bevy_math/bounding/bounded2d/

primitive_impls.rs

//! Contains [`Bounded2d`] implementations for [geometric primitives](crate::primitives).

use crate::{
    primitives::{
        Arc2d, BoxedPolygon, BoxedPolyline2d, Capsule2d, Circle, CircularSector, CircularSegment,
        Ellipse, Line2d, Plane2d, Polygon, Polyline2d, Rectangle, RegularPolygon, Rhombus,
        Segment2d, Triangle2d,
    },
    Dir2, Mat2, Rot2, Vec2,
};
use std::f32::consts::{FRAC_PI_2, PI, TAU};

use smallvec::SmallVec;

use super::{Aabb2d, Bounded2d, BoundingCircle};

impl Bounded2d for Circle {
    fn aabb_2d(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> Aabb2d {
        Aabb2d::new(translation, Vec2::splat(self.radius))
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, self.radius)
    }
}

// Compute the axis-aligned bounding points of a rotated arc, used for computing the AABB of arcs and derived shapes.
// The return type has room for 7 points so that the CircularSector code can add an additional point.
#[inline]
fn arc_bounding_points(arc: Arc2d, rotation: impl Into<Rot2>) -> SmallVec<[Vec2; 7]> {
    // Otherwise, the extreme points will always be either the endpoints or the axis-aligned extrema of the arc's circle.
    // We need to compute which axis-aligned extrema are actually contained within the rotated arc.
    let mut bounds = SmallVec::<[Vec2; 7]>::new();
    let rotation = rotation.into();
    bounds.push(rotation * arc.left_endpoint());
    bounds.push(rotation * arc.right_endpoint());

    // The half-angles are measured from a starting point of π/2, being the angle of Vec2::Y.
    // Compute the normalized angles of the endpoints with the rotation taken into account, and then
    // check if we are looking for an angle that is between or outside them.
    let left_angle = (FRAC_PI_2 + arc.half_angle + rotation.as_radians()).rem_euclid(TAU);
    let right_angle = (FRAC_PI_2 - arc.half_angle + rotation.as_radians()).rem_euclid(TAU);
    let inverted = left_angle < right_angle;
    for extremum in [Vec2::X, Vec2::Y, Vec2::NEG_X, Vec2::NEG_Y] {
        let angle = extremum.to_angle().rem_euclid(TAU);
        // If inverted = true, then right_angle > left_angle, so we are looking for an angle that is not between them.
        // There's a chance that this condition fails due to rounding error, if the endpoint angle is juuuust shy of the axis.
        // But in that case, the endpoint itself is within rounding error of the axis and will define the bounds just fine.
        #[allow(clippy::nonminimal_bool)]
        if !inverted && angle >= right_angle && angle <= left_angle
            || inverted && (angle >= right_angle || angle <= left_angle)
        {
            bounds.push(extremum * arc.radius);
        }
    }
    bounds
}

impl Bounded2d for Arc2d {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        // If our arc covers more than a circle, just return the bounding box of the circle.
        if self.half_angle >= PI {
            return Circle::new(self.radius).aabb_2d(translation, rotation);
        }

        Aabb2d::from_point_cloud(translation, 0.0, &arc_bounding_points(*self, rotation))
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        // There are two possibilities for the bounding circle.
        if self.is_major() {
            // If the arc is major, then the widest distance between two points is a diameter of the arc's circle;
            // therefore, that circle is the bounding radius.
            BoundingCircle::new(translation, self.radius)
        } else {
            // Otherwise, the widest distance between two points is the chord,
            // so a circle of that diameter around the midpoint will contain the entire arc.
            let center = rotation.into() * self.chord_midpoint();
            BoundingCircle::new(center + translation, self.half_chord_length())
        }
    }
}

impl Bounded2d for CircularSector {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        // If our sector covers more than a circle, just return the bounding box of the circle.
        if self.half_angle() >= PI {
            return Circle::new(self.radius()).aabb_2d(translation, rotation);
        }

        // Otherwise, we use the same logic as for Arc2d, above, just with the circle's center as an additional possibility.
        let mut bounds = arc_bounding_points(self.arc, rotation);
        bounds.push(Vec2::ZERO);

        Aabb2d::from_point_cloud(translation, 0.0, &bounds)
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        if self.arc.is_major() {
            // If the arc is major, that is, greater than a semicircle,
            // then bounding circle is just the circle defining the sector.
            BoundingCircle::new(translation, self.arc.radius)
        } else {
            // However, when the arc is minor,
            // we need our bounding circle to include both endpoints of the arc as well as the circle center.
            // This means we need the circumcircle of those three points.
            // The circumcircle will always have a greater curvature than the circle itself, so it will contain
            // the entire circular sector.
            Triangle2d::new(
                Vec2::ZERO,
                self.arc.left_endpoint(),
                self.arc.right_endpoint(),
            )
            .bounding_circle(translation, rotation)
        }
    }
}

impl Bounded2d for CircularSegment {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        self.arc.aabb_2d(translation, rotation)
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        self.arc.bounding_circle(translation, rotation)
    }
}

impl Bounded2d for Ellipse {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation: Rot2 = rotation.into();

        //           V = (hh * cos(beta), hh * sin(beta))
        //      #####*#####
        //   ###     |     ###
        //  #     hh |        #
        // #         *---------* U = (hw * cos(alpha), hw * sin(alpha))
        //  #            hw   #
        //   ###           ###
        //      ###########

        let (hw, hh) = (self.half_size.x, self.half_size.y);

        // Sine and cosine of rotation angle alpha.
        let (alpha_sin, alpha_cos) = rotation.sin_cos();

        // Sine and cosine of alpha + pi/2. We can avoid the trigonometric functions:
        // sin(beta) = sin(alpha + pi/2) = cos(alpha)
        // cos(beta) = cos(alpha + pi/2) = -sin(alpha)
        let (beta_sin, beta_cos) = (alpha_cos, -alpha_sin);

        // Compute points U and V, the extremes of the ellipse
        let (ux, uy) = (hw * alpha_cos, hw * alpha_sin);
        let (vx, vy) = (hh * beta_cos, hh * beta_sin);

        let half_size = Vec2::new(ux.hypot(vx), uy.hypot(vy));

        Aabb2d::new(translation, half_size)
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, self.semi_major())
    }
}

impl Bounded2d for Rhombus {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation_mat = rotation.into();

        let [rotated_x_half_diagonal, rotated_y_half_diagonal] = [
            rotation_mat * Vec2::new(self.half_diagonals.x, 0.0),
            rotation_mat * Vec2::new(0.0, self.half_diagonals.y),
        ];
        let aabb_half_extent = rotated_x_half_diagonal
            .abs()
            .max(rotated_y_half_diagonal.abs());

        Aabb2d {
            min: -aabb_half_extent + translation,
            max: aabb_half_extent + translation,
        }
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, self.circumradius())
    }
}

impl Bounded2d for Plane2d {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation: Rot2 = rotation.into();
        let normal = rotation * *self.normal;
        let facing_x = normal == Vec2::X || normal == Vec2::NEG_X;
        let facing_y = normal == Vec2::Y || normal == Vec2::NEG_Y;

        // Dividing `f32::MAX` by 2.0 is helpful so that we can do operations
        // like growing or shrinking the AABB without breaking things.
        let half_width = if facing_x { 0.0 } else { f32::MAX / 2.0 };
        let half_height = if facing_y { 0.0 } else { f32::MAX / 2.0 };
        let half_size = Vec2::new(half_width, half_height);

        Aabb2d::new(translation, half_size)
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, f32::MAX / 2.0)
    }
}

impl Bounded2d for Line2d {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation: Rot2 = rotation.into();
        let direction = rotation * *self.direction;

        // Dividing `f32::MAX` by 2.0 is helpful so that we can do operations
        // like growing or shrinking the AABB without breaking things.
        let max = f32::MAX / 2.0;
        let half_width = if direction.x == 0.0 { 0.0 } else { max };
        let half_height = if direction.y == 0.0 { 0.0 } else { max };
        let half_size = Vec2::new(half_width, half_height);

        Aabb2d::new(translation, half_size)
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, f32::MAX / 2.0)
    }
}

impl Bounded2d for Segment2d {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        // Rotate the segment by `rotation`
        let rotation: Rot2 = rotation.into();
        let direction = rotation * *self.direction;
        let half_size = (self.half_length * direction).abs();

        Aabb2d::new(translation, half_size)
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, self.half_length)
    }
}

impl<const N: usize> Bounded2d for Polyline2d<N> {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        Aabb2d::from_point_cloud(translation, rotation, &self.vertices)
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::from_point_cloud(translation, rotation, &self.vertices)
    }
}

impl Bounded2d for BoxedPolyline2d {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        Aabb2d::from_point_cloud(translation, rotation, &self.vertices)
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::from_point_cloud(translation, rotation, &self.vertices)
    }
}

impl Bounded2d for Triangle2d {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation: Rot2 = rotation.into();
        let [a, b, c] = self.vertices.map(|vtx| rotation * vtx);

        let min = Vec2::new(a.x.min(b.x).min(c.x), a.y.min(b.y).min(c.y));
        let max = Vec2::new(a.x.max(b.x).max(c.x), a.y.max(b.y).max(c.y));

        Aabb2d {
            min: min + translation,
            max: max + translation,
        }
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        let rotation: Rot2 = rotation.into();
        let [a, b, c] = self.vertices;

        // The points of the segment opposite to the obtuse or right angle if one exists
        let side_opposite_to_non_acute = if (b - a).dot(c - a) <= 0.0 {
            Some((b, c))
        } else if (c - b).dot(a - b) <= 0.0 {
            Some((c, a))
        } else if (a - c).dot(b - c) <= 0.0 {
            Some((a, b))
        } else {
            // The triangle is acute.
            None
        };

        // Find the minimum bounding circle. If the triangle is obtuse, the circle passes through two vertices.
        // Otherwise, it's the circumcircle and passes through all three.
        if let Some((point1, point2)) = side_opposite_to_non_acute {
            // The triangle is obtuse or right, so the minimum bounding circle's diameter is equal to the longest side.
            // We can compute the minimum bounding circle from the line segment of the longest side.
            let (segment, center) = Segment2d::from_points(point1, point2);
            segment.bounding_circle(rotation * center + translation, rotation)
        } else {
            // The triangle is acute, so the smallest bounding circle is the circumcircle.
            let (Circle { radius }, circumcenter) = self.circumcircle();
            BoundingCircle::new(rotation * circumcenter + translation, radius)
        }
    }
}

impl Bounded2d for Rectangle {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation: Rot2 = rotation.into();

        // Compute the AABB of the rotated rectangle by transforming the half-extents
        // by an absolute rotation matrix.
        let (sin, cos) = rotation.sin_cos();
        let abs_rot_mat = Mat2::from_cols_array(&[cos.abs(), sin.abs(), sin.abs(), cos.abs()]);
        let half_size = abs_rot_mat * self.half_size;

        Aabb2d::new(translation, half_size)
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        let radius = self.half_size.length();
        BoundingCircle::new(translation, radius)
    }
}

impl<const N: usize> Bounded2d for Polygon<N> {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        Aabb2d::from_point_cloud(translation, rotation, &self.vertices)
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::from_point_cloud(translation, rotation, &self.vertices)
    }
}

impl Bounded2d for BoxedPolygon {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        Aabb2d::from_point_cloud(translation, rotation, &self.vertices)
    }

    fn bounding_circle(&self, translation: Vec2, rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::from_point_cloud(translation, rotation, &self.vertices)
    }
}

impl Bounded2d for RegularPolygon {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation: Rot2 = rotation.into();

        let mut min = Vec2::ZERO;
        let mut max = Vec2::ZERO;

        for vertex in self.vertices(rotation.as_radians()) {
            min = min.min(vertex);
            max = max.max(vertex);
        }

        Aabb2d {
            min: min + translation,
            max: max + translation,
        }
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, self.circumcircle.radius)
    }
}

impl Bounded2d for Capsule2d {
    fn aabb_2d(&self, translation: Vec2, rotation: impl Into<Rot2>) -> Aabb2d {
        let rotation: Rot2 = rotation.into();

        // Get the line segment between the hemicircles of the rotated capsule
        let segment = Segment2d {
            // Multiplying a normalized vector (Vec2::Y) with a rotation returns a normalized vector.
            direction: rotation * Dir2::Y,
            half_length: self.half_length,
        };
        let (a, b) = (segment.point1(), segment.point2());

        // Expand the line segment by the capsule radius to get the capsule half-extents
        let min = a.min(b) - Vec2::splat(self.radius);
        let max = a.max(b) + Vec2::splat(self.radius);

        Aabb2d {
            min: min + translation,
            max: max + translation,
        }
    }

    fn bounding_circle(&self, translation: Vec2, _rotation: impl Into<Rot2>) -> BoundingCircle {
        BoundingCircle::new(translation, self.radius + self.half_length)
    }
}

#[cfg(test)]
mod tests {
    use std::f32::consts::{FRAC_PI_2, FRAC_PI_3, FRAC_PI_4, FRAC_PI_6, TAU};

    use approx::assert_abs_diff_eq;
    use glam::Vec2;

    use crate::{
        bounding::Bounded2d,
        primitives::{
            Arc2d, Capsule2d, Circle, CircularSector, CircularSegment, Ellipse, Line2d, Plane2d,
            Polygon, Polyline2d, Rectangle, RegularPolygon, Rhombus, Segment2d, Triangle2d,
        },
        Dir2,
    };

    #[test]
    fn circle() {
        let circle = Circle { radius: 1.0 };
        let translation = Vec2::new(2.0, 1.0);

        let aabb = circle.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, Vec2::new(1.0, 0.0));
        assert_eq!(aabb.max, Vec2::new(3.0, 2.0));

        let bounding_circle = circle.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 1.0);
    }

    #[test]
    // Arcs and circular segments have the same bounding shapes so they share test cases.
    fn arc_and_segment() {
        struct TestCase {
            name: &'static str,
            arc: Arc2d,
            translation: Vec2,
            rotation: f32,
            aabb_min: Vec2,
            aabb_max: Vec2,
            bounding_circle_center: Vec2,
            bounding_circle_radius: f32,
        }

        // The apothem of an arc covering 1/6th of a circle.
        let apothem = f32::sqrt(3.0) / 2.0;
        let tests = [
            // Test case: a basic minor arc
            TestCase {
                name: "1/6th circle untransformed",
                arc: Arc2d::from_radians(1.0, FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-0.5, apothem),
                aabb_max: Vec2::new(0.5, 1.0),
                bounding_circle_center: Vec2::new(0.0, apothem),
                bounding_circle_radius: 0.5,
            },
            // Test case: a smaller arc, verifying that radius scaling works
            TestCase {
                name: "1/6th circle with radius 0.5",
                arc: Arc2d::from_radians(0.5, FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-0.25, apothem / 2.0),
                aabb_max: Vec2::new(0.25, 0.5),
                bounding_circle_center: Vec2::new(0.0, apothem / 2.0),
                bounding_circle_radius: 0.25,
            },
            // Test case: a larger arc, verifying that radius scaling works
            TestCase {
                name: "1/6th circle with radius 2.0",
                arc: Arc2d::from_radians(2.0, FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-1.0, 2.0 * apothem),
                aabb_max: Vec2::new(1.0, 2.0),
                bounding_circle_center: Vec2::new(0.0, 2.0 * apothem),
                bounding_circle_radius: 1.0,
            },
            // Test case: translation of a minor arc
            TestCase {
                name: "1/6th circle translated",
                arc: Arc2d::from_radians(1.0, FRAC_PI_3),
                translation: Vec2::new(2.0, 3.0),
                rotation: 0.0,
                aabb_min: Vec2::new(1.5, 3.0 + apothem),
                aabb_max: Vec2::new(2.5, 4.0),
                bounding_circle_center: Vec2::new(2.0, 3.0 + apothem),
                bounding_circle_radius: 0.5,
            },
            // Test case: rotation of a minor arc
            TestCase {
                name: "1/6th circle rotated",
                arc: Arc2d::from_radians(1.0, FRAC_PI_3),
                translation: Vec2::ZERO,
                // Rotate left by 1/12 of a circle, so the right endpoint is on the y-axis.
                rotation: FRAC_PI_6,
                aabb_min: Vec2::new(-apothem, 0.5),
                aabb_max: Vec2::new(0.0, 1.0),
                // The exact coordinates here are not obvious, but can be computed by constructing
                // an altitude from the midpoint of the chord to the y-axis and using the right triangle
                // similarity theorem.
                bounding_circle_center: Vec2::new(-apothem / 2.0, apothem.powi(2)),
                bounding_circle_radius: 0.5,
            },
            // Test case: handling of axis-aligned extrema
            TestCase {
                name: "1/4er circle rotated to be axis-aligned",
                arc: Arc2d::from_radians(1.0, FRAC_PI_2),
                translation: Vec2::ZERO,
                // Rotate right by 1/8 of a circle, so the right endpoint is on the x-axis and the left endpoint is on the y-axis.
                rotation: -FRAC_PI_4,
                aabb_min: Vec2::ZERO,
                aabb_max: Vec2::splat(1.0),
                bounding_circle_center: Vec2::splat(0.5),
                bounding_circle_radius: f32::sqrt(2.0) / 2.0,
            },
            // Test case: a basic major arc
            TestCase {
                name: "5/6th circle untransformed",
                arc: Arc2d::from_radians(1.0, 5.0 * FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-1.0, -apothem),
                aabb_max: Vec2::new(1.0, 1.0),
                bounding_circle_center: Vec2::ZERO,
                bounding_circle_radius: 1.0,
            },
            // Test case: a translated major arc
            TestCase {
                name: "5/6th circle translated",
                arc: Arc2d::from_radians(1.0, 5.0 * FRAC_PI_3),
                translation: Vec2::new(2.0, 3.0),
                rotation: 0.0,
                aabb_min: Vec2::new(1.0, 3.0 - apothem),
                aabb_max: Vec2::new(3.0, 4.0),
                bounding_circle_center: Vec2::new(2.0, 3.0),
                bounding_circle_radius: 1.0,
            },
            // Test case: a rotated major arc, with inverted left/right angles
            TestCase {
                name: "5/6th circle rotated",
                arc: Arc2d::from_radians(1.0, 5.0 * FRAC_PI_3),
                translation: Vec2::ZERO,
                // Rotate left by 1/12 of a circle, so the left endpoint is on the y-axis.
                rotation: FRAC_PI_6,
                aabb_min: Vec2::new(-1.0, -1.0),
                aabb_max: Vec2::new(1.0, 1.0),
                bounding_circle_center: Vec2::ZERO,
                bounding_circle_radius: 1.0,
            },
        ];

        for test in tests {
            println!("subtest case: {}", test.name);
            let segment: CircularSegment = test.arc.into();

            let arc_aabb = test.arc.aabb_2d(test.translation, test.rotation);
            assert_abs_diff_eq!(test.aabb_min, arc_aabb.min);
            assert_abs_diff_eq!(test.aabb_max, arc_aabb.max);
            let segment_aabb = segment.aabb_2d(test.translation, test.rotation);
            assert_abs_diff_eq!(test.aabb_min, segment_aabb.min);
            assert_abs_diff_eq!(test.aabb_max, segment_aabb.max);

            let arc_bounding_circle = test.arc.bounding_circle(test.translation, test.rotation);
            assert_abs_diff_eq!(test.bounding_circle_center, arc_bounding_circle.center);
            assert_abs_diff_eq!(test.bounding_circle_radius, arc_bounding_circle.radius());
            let segment_bounding_circle = segment.bounding_circle(test.translation, test.rotation);
            assert_abs_diff_eq!(test.bounding_circle_center, segment_bounding_circle.center);
            assert_abs_diff_eq!(
                test.bounding_circle_radius,
                segment_bounding_circle.radius()
            );
        }
    }

    #[test]
    fn circular_sector() {
        struct TestCase {
            name: &'static str,
            arc: Arc2d,
            translation: Vec2,
            rotation: f32,
            aabb_min: Vec2,
            aabb_max: Vec2,
            bounding_circle_center: Vec2,
            bounding_circle_radius: f32,
        }

        // The apothem of an arc covering 1/6th of a circle.
        let apothem = f32::sqrt(3.0) / 2.0;
        let inv_sqrt_3 = f32::sqrt(3.0).recip();
        let tests = [
            // Test case: An sector whose arc is minor, but whose bounding circle is not the circumcircle of the endpoints and center
            TestCase {
                name: "1/3rd circle",
                arc: Arc2d::from_radians(1.0, TAU / 3.0),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-apothem, 0.0),
                aabb_max: Vec2::new(apothem, 1.0),
                bounding_circle_center: Vec2::new(0.0, 0.5),
                bounding_circle_radius: apothem,
            },
            // The remaining test cases are selected as for arc_and_segment.
            TestCase {
                name: "1/6th circle untransformed",
                arc: Arc2d::from_radians(1.0, FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-0.5, 0.0),
                aabb_max: Vec2::new(0.5, 1.0),
                // The bounding circle is a circumcircle of an equilateral triangle with side length 1.
                // The distance from the corner to the center of such a triangle is 1/sqrt(3).
                bounding_circle_center: Vec2::new(0.0, inv_sqrt_3),
                bounding_circle_radius: inv_sqrt_3,
            },
            TestCase {
                name: "1/6th circle with radius 0.5",
                arc: Arc2d::from_radians(0.5, FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-0.25, 0.0),
                aabb_max: Vec2::new(0.25, 0.5),
                bounding_circle_center: Vec2::new(0.0, inv_sqrt_3 / 2.0),
                bounding_circle_radius: inv_sqrt_3 / 2.0,
            },
            TestCase {
                name: "1/6th circle with radius 2.0",
                arc: Arc2d::from_radians(2.0, FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-1.0, 0.0),
                aabb_max: Vec2::new(1.0, 2.0),
                bounding_circle_center: Vec2::new(0.0, 2.0 * inv_sqrt_3),
                bounding_circle_radius: 2.0 * inv_sqrt_3,
            },
            TestCase {
                name: "1/6th circle translated",
                arc: Arc2d::from_radians(1.0, FRAC_PI_3),
                translation: Vec2::new(2.0, 3.0),
                rotation: 0.0,
                aabb_min: Vec2::new(1.5, 3.0),
                aabb_max: Vec2::new(2.5, 4.0),
                bounding_circle_center: Vec2::new(2.0, 3.0 + inv_sqrt_3),
                bounding_circle_radius: inv_sqrt_3,
            },
            TestCase {
                name: "1/6th circle rotated",
                arc: Arc2d::from_radians(1.0, FRAC_PI_3),
                translation: Vec2::ZERO,
                // Rotate left by 1/12 of a circle, so the right endpoint is on the y-axis.
                rotation: FRAC_PI_6,
                aabb_min: Vec2::new(-apothem, 0.0),
                aabb_max: Vec2::new(0.0, 1.0),
                // The x-coordinate is now the inradius of the equilateral triangle, which is sqrt(3)/2.
                bounding_circle_center: Vec2::new(-inv_sqrt_3 / 2.0, 0.5),
                bounding_circle_radius: inv_sqrt_3,
            },
            TestCase {
                name: "1/4er circle rotated to be axis-aligned",
                arc: Arc2d::from_radians(1.0, FRAC_PI_2),
                translation: Vec2::ZERO,
                // Rotate right by 1/8 of a circle, so the right endpoint is on the x-axis and the left endpoint is on the y-axis.
                rotation: -FRAC_PI_4,
                aabb_min: Vec2::ZERO,
                aabb_max: Vec2::splat(1.0),
                bounding_circle_center: Vec2::splat(0.5),
                bounding_circle_radius: f32::sqrt(2.0) / 2.0,
            },
            TestCase {
                name: "5/6th circle untransformed",
                arc: Arc2d::from_radians(1.0, 5.0 * FRAC_PI_3),
                translation: Vec2::ZERO,
                rotation: 0.0,
                aabb_min: Vec2::new(-1.0, -apothem),
                aabb_max: Vec2::new(1.0, 1.0),
                bounding_circle_center: Vec2::ZERO,
                bounding_circle_radius: 1.0,
            },
            TestCase {
                name: "5/6th circle translated",
                arc: Arc2d::from_radians(1.0, 5.0 * FRAC_PI_3),
                translation: Vec2::new(2.0, 3.0),
                rotation: 0.0,
                aabb_min: Vec2::new(1.0, 3.0 - apothem),
                aabb_max: Vec2::new(3.0, 4.0),
                bounding_circle_center: Vec2::new(2.0, 3.0),
                bounding_circle_radius: 1.0,
            },
            TestCase {
                name: "5/6th circle rotated",
                arc: Arc2d::from_radians(1.0, 5.0 * FRAC_PI_3),
                translation: Vec2::ZERO,
                // Rotate left by 1/12 of a circle, so the left endpoint is on the y-axis.
                rotation: FRAC_PI_6,
                aabb_min: Vec2::new(-1.0, -1.0),
                aabb_max: Vec2::new(1.0, 1.0),
                bounding_circle_center: Vec2::ZERO,
                bounding_circle_radius: 1.0,
            },
        ];

        for test in tests {
            println!("subtest case: {}", test.name);
            let sector: CircularSector = test.arc.into();

            let aabb = sector.aabb_2d(test.translation, test.rotation);
            assert_abs_diff_eq!(test.aabb_min, aabb.min);
            assert_abs_diff_eq!(test.aabb_max, aabb.max);

            let bounding_circle = sector.bounding_circle(test.translation, test.rotation);
            assert_abs_diff_eq!(test.bounding_circle_center, bounding_circle.center);
            assert_abs_diff_eq!(test.bounding_circle_radius, bounding_circle.radius());
        }
    }

    #[test]
    fn ellipse() {
        let ellipse = Ellipse::new(1.0, 0.5);
        let translation = Vec2::new(2.0, 1.0);

        let aabb = ellipse.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, Vec2::new(1.0, 0.5));
        assert_eq!(aabb.max, Vec2::new(3.0, 1.5));

        let bounding_circle = ellipse.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 1.0);
    }

    #[test]
    fn rhombus() {
        let rhombus = Rhombus::new(2.0, 1.0);
        let translation = Vec2::new(2.0, 1.0);

        let aabb = rhombus.aabb_2d(translation, std::f32::consts::FRAC_PI_4);
        assert_eq!(aabb.min, Vec2::new(1.2928932, 0.29289323));
        assert_eq!(aabb.max, Vec2::new(2.7071068, 1.7071068));

        let bounding_circle = rhombus.bounding_circle(translation, std::f32::consts::FRAC_PI_4);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 1.0);

        let rhombus = Rhombus::new(0.0, 0.0);
        let translation = Vec2::new(0.0, 0.0);

        let aabb = rhombus.aabb_2d(translation, std::f32::consts::FRAC_PI_4);
        assert_eq!(aabb.min, Vec2::new(0.0, 0.0));
        assert_eq!(aabb.max, Vec2::new(0.0, 0.0));

        let bounding_circle = rhombus.bounding_circle(translation, std::f32::consts::FRAC_PI_4);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 0.0);
    }

    #[test]
    fn plane() {
        let translation = Vec2::new(2.0, 1.0);

        let aabb1 = Plane2d::new(Vec2::X).aabb_2d(translation, 0.0);
        assert_eq!(aabb1.min, Vec2::new(2.0, -f32::MAX / 2.0));
        assert_eq!(aabb1.max, Vec2::new(2.0, f32::MAX / 2.0));

        let aabb2 = Plane2d::new(Vec2::Y).aabb_2d(translation, 0.0);
        assert_eq!(aabb2.min, Vec2::new(-f32::MAX / 2.0, 1.0));
        assert_eq!(aabb2.max, Vec2::new(f32::MAX / 2.0, 1.0));

        let aabb3 = Plane2d::new(Vec2::ONE).aabb_2d(translation, 0.0);
        assert_eq!(aabb3.min, Vec2::new(-f32::MAX / 2.0, -f32::MAX / 2.0));
        assert_eq!(aabb3.max, Vec2::new(f32::MAX / 2.0, f32::MAX / 2.0));

        let bounding_circle = Plane2d::new(Vec2::Y).bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), f32::MAX / 2.0);
    }

    #[test]
    fn line() {
        let translation = Vec2::new(2.0, 1.0);

        let aabb1 = Line2d { direction: Dir2::Y }.aabb_2d(translation, 0.0);
        assert_eq!(aabb1.min, Vec2::new(2.0, -f32::MAX / 2.0));
        assert_eq!(aabb1.max, Vec2::new(2.0, f32::MAX / 2.0));

        let aabb2 = Line2d { direction: Dir2::X }.aabb_2d(translation, 0.0);
        assert_eq!(aabb2.min, Vec2::new(-f32::MAX / 2.0, 1.0));
        assert_eq!(aabb2.max, Vec2::new(f32::MAX / 2.0, 1.0));

        let aabb3 = Line2d {
            direction: Dir2::from_xy(1.0, 1.0).unwrap(),
        }
        .aabb_2d(translation, 0.0);
        assert_eq!(aabb3.min, Vec2::new(-f32::MAX / 2.0, -f32::MAX / 2.0));
        assert_eq!(aabb3.max, Vec2::new(f32::MAX / 2.0, f32::MAX / 2.0));

        let bounding_circle = Line2d { direction: Dir2::Y }.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), f32::MAX / 2.0);
    }

    #[test]
    fn segment() {
        let translation = Vec2::new(2.0, 1.0);
        let segment = Segment2d::from_points(Vec2::new(-1.0, -0.5), Vec2::new(1.0, 0.5)).0;

        let aabb = segment.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, Vec2::new(1.0, 0.5));
        assert_eq!(aabb.max, Vec2::new(3.0, 1.5));

        let bounding_circle = segment.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 1.0_f32.hypot(0.5));
    }

    #[test]
    fn polyline() {
        let polyline = Polyline2d::<4>::new([
            Vec2::ONE,
            Vec2::new(-1.0, 1.0),
            Vec2::NEG_ONE,
            Vec2::new(1.0, -1.0),
        ]);
        let translation = Vec2::new(2.0, 1.0);

        let aabb = polyline.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, Vec2::new(1.0, 0.0));
        assert_eq!(aabb.max, Vec2::new(3.0, 2.0));

        let bounding_circle = polyline.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), std::f32::consts::SQRT_2);
    }

    #[test]
    fn acute_triangle() {
        let acute_triangle =
            Triangle2d::new(Vec2::new(0.0, 1.0), Vec2::NEG_ONE, Vec2::new(1.0, -1.0));
        let translation = Vec2::new(2.0, 1.0);

        let aabb = acute_triangle.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, Vec2::new(1.0, 0.0));
        assert_eq!(aabb.max, Vec2::new(3.0, 2.0));

        // For acute triangles, the center is the circumcenter
        let (Circle { radius }, circumcenter) = acute_triangle.circumcircle();
        let bounding_circle = acute_triangle.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, circumcenter + translation);
        assert_eq!(bounding_circle.radius(), radius);
    }

    #[test]
    fn obtuse_triangle() {
        let obtuse_triangle = Triangle2d::new(
            Vec2::new(0.0, 1.0),
            Vec2::new(-10.0, -1.0),
            Vec2::new(10.0, -1.0),
        );
        let translation = Vec2::new(2.0, 1.0);

        let aabb = obtuse_triangle.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, Vec2::new(-8.0, 0.0));
        assert_eq!(aabb.max, Vec2::new(12.0, 2.0));

        // For obtuse and right triangles, the center is the midpoint of the longest side (diameter of bounding circle)
        let bounding_circle = obtuse_triangle.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation - Vec2::Y);
        assert_eq!(bounding_circle.radius(), 10.0);
    }

    #[test]
    fn rectangle() {
        let rectangle = Rectangle::new(2.0, 1.0);
        let translation = Vec2::new(2.0, 1.0);

        let aabb = rectangle.aabb_2d(translation, std::f32::consts::FRAC_PI_4);
        let expected_half_size = Vec2::splat(1.0606601);
        assert_eq!(aabb.min, translation - expected_half_size);
        assert_eq!(aabb.max, translation + expected_half_size);

        let bounding_circle = rectangle.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 1.0_f32.hypot(0.5));
    }

    #[test]
    fn polygon() {
        let polygon = Polygon::<4>::new([
            Vec2::ONE,
            Vec2::new(-1.0, 1.0),
            Vec2::NEG_ONE,
            Vec2::new(1.0, -1.0),
        ]);
        let translation = Vec2::new(2.0, 1.0);

        let aabb = polygon.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, Vec2::new(1.0, 0.0));
        assert_eq!(aabb.max, Vec2::new(3.0, 2.0));

        let bounding_circle = polygon.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), std::f32::consts::SQRT_2);
    }

    #[test]
    fn regular_polygon() {
        let regular_polygon = RegularPolygon::new(1.0, 5);
        let translation = Vec2::new(2.0, 1.0);

        let aabb = regular_polygon.aabb_2d(translation, 0.0);
        assert!((aabb.min - (translation - Vec2::new(0.9510565, 0.8090169))).length() < 1e-6);
        assert!((aabb.max - (translation + Vec2::new(0.9510565, 1.0))).length() < 1e-6);

        let bounding_circle = regular_polygon.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 1.0);
    }

    #[test]
    fn capsule() {
        let capsule = Capsule2d::new(0.5, 2.0);
        let translation = Vec2::new(2.0, 1.0);

        let aabb = capsule.aabb_2d(translation, 0.0);
        assert_eq!(aabb.min, translation - Vec2::new(0.5, 1.5));
        assert_eq!(aabb.max, translation + Vec2::new(0.5, 1.5));

        let bounding_circle = capsule.bounding_circle(translation, 0.0);
        assert_eq!(bounding_circle.center, translation);
        assert_eq!(bounding_circle.radius(), 1.5);
    }
}