Implement bounding volume types (bevyengine#10946)

# Objective

Implement bounding volume trait and the 4 types from
bevyengine#10570. I will add intersection
tests in a future PR.

## Solution

Implement mostly everything as written in the issue, except:
- Intersection is no longer a method on the bounding volumes, but a
separate trait.
- I implemented a `visible_area` since it's the most common usecase to
care about the surface that could collide with cast rays.
  - Maybe we want both?

---

## Changelog

- Added bounding volume types to bevy_math

---------

Co-authored-by: Alice Cecile <alice.i.cecile@gmail.com>

crates/bevy_math/src/bounding/bounded2d.rs

@@ -0,0 +1,353 @@
+use super::BoundingVolume;
+use crate::prelude::Vec2;
+
+/// A trait with methods that return 2D bounded volumes for a shape
+pub trait Bounded2d {
+    /// Get an axis-aligned bounding box for the shape with the given translation and rotation.
+    /// The rotation is in radians, counterclockwise, with 0 meaning no rotation.
+    fn aabb_2d(&self, translation: Vec2, rotation: f32) -> Aabb2d;
+    /// Get a bounding circle for the shape
+    /// The rotation is in radians, counterclockwise, with 0 meaning no rotation.
+    fn bounding_circle(&self, translation: Vec2, rotation: f32) -> BoundingCircle;
+}
+
+/// A 2D axis-aligned bounding box, or bounding rectangle
+#[doc(alias = "BoundingRectangle")]
+#[derive(Clone, Debug)]
+pub struct Aabb2d {
+    /// The minimum, conventionally bottom-left, point of the box
+    pub min: Vec2,
+    /// The maximum, conventionally top-right, point of the box
+    pub max: Vec2,
+}
+
+impl BoundingVolume for Aabb2d {
+    type Position = Vec2;
+    type HalfSize = Vec2;
+
+    #[inline(always)]
+    fn center(&self) -> Self::Position {
+        (self.min + self.max) / 2.
+    }
+
+    #[inline(always)]
+    fn half_size(&self) -> Self::HalfSize {
+        (self.max - self.min) / 2.
+    }
+
+    #[inline(always)]
+    fn visible_area(&self) -> f32 {
+        let b = self.max - self.min;
+        b.x * b.y
+    }
+
+    #[inline(always)]
+    fn contains(&self, other: &Self) -> bool {
+        other.min.x >= self.min.x
+            && other.min.y >= self.min.y
+            && other.max.x <= self.max.x
+            && other.max.y <= self.max.y
+    }
+
+    #[inline(always)]
+    fn merge(&self, other: &Self) -> Self {
+        Self {
+            min: self.min.min(other.min),
+            max: self.max.max(other.max),
+        }
+    }
+
+    #[inline(always)]
+    fn grow(&self, amount: Self::HalfSize) -> Self {
+        let b = Self {
+            min: self.min - amount,
+            max: self.max + amount,
+        };
+        debug_assert!(b.min.x <= b.max.x && b.min.y <= b.max.y);
+        b
+    }
+
+    #[inline(always)]
+    fn shrink(&self, amount: Self::HalfSize) -> Self {
+        let b = Self {
+            min: self.min + amount,
+            max: self.max - amount,
+        };
+        debug_assert!(b.min.x <= b.max.x && b.min.y <= b.max.y);
+        b
+    }
+}
+
+#[cfg(test)]
+mod aabb2d_tests {
+    use super::Aabb2d;
+    use crate::{bounding::BoundingVolume, Vec2};
+
+    #[test]
+    fn center() {
+        let aabb = Aabb2d {
+            min: Vec2::new(-0.5, -1.),
+            max: Vec2::new(1., 1.),
+        };
+        assert!((aabb.center() - Vec2::new(0.25, 0.)).length() < std::f32::EPSILON);
+        let aabb = Aabb2d {
+            min: Vec2::new(5., -10.),
+            max: Vec2::new(10., -5.),
+        };
+        assert!((aabb.center() - Vec2::new(7.5, -7.5)).length() < std::f32::EPSILON);
+    }
+
+    #[test]
+    fn half_size() {
+        let aabb = Aabb2d {
+            min: Vec2::new(-0.5, -1.),
+            max: Vec2::new(1., 1.),
+        };
+        let half_size = aabb.half_size();
+        assert!((half_size - Vec2::new(0.75, 1.)).length() < std::f32::EPSILON);
+    }
+
+    #[test]
+    fn area() {
+        let aabb = Aabb2d {
+            min: Vec2::new(-1., -1.),
+            max: Vec2::new(1., 1.),
+        };
+        assert!((aabb.visible_area() - 4.).abs() < std::f32::EPSILON);
+        let aabb = Aabb2d {
+            min: Vec2::new(0., 0.),
+            max: Vec2::new(1., 0.5),
+        };
+        assert!((aabb.visible_area() - 0.5).abs() < std::f32::EPSILON);
+    }
+
+    #[test]
+    fn contains() {
+        let a = Aabb2d {
+            min: Vec2::new(-1., -1.),
+            max: Vec2::new(1., 1.),
+        };
+        let b = Aabb2d {
+            min: Vec2::new(-2., -1.),
+            max: Vec2::new(1., 1.),
+        };
+        assert!(!a.contains(&b));
+        let b = Aabb2d {
+            min: Vec2::new(-0.25, -0.8),
+            max: Vec2::new(1., 1.),
+        };
+        assert!(a.contains(&b));
+    }
+
+    #[test]
+    fn merge() {
+        let a = Aabb2d {
+            min: Vec2::new(-1., -1.),
+            max: Vec2::new(1., 0.5),
+        };
+        let b = Aabb2d {
+            min: Vec2::new(-2., -0.5),
+            max: Vec2::new(0.75, 1.),
+        };
+        let merged = a.merge(&b);
+        assert!((merged.min - Vec2::new(-2., -1.)).length() < std::f32::EPSILON);
+        assert!((merged.max - Vec2::new(1., 1.)).length() < std::f32::EPSILON);
+        assert!(merged.contains(&a));
+        assert!(merged.contains(&b));
+        assert!(!a.contains(&merged));
+        assert!(!b.contains(&merged));
+    }
+
+    #[test]
+    fn grow() {
+        let a = Aabb2d {
+            min: Vec2::new(-1., -1.),
+            max: Vec2::new(1., 1.),
+        };
+        let padded = a.grow(Vec2::ONE);
+        assert!((padded.min - Vec2::new(-2., -2.)).length() < std::f32::EPSILON);
+        assert!((padded.max - Vec2::new(2., 2.)).length() < std::f32::EPSILON);
+        assert!(padded.contains(&a));
+        assert!(!a.contains(&padded));
+    }
+
+    #[test]
+    fn shrink() {
+        let a = Aabb2d {
+            min: Vec2::new(-2., -2.),
+            max: Vec2::new(2., 2.),
+        };
+        let shrunk = a.shrink(Vec2::ONE);
+        assert!((shrunk.min - Vec2::new(-1., -1.)).length() < std::f32::EPSILON);
+        assert!((shrunk.max - Vec2::new(1., 1.)).length() < std::f32::EPSILON);
+        assert!(a.contains(&shrunk));
+        assert!(!shrunk.contains(&a));
+    }
+}
+
+use crate::primitives::Circle;
+
+/// A bounding circle
+#[derive(Clone, Debug)]
+pub struct BoundingCircle {
+    /// The center of the bounding circle
+    pub center: Vec2,
+    /// The circle
+    pub circle: Circle,
+}
+
+impl BoundingCircle {
+    /// Construct a bounding circle from its center and radius
+    #[inline(always)]
+    pub fn new(center: Vec2, radius: f32) -> Self {
+        debug_assert!(radius >= 0.);
+        Self {
+            center,
+            circle: Circle { radius },
+        }
+    }
+
+    /// Get the radius of the bounding circle
+    #[inline(always)]
+    pub fn radius(&self) -> f32 {
+        self.circle.radius
+    }
+}
+
+impl BoundingVolume for BoundingCircle {
+    type Position = Vec2;
+    type HalfSize = f32;
+
+    #[inline(always)]
+    fn center(&self) -> Self::Position {
+        self.center
+    }
+
+    #[inline(always)]
+    fn half_size(&self) -> Self::HalfSize {
+        self.radius()
+    }
+
+    #[inline(always)]
+    fn visible_area(&self) -> f32 {
+        std::f32::consts::PI * self.radius() * self.radius()
+    }
+
+    #[inline(always)]
+    fn contains(&self, other: &Self) -> bool {
+        let diff = self.radius() - other.radius();
+        self.center.distance_squared(other.center) <= diff.powi(2).copysign(diff)
+    }
+
+    #[inline(always)]
+    fn merge(&self, other: &Self) -> Self {
+        let diff = other.center - self.center;
+        let length = diff.length();
+        if self.radius() >= length + other.radius() {
+            return self.clone();
+        }
+        if other.radius() >= length + self.radius() {
+            return other.clone();
+        }
+        let dir = diff / length;
+        Self::new(
+            (self.center + other.center) / 2. + dir * ((other.radius() - self.radius()) / 2.),
+            (length + self.radius() + other.radius()) / 2.,
+        )
+    }
+
+    #[inline(always)]
+    fn grow(&self, amount: Self::HalfSize) -> Self {
+        debug_assert!(amount >= 0.);
+        Self::new(self.center, self.radius() + amount)
+    }
+
+    #[inline(always)]
+    fn shrink(&self, amount: Self::HalfSize) -> Self {
+        debug_assert!(amount >= 0.);
+        debug_assert!(self.radius() >= amount);
+        Self::new(self.center, self.radius() - amount)
+    }
+}
+
+#[cfg(test)]
+mod bounding_circle_tests {
+    use super::BoundingCircle;
+    use crate::{bounding::BoundingVolume, Vec2};
+
+    #[test]
+    fn area() {
+        let circle = BoundingCircle::new(Vec2::ONE, 5.);
+        // Since this number is messy we check it with a higher threshold
+        assert!((circle.visible_area() - 78.5398).abs() < 0.001);
+    }
+
+    #[test]
+    fn contains() {
+        let a = BoundingCircle::new(Vec2::ONE, 5.);
+        let b = BoundingCircle::new(Vec2::new(5.5, 1.), 1.);
+        assert!(!a.contains(&b));
+        let b = BoundingCircle::new(Vec2::new(1., -3.5), 0.5);
+        assert!(a.contains(&b));
+    }
+
+    #[test]
+    fn contains_identical() {
+        let a = BoundingCircle::new(Vec2::ONE, 5.);
+        assert!(a.contains(&a));
+    }
+
+    #[test]
+    fn merge() {
+        // When merging two circles that don't contain each other, we find a center position that
+        // contains both
+        let a = BoundingCircle::new(Vec2::ONE, 5.);
+        let b = BoundingCircle::new(Vec2::new(1., -4.), 1.);
+        let merged = a.merge(&b);
+        assert!((merged.center - Vec2::new(1., 0.5)).length() < std::f32::EPSILON);
+        assert!((merged.radius() - 5.5).abs() < std::f32::EPSILON);
+        assert!(merged.contains(&a));
+        assert!(merged.contains(&b));
+        assert!(!a.contains(&merged));
+        assert!(!b.contains(&merged));
+
+        // When one circle contains the other circle, we use the bigger circle
+        let b = BoundingCircle::new(Vec2::ZERO, 3.);
+        assert!(a.contains(&b));
+        let merged = a.merge(&b);
+        assert_eq!(merged.center, a.center);
+        assert_eq!(merged.radius(), a.radius());
+
+        // When two circles are at the same point, we use the bigger radius
+        let b = BoundingCircle::new(Vec2::ONE, 6.);
+        let merged = a.merge(&b);
+        assert_eq!(merged.center, a.center);
+        assert_eq!(merged.radius(), b.radius());
+    }
+
+    #[test]
+    fn merge_identical() {
+        let a = BoundingCircle::new(Vec2::ONE, 5.);
+        let merged = a.merge(&a);
+        assert_eq!(merged.center, a.center);
+        assert_eq!(merged.radius(), a.radius());
+    }
+
+    #[test]
+    fn grow() {
+        let a = BoundingCircle::new(Vec2::ONE, 5.);
+        let padded = a.grow(1.25);
+        assert!((padded.radius() - 6.25).abs() < std::f32::EPSILON);
+        assert!(padded.contains(&a));
+        assert!(!a.contains(&padded));
+    }
+
+    #[test]
+    fn shrink() {
+        let a = BoundingCircle::new(Vec2::ONE, 5.);
+        let shrunk = a.shrink(0.5);
+        assert!((shrunk.radius() - 4.5).abs() < std::f32::EPSILON);
+        assert!(a.contains(&shrunk));
+        assert!(!shrunk.contains(&a));
+    }
+}