a1, a2, cb);
    }

    fn for_each_extremum_inner<F>(&self, a1: Angle<S>, a2: Angle<S>, cb: &mut F)
    where
        F: FnMut(S),
    {
        let sweep = self.sweep_angle.radians;
        let abs_sweep = S::abs(sweep);
        let sign = S::signum(sweep);

        let mut a1 = (a1 - self.start_angle).positive().radians;
        let mut a2 = (a2 - self.start_angle).positive().radians;
        if a1 * sign > a2 * sign {
            swap(&mut a1, &mut a2);
        }

        let two_pi = S::TWO * S::PI();
        if sweep >= S::ZERO {
            if a1 < abs_sweep {
                cb(a1 / abs_sweep);
            }
            if a2 < abs_sweep {
                cb(a2 / abs_sweep);
            }
        } else {
            if a1 > two_pi - abs_sweep {
                cb(a1 / abs_sweep);
            }
            if a2 > two_pi - abs_sweep {
                cb(a2 / abs_sweep);
            }
        }
    }

    pub fn bounding_range_x(&self) -> (S, S) {
        let r = self.bounding_box();
        (r.min.x, r.max.x)
    }

    pub fn bounding_range_y(&self) -> (S, S) {
        let r = self.bounding_box();
        (r.min.y, r.max.y)
    }

    pub fn fast_bounding_range_x(&self) -> (S, S) {
        let r = self.fast_bounding_box();
        (r.min.x, r.max.x)
    }

    pub fn fast_bounding_range_y(&self) -> (S, S) {
        let r = self.fast_bounding_box();
        (r.min.y, r.max.y)
    }

    pub fn approximate_length(&self, tolerance: S) -> S {
        let mut len = S::ZERO;
        self.for_each_flattened(tolerance, &mut |segment| {
            len += segment.length();
        });

        len
    }

    #[inline]
    fn tangent_at_angle(&self, angle: Angle<S>) -> Vector<S> {
        let a = angle.get();
        Rotation::new(self.x_rotation).transform_vector(vector(
            -self.radii.x * Float::sin(a),
            self.radii.y * Float::cos(a),
        ))
    }
}

impl<S: Scalar> From<SvgArc<S>> for Arc<S> {
    fn from(svg: SvgArc<S>) -> Self {
        svg.to_arc()
    }
}

impl<S: Scalar> SvgArc<S> {
    /// Converts this arc from endpoints to center notation.
    pub fn to_arc(&self) -> Arc<S> {
        Arc::from_svg_arc(self)
    }

    /// Per SVG spec, this arc should be rendered as a line_to segment.
    ///
    /// Do not convert an `SvgArc` into an `arc` if this returns true.
    pub fn is_straight_line(&self) -> bool {
        S::abs(self.radii.x) <= S::EPSILON
            || S::abs(self.radii.y) <= S::EPSILON
            || self.from == self.to
    }

    /// Approximates the arc with a sequence of quadratic bézier segments.
    pub fn for_each_quadratic_bezier<F>(&self, cb: &mut F)
    where
        F: FnMut(&QuadraticBezierSegment<S>),
    {
        if self.is_straight_line() {
            cb(&QuadraticBezierSegment {
                from: self.from,
                ctrl: self.from,
                to: self.to,
            });
            return;
        }

        Arc::from_svg_arc(self).for_each_quadratic_bezier(cb);
    }

    /// Approximates the arc with a sequence of quadratic bézier segments.
    pub fn for_each_quadratic_bezier_with_t<F>(&self, cb: &mut F)
    where
        F: FnMut(&QuadraticBezierSegment<S>, Range<S>),
    {
        if self.is_straight_line() {
            cb(
                &QuadraticBezierSegment {
                    from: self.from,
                    ctrl: self.from,
                    to: self.to,
                },
                S::ZERO..S::ONE,
            );
            return;
        }

        Arc::from_svg_arc(self).for_each_quadratic_bezier_with_t(cb);
    }

    /// Approximates the arc with a sequence of cubic bézier segments.
    pub fn for_each_cubic_bezier<F>(&self, cb: &mut F)
    where
        F: FnMut(&CubicBezierSegment<S>),
    {
        if self.is_straight_line() {
            cb(&CubicBezierSegment {
                from: self.from,
                ctrl1: self.from,
                ctrl2: self.to,
                to: self.to,
            });
            return;
        }

        Arc::from_svg_arc(self).for_each_cubic_bezier(cb);
    }

    /// Approximates the curve with sequence of line segments.
    ///
    /// The `tolerance` parameter defines the maximum distance between the curve and
    /// its approximation.
    pub fn for_each_flattened<F: FnMut(&LineSegment<S>)>(&self, tolerance: S, cb: &mut F) {
        if self.is_straight_line() {
            cb(&LineSegment {
                from: self.from,
                to: self.to,
            });
            return;
        }

        Arc::from_svg_arc(self).for_each_flattened(tolerance, cb);
    }

    /// Approximates the curve with sequence of line segments.
    ///
    /// The `tolerance` parameter defines the maximum distance between the curve and
    /// its approximation.
    ///
    /// The end of the t parameter range at the final segment is guaranteed to be equal to `1.0`.
    pub fn for_each_flattened_with_t<F: FnMut(&LineSegment<S>, Range<S>)>(
        &self,
        tolerance: S,
        cb: &mut F,
    ) {
        if self.is_straight_line() {
            cb(
                &LineSegment {
                    from: self.from,
                    to: self.to,
                },
                S::ZERO..S::ONE,
            );
            return;
        }

        Arc::from_svg_arc(self).for_each_flattened_with_t(tolerance, cb);
    }
}

/// Flag parameters for arcs as described by the SVG specification.
///
/// For most situations using the SVG arc notation, there are four different arcs
/// (two different ellipses, each with two different arc sweeps) that satisfy the
/// arc parameters. The `large_arc` and `sweep` flags indicate which one of the
/// four arcs are drawn, as follows:
///
/// See more examples in the [SVG specification](https://svgwg.org/specs/paths/)
#[derive(Copy, Clone, Debug, PartialEq, Default)]
#[cfg_attr(feature = "serialization", derive(Serialize, Deserialize))]
pub struct ArcFlags {
    /// Of the four candidate arc sweeps, two will represent an arc sweep of greater
    /// than or equal to 180 degrees (the "large-arc"), and two will represent an arc
    /// sweep of less than or equal to 180 degrees (the "small arc"). If `large_arc`
    /// is `true`, then one of the two larger arc sweeps will be chosen; otherwise, if
    /// `large_arc` is `false`, one of the smaller arc sweeps will be chosen.
    pub large_arc: bool,
    /// If `sweep` is `true`, then the arc will be drawn in a "positive-angle" direction
    /// (the ellipse formula `x=cx+rx*cos(theta)` and `y=cy+ry*sin(theta)` is evaluated
    /// such that theta starts at an angle corresponding to the current point and increases
    /// positively until the arc reaches the destination position). A value of `false`
    /// causes the arc to be drawn in a "negative-angle" direction (theta starts at an
    /// angle value corresponding to the current point and decreases until the arc reaches
    /// the destination position).
    pub sweep: bool,
}

fn arc_to_quadratic_beziers_with_t<S, F>(arc: &Arc<S>, callback: &mut F)
where
    S: Scalar,
    F: FnMut(&QuadraticBezierSegment<S>, Range<S>),
{
    let sign = arc.sweep_angle.get().signum();
    let sweep_angle = S::abs(arc.sweep_angle.get()).min(S::PI() * S::TWO);

    let n_steps = S::ceil(sweep_angle / S::FRAC_PI_4());
    let step = Angle::radians(sweep_angle / n_steps * sign);

    let mut t0 = S::ZERO;
    let dt = S::ONE / n_steps;

    let n = cast::<S, i32>(n_steps).unwrap();
    for i in 0..n {
        let a1 = arc.start_angle + step * cast(i).unwrap();
        let a2 = arc.start_angle + step * cast(i + 1).unwrap();

        let v1 = sample_ellipse(arc.radii, arc.x_rotation, a1).to_vector();
        let v2 = sample_ellipse(arc.radii, arc.x_rotation, a2).to_vector();
        let from = arc.center + v1;
        let to = arc.center + v2;
        let l1 = Line {
            point: from,
            vector: arc.tangent_at_angle(a1),
        };
        let l2 = Line {
            point: to,
            vector: arc.tangent_at_angle(a2),
        };
        let ctrl = l2.intersection(&l1).unwrap_or(from);

        let t1 = if i + 1 == n { S::ONE } else { t0 + dt };

        callback(&QuadraticBezierSegment { from, ctrl, to }, t0..t1);
        t0 = t1;
    }
}

fn arc_to_cubic_beziers<S, F>(arc: &Arc<S>, callback: &mut F)
where
    S: Scalar,
    F: FnMut(&CubicBezierSegment<S>),
{
    let sign = arc.sweep_angle.get().signum();
    let sweep_angle = S::abs(arc.sweep_angle.get()).min(S::PI() * S::TWO);

    let n_steps = S::ceil(sweep_angle / S::FRAC_PI_2());
    let step = Angle::radians(sweep_angle / n_steps * sign);

    for i in 0..cast::<S, i32>(n_steps).unwrap() {
        let a1 = arc.start_angle + step * cast(i).unwrap();
        let a2 = arc.start_angle + step * cast(i + 1).unwrap();

        let v1 = sample_ellipse(arc.radii, arc.x_rotation, a1).to_vector();
        let v2 = sample_ellipse(arc.radii, arc.x_rotation, a2).to_vector();
        let from = arc.center + v1;
        let to = arc.center + v2;

        // From http://www.spaceroots.org/documents/ellipse/elliptical-arc.pdf
        // Note that the parameterization used by Arc (see sample_ellipse for
        // example) is the same as the eta-parameterization used at the link.
        let delta_a = a2 - a1;
        let tan_da = Float::tan(delta_a.get() * S::HALF);
        let alpha_sqrt = S::sqrt(S::FOUR + S::THREE * tan_da * tan_da);
        let alpha = Float::sin(delta_a.get()) * (alpha_sqrt - S::ONE) / S::THREE;
        let ctrl1 = from + arc.tangent_at_angle(a1) * alpha;
        let ctrl2 = to - arc.tangent_at_angle(a2) * alpha;

        callback(&CubicBezierSegment {
            from,
            ctrl1,
            ctrl2,
            to,
        });
    }
}

fn sample_ellipse<S: Scalar>(radii: Vector<S>, x_rotation: Angle<S>, angle: Angle<S>) -> Point<S> {
    Rotation::new(x_rotation).transform_point(point(
        radii.x * Float::cos(angle.get()),
        radii.y * Float::sin(angle.get()),
    ))
}

impl<S: Scalar> Segment for Arc<S> {
    type Scalar = S;
    fn from(&self) -> Point<S> {
        self.from()
    }
    fn to(&self) -> Point<S> {
        self.to()
    }
    fn sample(&self, t: S) -> Point<S> {
        self.sample(t)
    }
    fn x(&self, t: S) -> S {
        self.x(t)
    }
    fn y(&self, t: S) -> S {
        self.y(t)
    }
    fn derivative(&self, t: S) -> Vector<S> {
        self.sample_tangent(t)
    }
    fn split(&self, t: S) -> (Self, Self) {
        self.split(t)
    }
    fn before_split(&self, t: S) -> Self {
        self.before_split(t)
    }
    fn after_split(&self, t: S) -> Self {
        self.after_split(t)
    }
    fn split_range(&self, t_range: Range<S>) -> Self {
        self.split_range(t_range)
    }
    fn flip(&self) -> Self {
        self.flip()
    }
    fn approximate_length(&self, tolerance: S) -> S {
        self.approximate_length(tolerance)
    }

    fn for_each_flattened_with_t(
        &self,
        tolerance: Self::Scalar,
        callback: &mut dyn FnMut(&LineSegment<S>, Range<S>),
    ) {
        self.for_each_flattened_with_t(tolerance, &mut |s, t| callback(s, t));
    }
}

impl<S: Scalar> BoundingBox for Arc<S> {
    type Scalar = S;
    fn bounding_range_x(&self) -> (S, S) {
        self.bounding_range_x()
    }
    fn bounding_range_y(&self) -> (S, S) {
        self.bounding_range_y()
    }
    fn fast_bounding_range_x(&self) -> (S, S) {
        self.fast_bounding_range_x()
    }
    fn fast_bounding_range_y(&self) -> (S, S) {
        self.fast_bounding_range_y()
    }
}

/// Flattening iterator for arcs.
///
/// The iterator starts at the first point *after* the origin of the curve and ends at the
/// destination.
pub struct Flattened<S> {
    arc: Arc<S>,
    tolerance: S,
    done: bool,
}

impl<S: Scalar> Flattened<S> {
    pub(crate) fn new(arc: Arc<S>, tolerance: S) -> Self {
        assert!(tolerance > S::ZERO);
        Flattened {
            arc,
            tolerance,
            done: false,
        }
    }
}
impl<S: Scalar> Iterator for Flattened<S> {
    type Item = Point<S>;
    fn next(&mut self) -> Option<Point<S>> {
        if self.done {
            return None;
        }

        let t = self.arc.flattening_step(self.tolerance);
        if t >= S::ONE {
            self.done = true;
            return Some(self.arc.to());
        }
        self.arc = self.arc.after_split(t);

        Some(self.arc.from())
    }
}

#[test]
fn test_from_svg_arc() {
    use crate::vector;
    use euclid::approxeq::ApproxEq;

    let flags = ArcFlags {
        large_arc: false,
        sweep: false,
    };

    test_endpoints(&SvgArc {
        from: point(0.0, -10.0),
        to: point(10.0, 0.0),
        radii: vector(10.0, 10.0),
        x_rotation: Angle::radians(0.0),
        flags,
    });

    test_endpoints(&SvgArc {
        from: point(0.0, -10.0),
        to: point(10.0, 0.0),
        radii: vector(100.0, 10.0),
        x_rotation: Angle::radians(0.0),
        flags,
    });

    test_endpoints(&SvgArc {
        from: point(0.0, -10.0),
        to: point(10.0, 0.0),
        radii: vector(10.0, 30.0),
        x_rotation: Angle::radians(1.0),
        flags,
    });

    test_endpoints(&SvgArc {
        from: point(5.0, -10.0),
        to: point(5.0, 5.0),
        radii: vector(10.0, 30.0),
        x_rotation: Angle::radians(-2.0),
        flags,
    });

    // This arc has invalid radii (too small to connect the two endpoints),
    // but the conversion needs to be able to cope with that.
    test_endpoints(&SvgArc {
        from: point(0.0, 0.0),
        to: point(80.0, 60.0),
        radii: vector(40.0, 40.0),
        x_rotation: Angle::radians(0.0),
        flags,
    });

    fn test_endpoints(svg_arc: &SvgArc<f64>) {
        do_test_endpoints(&SvgArc {
            flags: ArcFlags {
                large_arc: false,
                sweep: false,
            },
            ..svg_arc.clone()
        });

        do_test_endpoints(&SvgArc {
            flags: ArcFlags {
                large_arc: true,
                sweep: false,
            },
            ..svg_arc.clone()
        });

        do_test_endpoints(&SvgArc {
            flags: ArcFlags {
                large_arc: false,
                sweep: true,
            },
            ..svg_arc.clone()
        });

        do_test_endpoints(&SvgArc {
            flags: ArcFlags {
                large_arc: true,
                sweep: true,
            },
            ..svg_arc.clone()
        });
    }

    fn do_test_endpoints(svg_arc: &SvgArc<f64>) {
        let eps = point(0.01, 0.01);
        let arc = svg_arc.to_arc();
        assert!(
            arc.from().approx_eq_eps(&svg_arc.from, &eps),
            "unexpected arc.from: {:?} == {:?}, flags: {:?}",
            arc.from(),
            svg_arc.from,
            svg_arc.flags,
        );
        assert!(
            arc.to().approx_eq_eps(&svg_arc.to, &eps),
            "unexpected arc.from: {:?} == {:?}, flags: {:?}",
            arc.to(),
            svg_arc.to,
            svg_arc.flags,
        );
    }
}

#[test]
fn test_to_quadratics_and_cubics() {
    use euclid::approxeq::ApproxEq;

    fn do_test(arc: &Arc<f32>, expected_quadratic_count: u32, expected_cubic_count: u32) {
        let last = arc.to();
        {
            let mut prev = arc.from();
            let mut count = 0;
            arc.for_each_quadratic_bezier(&mut |c| {
                assert!(c.from.approx_eq(&prev));
                prev = c.to;
                count += 1;
            });
            assert!(prev.approx_eq(&last));
            assert_eq!(count, expected_quadratic_count);
        }
        {
            let mut prev = arc.from();
            let mut count = 0;
            arc.for_each_cubic_bezier(&mut |c| {
                assert!(c.from.approx_eq(&prev));
                prev = c.to;
                count += 1;
            });
            assert!(prev.approx_eq(&last));
            assert_eq!(count, expected_cubic_count);
        }
    }

    do_test(
        &Arc {
            center: point(2.0, 3.0),
            radii: vector(10.0, 3.0),
            start_angle: Angle::radians(0.1),
            sweep_angle: Angle::radians(3.0),
            x_rotation: Angle::radians(0.5),
        },
        4,
        2,
    );

    do_test(
        &Arc {
            center: point(4.0, 5.0),
            radii: vector(3.0, 5.0),
            start_angle: Angle::radians(2.0),
            sweep_angle: Angle::radians(-3.0),
            x_rotation: Angle::radians(1.3),
        },
        4,
        2,
    );

    do_test(
        &Arc {
            center: point(0.0, 0.0),
            radii: vector(100.0, 0.01),
            start_angle: Angle::radians(-1.0),
            sweep_angle: Angle::radians(0.1),
            x_rotation: Angle::radians(0.3),
        },
        1,
        1,
    );

    do_test(
        &Arc {
            center: point(0.0, 0.0),
            radii: vector(1.0, 1.0),
            start_angle: Angle::radians(3.0),
            sweep_angle: Angle::radians(-0.1),
            x_rotation: Angle::radians(-0.3),
        },
        1,
        1,
    );
}

#[test]
fn test_bounding_box() {
    use euclid::approxeq::ApproxEq;

    fn approx_eq(r1: Box2D<f32>, r2: Box2D<f32>) -> bool {
        if !r1.min.x.approx_eq(&r2.min.x)
            || !r1.max.x.approx_eq(&r2.max.x)
            || !r1.min.y.approx_eq(&r2.min.y)
            || !r1.max.y.approx_eq(&r2.max.y)
        {
            std::println!("\n   left: {r1:?}\n   right: {r2:?}");
            return false;
        }

        true
    }

    let r = Arc {
        center: point(0.0, 0.0),
        radii: vector(1.0, 1.0),
        start_angle: Angle::radians(0.0),
        sweep_angle: Angle::pi(),
        x_rotation: Angle::zero(),
    }
    .bounding_box();
    assert!(approx_eq(
        r,
        Box2D {
            min: point(-1.0, 0.0),
            max: point(1.0, 1.0)
        }
    ));

    let r = Arc {
        center: point(0.0, 0.0),
        radii: vector(1.0, 1.0),
        start_angle: Angle::radians(0.0),
        sweep_angle: Angle::pi(),
        x_rotation: Angle::pi(),
    }
    .bounding_box();
    assert!(approx_eq(
        r,
        Box2D {
            min: point(-1.0, -1.0),
            max: point(1.0, 0.0)
        }
    ));

    let r = Arc {
        center: point(0.0, 0.0),
        radii: vector(2.0, 1.0),
        start_angle: Angle::radians(0.0),
        sweep_angle: Angle::pi(),
        x_rotation: Angle::pi() * 0.5,
    }
    .bounding_box();
    assert!(approx_eq(
        r,
        Box2D {
            min: point(-1.0, -2.0),
            max: point(0.0, 2.0)
        }
    ));

    let r = Arc {
        center: point(1.0, 1.0),
        radii: vector(1.0, 1.0),
        start_angle: Angle::pi(),
        sweep_angle: Angle::pi(),
        x_rotation: -Angle::pi() * 0.25,
    }
    .bounding_box();
    assert!(approx_eq(
        r,
        Box2D {
            min: point(0.0, 0.0),
            max: point(1.707107, 1.707107)
        }
    ));

    let mut angle = Angle::zero();
    for _ in 0..10 {
        std::println!("angle: {angle:?}");
        let r = Arc {
            center: point(0.0, 0.0),
            radii: vector(4.0, 4.0),
            start_angle: angle,
            sweep_angle: Angle::pi() * 2.0,
            x_rotation: Angle::pi() * 0.25,
        }
        .bounding_box();
        assert!(approx_eq(
            r,
            Box2D {
                min: point(-4.0, -4.0),
                max: point(4.0, 4.0)
            }
        ));
        angle += Angle::pi() * 2.0 / 10.0;
    }

    let mut angle = Angle::zero();
    for _ in 0..10 {
        std::println!("angle: {angle:?}");
        let r = Arc {
            center: point(0.0, 0.0),
            radii: vector(4.0, 4.0),
            start_angle: Angle::zero(),
            sweep_angle: Angle::pi() * 2.0,
            x_rotation: angle,
        }
        .bounding_box();
        assert!(approx_eq(
            r,
            Box2D {
                min: point(-4.0, -4.0),
                max: point(4.0, 4.0)
            }
        ));
        angle += Angle::pi() * 2.0 / 10.0;
    }
}

#[test]
fn negative_flattening_step() {
    // These parameters were running into a precision issue which led the
    // flattening step to never converge towards 1 and cause an infinite loop.

    let arc = Arc {
        center: point(-100.0, -150.0),
        radii: vector(50.0, 50.0),
        start_angle: Angle::radians(0.982944787),
        sweep_angle: Angle::radians(-898.0),
        x_rotation: Angle::zero(),
    };

    arc.for_each_flattened(0.100000001, &mut |_| {});

    // There was also an issue with negative sweep_angle leading to a negative step
    // causing the arc to be approximated with a single line segment.

    let arc = Arc {
        center: point(0.0, 0.0),
        radii: vector(100.0, 10.0),
        start_angle: Angle::radians(0.2),
        sweep_angle: Angle::radians(-2.0),
        x_rotation: Angle::zero(),
    };

    let flattened: std::vec::Vec<_> = arc.flattened(0.1).collect();

    assert!(flattened.len() > 1);
}