bevy_math/primitives/

dim3.rs

use core::f32::consts::{FRAC_PI_3, PI};

use super::{Circle, Measured2d, Measured3d, Primitive2d, Primitive3d};
use crate::{ops, ops::FloatPow, Dir3, InvalidDirectionError, Isometry3d, Mat3, Vec2, Vec3};

#[cfg(feature = "bevy_reflect")]
use bevy_reflect::{std_traits::ReflectDefault, Reflect};
#[cfg(all(feature = "serialize", feature = "bevy_reflect"))]
use bevy_reflect::{ReflectDeserialize, ReflectSerialize};
use glam::Quat;

/// A sphere primitive, representing the set of all points some distance from the origin
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Sphere {
    /// The radius of the sphere
    pub radius: f32,
}
impl Primitive3d for Sphere {}

impl Default for Sphere {
    /// Returns the default [`Sphere`] with a radius of `0.5`.
    fn default() -> Self {
        Self { radius: 0.5 }
    }
}

impl Sphere {
    /// Create a new [`Sphere`] from a `radius`
    #[inline(always)]
    pub const fn new(radius: f32) -> Self {
        Self { radius }
    }

    /// Get the diameter of the sphere
    #[inline(always)]
    pub fn diameter(&self) -> f32 {
        2.0 * self.radius
    }

    /// Finds the point on the sphere that is closest to the given `point`.
    ///
    /// If the point is outside the sphere, the returned point will be on the surface of the sphere.
    /// Otherwise, it will be inside the sphere and returned as is.
    #[inline(always)]
    pub fn closest_point(&self, point: Vec3) -> Vec3 {
        let distance_squared = point.length_squared();

        if distance_squared <= self.radius.squared() {
            // The point is inside the sphere.
            point
        } else {
            // The point is outside the sphere.
            // Find the closest point on the surface of the sphere.
            let dir_to_point = point / distance_squared.sqrt();
            self.radius * dir_to_point
        }
    }
}

impl Measured3d for Sphere {
    /// Get the surface area of the sphere
    #[inline(always)]
    fn area(&self) -> f32 {
        4.0 * PI * self.radius.squared()
    }

    /// Get the volume of the sphere
    #[inline(always)]
    fn volume(&self) -> f32 {
        4.0 * FRAC_PI_3 * self.radius.cubed()
    }
}

/// A bounded plane in 3D space. It forms a surface starting from the origin with a defined height and width.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Plane3d {
    /// The normal of the plane. The plane will be placed perpendicular to this direction
    pub normal: Dir3,
    /// Half of the width and height of the plane
    pub half_size: Vec2,
}
impl Primitive3d for Plane3d {}

impl Default for Plane3d {
    /// Returns the default [`Plane3d`] with a normal pointing in the `+Y` direction, width and height of `1.0`.
    fn default() -> Self {
        Self {
            normal: Dir3::Y,
            half_size: Vec2::splat(0.5),
        }
    }
}

impl Plane3d {
    /// Create a new `Plane3d` from a normal and a half size
    ///
    /// # Panics
    ///
    /// Panics if the given `normal` is zero (or very close to zero), or non-finite.
    #[inline(always)]
    pub fn new(normal: Vec3, half_size: Vec2) -> Self {
        Self {
            normal: Dir3::new(normal).expect("normal must be nonzero and finite"),
            half_size,
        }
    }

    /// Create a new `Plane3d` based on three points and compute the geometric center
    /// of those points.
    ///
    /// The direction of the plane normal is determined by the winding order
    /// of the triangular shape formed by the points.
    ///
    /// # Panics
    ///
    /// Panics if a valid normal can not be computed, for example when the points
    /// are *collinear* and lie on the same line.
    #[inline(always)]
    pub fn from_points(a: Vec3, b: Vec3, c: Vec3) -> (Self, Vec3) {
        let normal = Dir3::new((b - a).cross(c - a)).expect(
            "finite plane must be defined by three finite points that don't lie on the same line",
        );
        let translation = (a + b + c) / 3.0;

        (
            Self {
                normal,
                ..Default::default()
            },
            translation,
        )
    }
}

/// An unbounded plane in 3D space. It forms a separating surface through the origin,
/// stretching infinitely far
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct InfinitePlane3d {
    /// The normal of the plane. The plane will be placed perpendicular to this direction
    pub normal: Dir3,
}
impl Primitive3d for InfinitePlane3d {}

impl Default for InfinitePlane3d {
    /// Returns the default [`InfinitePlane3d`] with a normal pointing in the `+Y` direction.
    fn default() -> Self {
        Self { normal: Dir3::Y }
    }
}

impl InfinitePlane3d {
    /// Create a new `InfinitePlane3d` from a normal
    ///
    /// # Panics
    ///
    /// Panics if the given `normal` is zero (or very close to zero), or non-finite.
    #[inline(always)]
    pub fn new<T: TryInto<Dir3>>(normal: T) -> Self
    where
        <T as TryInto<Dir3>>::Error: core::fmt::Debug,
    {
        Self {
            normal: normal
                .try_into()
                .expect("normal must be nonzero and finite"),
        }
    }

    /// Create a new `InfinitePlane3d` based on three points and compute the geometric center
    /// of those points.
    ///
    /// The direction of the plane normal is determined by the winding order
    /// of the triangular shape formed by the points.
    ///
    /// # Panics
    ///
    /// Panics if a valid normal can not be computed, for example when the points
    /// are *collinear* and lie on the same line.
    #[inline(always)]
    pub fn from_points(a: Vec3, b: Vec3, c: Vec3) -> (Self, Vec3) {
        let normal = Dir3::new((b - a).cross(c - a)).expect(
            "infinite plane must be defined by three finite points that don't lie on the same line",
        );
        let translation = (a + b + c) / 3.0;

        (Self { normal }, translation)
    }

    /// Computes the shortest distance between a plane transformed with the given `isometry` and a
    /// `point`. The result is a signed value; it's positive if the point lies in the half-space
    /// that the plane's normal vector points towards.
    #[inline]
    pub fn signed_distance(&self, isometry: impl Into<Isometry3d>, point: Vec3) -> f32 {
        let isometry = isometry.into();
        self.normal.dot(isometry.inverse() * point)
    }

    /// Injects the `point` into this plane transformed with the given `isometry`.
    ///
    /// This projects the point orthogonally along the shortest path onto the plane.
    #[inline]
    pub fn project_point(&self, isometry: impl Into<Isometry3d>, point: Vec3) -> Vec3 {
        point - self.normal * self.signed_distance(isometry, point)
    }

    /// Computes an [`Isometry3d`] which transforms points from the plane in 3D space with the given
    /// `origin` to the XY-plane.
    ///
    /// ## Guarantees
    ///
    /// * the transformation is a [congruence] meaning it will preserve all distances and angles of
    ///   the transformed geometry
    /// * uses the least rotation possible to transform the geometry
    /// * if two geometries are transformed with the same isometry, then the relations between
    ///   them, like distances, are also preserved
    /// * compared to projections, the transformation is lossless (up to floating point errors)
    ///   reversible
    ///
    /// ## Non-Guarantees
    ///
    /// * the rotation used is generally not unique
    /// * the orientation of the transformed geometry in the XY plane might be arbitrary, to
    ///   enforce some kind of alignment the user has to use an extra transformation ontop of this
    ///   one
    ///
    /// See [`isometries_xy`] for example usescases.
    ///
    /// [congruence]: https://en.wikipedia.org/wiki/Congruence_(geometry)
    /// [`isometries_xy`]: `InfinitePlane3d::isometries_xy`
    #[inline]
    pub fn isometry_into_xy(&self, origin: Vec3) -> Isometry3d {
        let rotation = Quat::from_rotation_arc(self.normal.as_vec3(), Vec3::Z);
        let transformed_origin = rotation * origin;
        Isometry3d::new(-Vec3::Z * transformed_origin.z, rotation)
    }

    /// Computes an [`Isometry3d`] which transforms points from the XY-plane to this plane with the
    /// given `origin`.
    ///
    /// ## Guarantees
    ///
    /// * the transformation is a [congruence] meaning it will preserve all distances and angles of
    ///   the transformed geometry
    /// * uses the least rotation possible to transform the geometry
    /// * if two geometries are transformed with the same isometry, then the relations between
    ///   them, like distances, are also preserved
    /// * compared to projections, the transformation is lossless (up to floating point errors)
    ///   reversible
    ///
    /// ## Non-Guarantees
    ///
    /// * the rotation used is generally not unique
    /// * the orientation of the transformed geometry in the XY plane might be arbitrary, to
    ///   enforce some kind of alignment the user has to use an extra transformation ontop of this
    ///   one
    ///
    /// See [`isometries_xy`] for example usescases.
    ///
    /// [congruence]: https://en.wikipedia.org/wiki/Congruence_(geometry)
    /// [`isometries_xy`]: `InfinitePlane3d::isometries_xy`
    #[inline]
    pub fn isometry_from_xy(&self, origin: Vec3) -> Isometry3d {
        self.isometry_into_xy(origin).inverse()
    }

    /// Computes both [isometries] which transforms points from the plane in 3D space with the
    /// given `origin` to the XY-plane and back.
    ///
    /// [isometries]: `Isometry3d`
    ///
    /// # Example
    ///
    /// The projection and its inverse can be used to run 2D algorithms on flat shapes in 3D. The
    /// workflow would usually look like this:
    ///
    /// ```
    /// # use bevy_math::{Vec3, Dir3};
    /// # use bevy_math::primitives::InfinitePlane3d;
    ///
    /// let triangle_3d @ [a, b, c] = [Vec3::X, Vec3::Y, Vec3::Z];
    /// let center = (a + b + c) / 3.0;
    ///
    /// let plane = InfinitePlane3d::new(Vec3::ONE);
    ///
    /// let (to_xy, from_xy) = plane.isometries_xy(center);
    ///
    /// let triangle_2d = triangle_3d.map(|vec3| to_xy * vec3).map(|vec3| vec3.truncate());
    ///
    /// // apply some algorithm to `triangle_2d`
    ///
    /// let triangle_3d = triangle_2d.map(|vec2| vec2.extend(0.0)).map(|vec3| from_xy * vec3);
    /// ```
    #[inline]
    pub fn isometries_xy(&self, origin: Vec3) -> (Isometry3d, Isometry3d) {
        let projection = self.isometry_into_xy(origin);
        (projection, projection.inverse())
    }
}

/// An infinite line going through the origin along a direction in 3D space.
///
/// For a finite line: [`Segment3d`]
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug, PartialEq))]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Line3d {
    /// The direction of the line
    pub direction: Dir3,
}
impl Primitive3d for Line3d {}

/// A segment of a line going through the origin along a direction in 3D space.
#[doc(alias = "LineSegment3d")]
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug, PartialEq))]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Segment3d {
    /// The direction of the line
    pub direction: Dir3,
    /// Half the length of the line segment. The segment extends by this amount in both
    /// the given direction and its opposite direction
    pub half_length: f32,
}
impl Primitive3d for Segment3d {}

impl Segment3d {
    /// Create a new `Segment3d` from a direction and full length of the segment
    #[inline(always)]
    pub fn new(direction: Dir3, length: f32) -> Self {
        Self {
            direction,
            half_length: length / 2.0,
        }
    }

    /// Create a new `Segment3d` from its endpoints and compute its geometric center
    ///
    /// # Panics
    ///
    /// Panics if `point1 == point2`
    #[inline(always)]
    pub fn from_points(point1: Vec3, point2: Vec3) -> (Self, Vec3) {
        let diff = point2 - point1;
        let length = diff.length();

        (
            // We are dividing by the length here, so the vector is normalized.
            Self::new(Dir3::new_unchecked(diff / length), length),
            (point1 + point2) / 2.,
        )
    }

    /// Get the position of the first point on the line segment
    #[inline(always)]
    pub fn point1(&self) -> Vec3 {
        *self.direction * -self.half_length
    }

    /// Get the position of the second point on the line segment
    #[inline(always)]
    pub fn point2(&self) -> Vec3 {
        *self.direction * self.half_length
    }
}

/// A series of connected line segments in 3D space.
///
/// For a version without generics: [`BoxedPolyline3d`]
#[derive(Clone, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "bevy_reflect", derive(Reflect), reflect(Debug, PartialEq))]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Polyline3d<const N: usize> {
    /// The vertices of the polyline
    #[cfg_attr(feature = "serialize", serde(with = "super::serde::array"))]
    pub vertices: [Vec3; N],
}
impl<const N: usize> Primitive3d for Polyline3d<N> {}

impl<const N: usize> FromIterator<Vec3> for Polyline3d<N> {
    fn from_iter<I: IntoIterator<Item = Vec3>>(iter: I) -> Self {
        let mut vertices: [Vec3; N] = [Vec3::ZERO; N];

        for (index, i) in iter.into_iter().take(N).enumerate() {
            vertices[index] = i;
        }
        Self { vertices }
    }
}

impl<const N: usize> Polyline3d<N> {
    /// Create a new `Polyline3d` from its vertices
    pub fn new(vertices: impl IntoIterator<Item = Vec3>) -> Self {
        Self::from_iter(vertices)
    }
}

/// A series of connected line segments in 3D space, allocated on the heap
/// in a `Box<[Vec3]>`.
///
/// For a version without alloc: [`Polyline3d`]
#[derive(Clone, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
pub struct BoxedPolyline3d {
    /// The vertices of the polyline
    pub vertices: Box<[Vec3]>,
}
impl Primitive3d for BoxedPolyline3d {}

impl FromIterator<Vec3> for BoxedPolyline3d {
    fn from_iter<I: IntoIterator<Item = Vec3>>(iter: I) -> Self {
        let vertices: Vec<Vec3> = iter.into_iter().collect();
        Self {
            vertices: vertices.into_boxed_slice(),
        }
    }
}

impl BoxedPolyline3d {
    /// Create a new `BoxedPolyline3d` from its vertices
    pub fn new(vertices: impl IntoIterator<Item = Vec3>) -> Self {
        Self::from_iter(vertices)
    }
}

/// A cuboid primitive, which is like a cube, except that the x, y, and z dimensions are not
/// required to be the same.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Cuboid {
    /// Half of the width, height and depth of the cuboid
    pub half_size: Vec3,
}
impl Primitive3d for Cuboid {}

impl Default for Cuboid {
    /// Returns the default [`Cuboid`] with a width, height, and depth of `1.0`.
    fn default() -> Self {
        Self {
            half_size: Vec3::splat(0.5),
        }
    }
}

impl Cuboid {
    /// Create a new `Cuboid` from a full x, y, and z length
    #[inline(always)]
    pub fn new(x_length: f32, y_length: f32, z_length: f32) -> Self {
        Self::from_size(Vec3::new(x_length, y_length, z_length))
    }

    /// Create a new `Cuboid` from a given full size
    #[inline(always)]
    pub fn from_size(size: Vec3) -> Self {
        Self {
            half_size: size / 2.0,
        }
    }

    /// Create a new `Cuboid` from two corner points
    #[inline(always)]
    pub fn from_corners(point1: Vec3, point2: Vec3) -> Self {
        Self {
            half_size: (point2 - point1).abs() / 2.0,
        }
    }

    /// Create a `Cuboid` from a single length.
    /// The resulting `Cuboid` will be the same size in every direction.
    #[inline(always)]
    pub fn from_length(length: f32) -> Self {
        Self {
            half_size: Vec3::splat(length / 2.0),
        }
    }

    /// Get the size of the cuboid
    #[inline(always)]
    pub fn size(&self) -> Vec3 {
        2.0 * self.half_size
    }

    /// Finds the point on the cuboid that is closest to the given `point`.
    ///
    /// If the point is outside the cuboid, the returned point will be on the surface of the cuboid.
    /// Otherwise, it will be inside the cuboid and returned as is.
    #[inline(always)]
    pub fn closest_point(&self, point: Vec3) -> Vec3 {
        // Clamp point coordinates to the cuboid
        point.clamp(-self.half_size, self.half_size)
    }
}

impl Measured3d for Cuboid {
    /// Get the surface area of the cuboid
    #[inline(always)]
    fn area(&self) -> f32 {
        8.0 * (self.half_size.x * self.half_size.y
            + self.half_size.y * self.half_size.z
            + self.half_size.x * self.half_size.z)
    }

    /// Get the volume of the cuboid
    #[inline(always)]
    fn volume(&self) -> f32 {
        8.0 * self.half_size.x * self.half_size.y * self.half_size.z
    }
}

/// A cylinder primitive centered on the origin
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Cylinder {
    /// The radius of the cylinder
    pub radius: f32,
    /// The half height of the cylinder
    pub half_height: f32,
}
impl Primitive3d for Cylinder {}

impl Default for Cylinder {
    /// Returns the default [`Cylinder`] with a radius of `0.5` and a height of `1.0`.
    fn default() -> Self {
        Self {
            radius: 0.5,
            half_height: 0.5,
        }
    }
}

impl Cylinder {
    /// Create a new `Cylinder` from a radius and full height
    #[inline(always)]
    pub fn new(radius: f32, height: f32) -> Self {
        Self {
            radius,
            half_height: height / 2.0,
        }
    }

    /// Get the base of the cylinder as a [`Circle`]
    #[inline(always)]
    pub fn base(&self) -> Circle {
        Circle {
            radius: self.radius,
        }
    }

    /// Get the surface area of the side of the cylinder,
    /// also known as the lateral area
    #[inline(always)]
    #[doc(alias = "side_area")]
    pub fn lateral_area(&self) -> f32 {
        4.0 * PI * self.radius * self.half_height
    }

    /// Get the surface area of one base of the cylinder
    #[inline(always)]
    pub fn base_area(&self) -> f32 {
        PI * self.radius.squared()
    }
}

impl Measured3d for Cylinder {
    /// Get the total surface area of the cylinder
    #[inline(always)]
    fn area(&self) -> f32 {
        2.0 * PI * self.radius * (self.radius + 2.0 * self.half_height)
    }

    /// Get the volume of the cylinder
    #[inline(always)]
    fn volume(&self) -> f32 {
        self.base_area() * 2.0 * self.half_height
    }
}

/// A 3D capsule primitive centered on the origin
/// A three-dimensional capsule is defined as a surface at a distance (radius) from a line
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Capsule3d {
    /// The radius of the capsule
    pub radius: f32,
    /// Half the height of the capsule, excluding the hemispheres
    pub half_length: f32,
}
impl Primitive3d for Capsule3d {}

impl Default for Capsule3d {
    /// Returns the default [`Capsule3d`] with a radius of `0.5` and a segment length of `1.0`.
    /// The total height is `2.0`.
    fn default() -> Self {
        Self {
            radius: 0.5,
            half_length: 0.5,
        }
    }
}

impl Capsule3d {
    /// Create a new `Capsule3d` from a radius and length
    pub fn new(radius: f32, length: f32) -> Self {
        Self {
            radius,
            half_length: length / 2.0,
        }
    }

    /// Get the part connecting the hemispherical ends
    /// of the capsule as a [`Cylinder`]
    #[inline(always)]
    pub fn to_cylinder(&self) -> Cylinder {
        Cylinder {
            radius: self.radius,
            half_height: self.half_length,
        }
    }
}

impl Measured3d for Capsule3d {
    /// Get the surface area of the capsule
    #[inline(always)]
    fn area(&self) -> f32 {
        // Modified version of 2pi * r * (2r + h)
        4.0 * PI * self.radius * (self.radius + self.half_length)
    }

    /// Get the volume of the capsule
    #[inline(always)]
    fn volume(&self) -> f32 {
        // Modified version of pi * r^2 * (4/3 * r + a)
        let diameter = self.radius * 2.0;
        PI * self.radius * diameter * (diameter / 3.0 + self.half_length)
    }
}

/// A cone primitive centered on the midpoint between the tip of the cone and the center of its base.
///
/// The cone is oriented with its tip pointing towards the Y axis.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Cone {
    /// The radius of the base
    pub radius: f32,
    /// The height of the cone
    pub height: f32,
}
impl Primitive3d for Cone {}

impl Default for Cone {
    /// Returns the default [`Cone`] with a base radius of `0.5` and a height of `1.0`.
    fn default() -> Self {
        Self {
            radius: 0.5,
            height: 1.0,
        }
    }
}

impl Cone {
    /// Create a new [`Cone`] from a radius and height.
    pub fn new(radius: f32, height: f32) -> Self {
        Self { radius, height }
    }
    /// Get the base of the cone as a [`Circle`]
    #[inline(always)]
    pub fn base(&self) -> Circle {
        Circle {
            radius: self.radius,
        }
    }

    /// Get the slant height of the cone, the length of the line segment
    /// connecting a point on the base to the apex
    #[inline(always)]
    #[doc(alias = "side_length")]
    pub fn slant_height(&self) -> f32 {
        ops::hypot(self.radius, self.height)
    }

    /// Get the surface area of the side of the cone,
    /// also known as the lateral area
    #[inline(always)]
    #[doc(alias = "side_area")]
    pub fn lateral_area(&self) -> f32 {
        PI * self.radius * self.slant_height()
    }

    /// Get the surface area of the base of the cone
    #[inline(always)]
    pub fn base_area(&self) -> f32 {
        PI * self.radius.squared()
    }
}

impl Measured3d for Cone {
    /// Get the total surface area of the cone
    #[inline(always)]
    fn area(&self) -> f32 {
        self.base_area() + self.lateral_area()
    }

    /// Get the volume of the cone
    #[inline(always)]
    fn volume(&self) -> f32 {
        (self.base_area() * self.height) / 3.0
    }
}

/// A conical frustum primitive.
/// A conical frustum can be created
/// by slicing off a section of a cone.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct ConicalFrustum {
    /// The radius of the top of the frustum
    pub radius_top: f32,
    /// The radius of the base of the frustum
    pub radius_bottom: f32,
    /// The height of the frustum
    pub height: f32,
}
impl Primitive3d for ConicalFrustum {}

impl Default for ConicalFrustum {
    /// Returns the default [`ConicalFrustum`] with a top radius of `0.25`, bottom radius of `0.5`, and a height of `0.5`.
    fn default() -> Self {
        Self {
            radius_top: 0.25,
            radius_bottom: 0.5,
            height: 0.5,
        }
    }
}

/// The type of torus determined by the minor and major radii
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum TorusKind {
    /// A torus that has a ring.
    /// The major radius is greater than the minor radius
    Ring,
    /// A torus that has no hole but also doesn't intersect itself.
    /// The major radius is equal to the minor radius
    Horn,
    /// A self-intersecting torus.
    /// The major radius is less than the minor radius
    Spindle,
    /// A torus with non-geometric properties like
    /// a minor or major radius that is non-positive,
    /// infinite, or `NaN`
    Invalid,
}

/// A torus primitive, often representing a ring or donut shape
/// The set of points some distance from a circle centered at the origin
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Torus {
    /// The radius of the tube of the torus
    #[doc(
        alias = "ring_radius",
        alias = "tube_radius",
        alias = "cross_section_radius"
    )]
    pub minor_radius: f32,
    /// The distance from the center of the torus to the center of the tube
    #[doc(alias = "radius_of_revolution")]
    pub major_radius: f32,
}
impl Primitive3d for Torus {}

impl Default for Torus {
    /// Returns the default [`Torus`] with a minor radius of `0.25` and a major radius of `0.75`.
    fn default() -> Self {
        Self {
            minor_radius: 0.25,
            major_radius: 0.75,
        }
    }
}

impl Torus {
    /// Create a new `Torus` from an inner and outer radius.
    ///
    /// The inner radius is the radius of the hole, and the outer radius
    /// is the radius of the entire object
    #[inline(always)]
    pub fn new(inner_radius: f32, outer_radius: f32) -> Self {
        let minor_radius = (outer_radius - inner_radius) / 2.0;
        let major_radius = outer_radius - minor_radius;

        Self {
            minor_radius,
            major_radius,
        }
    }

    /// Get the inner radius of the torus.
    /// For a ring torus, this corresponds to the radius of the hole,
    /// or `major_radius - minor_radius`
    #[inline(always)]
    pub fn inner_radius(&self) -> f32 {
        self.major_radius - self.minor_radius
    }

    /// Get the outer radius of the torus.
    /// This corresponds to the overall radius of the entire object,
    /// or `major_radius + minor_radius`
    #[inline(always)]
    pub fn outer_radius(&self) -> f32 {
        self.major_radius + self.minor_radius
    }

    /// Get the [`TorusKind`] determined by the minor and major radii.
    ///
    /// The torus can either be a *ring torus* that has a hole,
    /// a *horn torus* that doesn't have a hole but also isn't self-intersecting,
    /// or a *spindle torus* that is self-intersecting.
    ///
    /// If the minor or major radius is non-positive, infinite, or `NaN`,
    /// [`TorusKind::Invalid`] is returned
    #[inline(always)]
    pub fn kind(&self) -> TorusKind {
        // Invalid if minor or major radius is non-positive, infinite, or NaN
        if self.minor_radius <= 0.0
            || !self.minor_radius.is_finite()
            || self.major_radius <= 0.0
            || !self.major_radius.is_finite()
        {
            return TorusKind::Invalid;
        }

        match self.major_radius.partial_cmp(&self.minor_radius).unwrap() {
            core::cmp::Ordering::Greater => TorusKind::Ring,
            core::cmp::Ordering::Equal => TorusKind::Horn,
            core::cmp::Ordering::Less => TorusKind::Spindle,
        }
    }
}

impl Measured3d for Torus {
    /// Get the surface area of the torus. Note that this only produces
    /// the expected result when the torus has a ring and isn't self-intersecting
    #[inline(always)]
    fn area(&self) -> f32 {
        4.0 * PI.squared() * self.major_radius * self.minor_radius
    }

    /// Get the volume of the torus. Note that this only produces
    /// the expected result when the torus has a ring and isn't self-intersecting
    #[inline(always)]
    fn volume(&self) -> f32 {
        2.0 * PI.squared() * self.major_radius * self.minor_radius.squared()
    }
}

/// A 3D triangle primitive.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Triangle3d {
    /// The vertices of the triangle.
    pub vertices: [Vec3; 3],
}

impl Primitive3d for Triangle3d {}

impl Default for Triangle3d {
    /// Returns the default [`Triangle3d`] with the vertices `[0.0, 0.5, 0.0]`, `[-0.5, -0.5, 0.0]`, and `[0.5, -0.5, 0.0]`.
    fn default() -> Self {
        Self {
            vertices: [
                Vec3::new(0.0, 0.5, 0.0),
                Vec3::new(-0.5, -0.5, 0.0),
                Vec3::new(0.5, -0.5, 0.0),
            ],
        }
    }
}

impl Triangle3d {
    /// Create a new [`Triangle3d`] from points `a`, `b`, and `c`.
    #[inline(always)]
    pub fn new(a: Vec3, b: Vec3, c: Vec3) -> Self {
        Self {
            vertices: [a, b, c],
        }
    }

    /// Get the normal of the triangle in the direction of the right-hand rule, assuming
    /// the vertices are ordered in a counter-clockwise direction.
    ///
    /// The normal is computed as the cross product of the vectors `ab` and `ac`.
    ///
    /// # Errors
    ///
    /// Returns [`Err(InvalidDirectionError)`](InvalidDirectionError) if the length
    /// of the given vector is zero (or very close to zero), infinite, or `NaN`.
    #[inline(always)]
    pub fn normal(&self) -> Result<Dir3, InvalidDirectionError> {
        let [a, b, c] = self.vertices;
        let ab = b - a;
        let ac = c - a;
        Dir3::new(ab.cross(ac))
    }

    /// Checks if the triangle is degenerate, meaning it has zero area.
    ///
    /// A triangle is degenerate if the cross product of the vectors `ab` and `ac` has a length less than `10e-7`.
    /// This indicates that the three vertices are collinear or nearly collinear.
    #[inline(always)]
    pub fn is_degenerate(&self) -> bool {
        let [a, b, c] = self.vertices;
        let ab = b - a;
        let ac = c - a;
        ab.cross(ac).length() < 10e-7
    }

    /// Checks if the triangle is acute, meaning all angles are less than 90 degrees
    #[inline(always)]
    pub fn is_acute(&self) -> bool {
        let [a, b, c] = self.vertices;
        let ab = b - a;
        let bc = c - b;
        let ca = a - c;

        // a^2 + b^2 < c^2 for an acute triangle
        let mut side_lengths = [
            ab.length_squared(),
            bc.length_squared(),
            ca.length_squared(),
        ];
        side_lengths.sort_by(|a, b| a.partial_cmp(b).unwrap());
        side_lengths[0] + side_lengths[1] > side_lengths[2]
    }

    /// Checks if the triangle is obtuse, meaning one angle is greater than 90 degrees
    #[inline(always)]
    pub fn is_obtuse(&self) -> bool {
        let [a, b, c] = self.vertices;
        let ab = b - a;
        let bc = c - b;
        let ca = a - c;

        // a^2 + b^2 > c^2 for an obtuse triangle
        let mut side_lengths = [
            ab.length_squared(),
            bc.length_squared(),
            ca.length_squared(),
        ];
        side_lengths.sort_by(|a, b| a.partial_cmp(b).unwrap());
        side_lengths[0] + side_lengths[1] < side_lengths[2]
    }

    /// Reverse the triangle by swapping the first and last vertices.
    #[inline(always)]
    pub fn reverse(&mut self) {
        self.vertices.swap(0, 2);
    }

    /// This triangle but reversed.
    #[inline(always)]
    #[must_use]
    pub fn reversed(mut self) -> Triangle3d {
        self.reverse();
        self
    }

    /// Get the centroid of the triangle.
    ///
    /// This function finds the geometric center of the triangle by averaging the vertices:
    /// `centroid = (a + b + c) / 3`.
    #[doc(alias("center", "barycenter", "baricenter"))]
    #[inline(always)]
    pub fn centroid(&self) -> Vec3 {
        (self.vertices[0] + self.vertices[1] + self.vertices[2]) / 3.0
    }

    /// Get the largest side of the triangle.
    ///
    /// Returns the two points that form the largest side of the triangle.
    #[inline(always)]
    pub fn largest_side(&self) -> (Vec3, Vec3) {
        let [a, b, c] = self.vertices;
        let ab = b - a;
        let bc = c - b;
        let ca = a - c;

        let mut largest_side_points = (a, b);
        let mut largest_side_length = ab.length();

        if bc.length() > largest_side_length {
            largest_side_points = (b, c);
            largest_side_length = bc.length();
        }

        if ca.length() > largest_side_length {
            largest_side_points = (a, c);
        }

        largest_side_points
    }

    /// Get the circumcenter of the triangle.
    #[inline(always)]
    pub fn circumcenter(&self) -> Vec3 {
        if self.is_degenerate() {
            // If the triangle is degenerate, the circumcenter is the midpoint of the largest side.
            let (p1, p2) = self.largest_side();
            return (p1 + p2) / 2.0;
        }

        let [a, b, c] = self.vertices;
        let ab = b - a;
        let ac = c - a;
        let n = ab.cross(ac);

        // Reference: https://gamedev.stackexchange.com/questions/60630/how-do-i-find-the-circumcenter-of-a-triangle-in-3d
        a + ((ac.length_squared() * n.cross(ab) + ab.length_squared() * ac.cross(ab).cross(ac))
            / (2.0 * n.length_squared()))
    }
}

impl Measured2d for Triangle3d {
    /// Get the area of the triangle.
    #[inline(always)]
    fn area(&self) -> f32 {
        let [a, b, c] = self.vertices;
        let ab = b - a;
        let ac = c - a;
        ab.cross(ac).length() / 2.0
    }

    /// Get the perimeter of the triangle.
    #[inline(always)]
    fn perimeter(&self) -> f32 {
        let [a, b, c] = self.vertices;
        a.distance(b) + b.distance(c) + c.distance(a)
    }
}

/// A tetrahedron primitive.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(
    feature = "bevy_reflect",
    derive(Reflect),
    reflect(Debug, PartialEq, Default)
)]
#[cfg_attr(
    all(feature = "serialize", feature = "bevy_reflect"),
    reflect(Serialize, Deserialize)
)]
pub struct Tetrahedron {
    /// The vertices of the tetrahedron.
    pub vertices: [Vec3; 4],
}
impl Primitive3d for Tetrahedron {}

impl Default for Tetrahedron {
    /// Returns the default [`Tetrahedron`] with the vertices
    /// `[0.5, 0.5, 0.5]`, `[-0.5, 0.5, -0.5]`, `[-0.5, -0.5, 0.5]` and `[0.5, -0.5, -0.5]`.
    fn default() -> Self {
        Self {
            vertices: [
                Vec3::new(0.5, 0.5, 0.5),
                Vec3::new(-0.5, 0.5, -0.5),
                Vec3::new(-0.5, -0.5, 0.5),
                Vec3::new(0.5, -0.5, -0.5),
            ],
        }
    }
}

impl Tetrahedron {
    /// Create a new [`Tetrahedron`] from points `a`, `b`, `c` and `d`.
    #[inline(always)]
    pub fn new(a: Vec3, b: Vec3, c: Vec3, d: Vec3) -> Self {
        Self {
            vertices: [a, b, c, d],
        }
    }

    /// Get the signed volume of the tetrahedron.
    ///
    /// If it's negative, the normal vector of the face defined by
    /// the first three points using the right-hand rule points
    /// away from the fourth vertex.
    #[inline(always)]
    pub fn signed_volume(&self) -> f32 {
        let [a, b, c, d] = self.vertices;
        let ab = b - a;
        let ac = c - a;
        let ad = d - a;
        Mat3::from_cols(ab, ac, ad).determinant() / 6.0
    }

    /// Get the centroid of the tetrahedron.
    ///
    /// This function finds the geometric center of the tetrahedron
    /// by averaging the vertices: `centroid = (a + b + c + d) / 4`.
    #[doc(alias("center", "barycenter", "baricenter"))]
    #[inline(always)]
    pub fn centroid(&self) -> Vec3 {
        (self.vertices[0] + self.vertices[1] + self.vertices[2] + self.vertices[3]) / 4.0
    }

    /// Get the triangles that form the faces of this tetrahedron.
    ///
    /// Note that the orientations of the faces are determined by that of the tetrahedron; if the
    /// signed volume of this tetrahedron is positive, then the triangles' normals will point
    /// outward, and if the signed volume is negative they will point inward.
    #[inline(always)]
    pub fn faces(&self) -> [Triangle3d; 4] {
        let [a, b, c, d] = self.vertices;
        [
            Triangle3d::new(b, c, d),
            Triangle3d::new(a, c, d).reversed(),
            Triangle3d::new(a, b, d),
            Triangle3d::new(a, b, c).reversed(),
        ]
    }
}

impl Measured3d for Tetrahedron {
    /// Get the surface area of the tetrahedron.
    #[inline(always)]
    fn area(&self) -> f32 {
        let [a, b, c, d] = self.vertices;
        let ab = b - a;
        let ac = c - a;
        let ad = d - a;
        let bc = c - b;
        let bd = d - b;
        (ab.cross(ac).length()
            + ab.cross(ad).length()
            + ac.cross(ad).length()
            + bc.cross(bd).length())
            / 2.0
    }

    /// Get the volume of the tetrahedron.
    #[inline(always)]
    fn volume(&self) -> f32 {
        self.signed_volume().abs()
    }
}

/// A 3D shape representing an extruded 2D `base_shape`.
///
/// Extruding a shape effectively "thickens" a 2D shapes,
/// creating a shape with the same cross-section over the entire depth.
///
/// The resulting volumes are prisms.
/// For example, a triangle becomes a triangular prism, while a circle becomes a cylinder.
#[doc(alias = "Prism")]
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serialize", derive(serde::Serialize, serde::Deserialize))]
pub struct Extrusion<T: Primitive2d> {
    /// The base shape of the extrusion
    pub base_shape: T,
    /// Half of the depth of the extrusion
    pub half_depth: f32,
}
impl<T: Primitive2d> Primitive3d for Extrusion<T> {}

impl<T: Primitive2d> Extrusion<T> {
    /// Create a new `Extrusion<T>` from a given `base_shape` and `depth`
    pub fn new(base_shape: T, depth: f32) -> Self {
        Self {
            base_shape,
            half_depth: depth / 2.,
        }
    }
}

impl<T: Primitive2d + Measured2d> Measured3d for Extrusion<T> {
    /// Get the surface area of the extrusion
    fn area(&self) -> f32 {
        2. * (self.base_shape.area() + self.half_depth * self.base_shape.perimeter())
    }

    /// Get the volume of the extrusion
    fn volume(&self) -> f32 {
        2. * self.base_shape.area() * self.half_depth
    }
}

#[cfg(test)]
mod tests {
    // Reference values were computed by hand and/or with external tools

    use super::*;
    use crate::{InvalidDirectionError, Quat};
    use approx::assert_relative_eq;

    #[test]
    fn direction_creation() {
        assert_eq!(Dir3::new(Vec3::X * 12.5), Ok(Dir3::X));
        assert_eq!(
            Dir3::new(Vec3::new(0.0, 0.0, 0.0)),
            Err(InvalidDirectionError::Zero)
        );
        assert_eq!(
            Dir3::new(Vec3::new(f32::INFINITY, 0.0, 0.0)),
            Err(InvalidDirectionError::Infinite)
        );
        assert_eq!(
            Dir3::new(Vec3::new(f32::NEG_INFINITY, 0.0, 0.0)),
            Err(InvalidDirectionError::Infinite)
        );
        assert_eq!(
            Dir3::new(Vec3::new(f32::NAN, 0.0, 0.0)),
            Err(InvalidDirectionError::NaN)
        );
        assert_eq!(Dir3::new_and_length(Vec3::X * 6.5), Ok((Dir3::X, 6.5)));

        // Test rotation
        assert!(
            (Quat::from_rotation_z(core::f32::consts::FRAC_PI_2) * Dir3::X)
                .abs_diff_eq(Vec3::Y, 10e-6)
        );
    }

    #[test]
    fn cuboid_closest_point() {
        let cuboid = Cuboid::new(2.0, 2.0, 2.0);
        assert_eq!(cuboid.closest_point(Vec3::X * 10.0), Vec3::X);
        assert_eq!(cuboid.closest_point(Vec3::NEG_ONE * 10.0), Vec3::NEG_ONE);
        assert_eq!(
            cuboid.closest_point(Vec3::new(0.25, 0.1, 0.3)),
            Vec3::new(0.25, 0.1, 0.3)
        );
    }

    #[test]
    fn sphere_closest_point() {
        let sphere = Sphere { radius: 1.0 };
        assert_eq!(sphere.closest_point(Vec3::X * 10.0), Vec3::X);
        assert_eq!(
            sphere.closest_point(Vec3::NEG_ONE * 10.0),
            Vec3::NEG_ONE.normalize()
        );
        assert_eq!(
            sphere.closest_point(Vec3::new(0.25, 0.1, 0.3)),
            Vec3::new(0.25, 0.1, 0.3)
        );
    }

    #[test]
    fn sphere_math() {
        let sphere = Sphere { radius: 4.0 };
        assert_eq!(sphere.diameter(), 8.0, "incorrect diameter");
        assert_eq!(sphere.area(), 201.06193, "incorrect area");
        assert_eq!(sphere.volume(), 268.08257, "incorrect volume");
    }

    #[test]
    fn plane_from_points() {
        let (plane, translation) = Plane3d::from_points(Vec3::X, Vec3::Z, Vec3::NEG_X);
        assert_eq!(*plane.normal, Vec3::NEG_Y, "incorrect normal");
        assert_eq!(plane.half_size, Vec2::new(0.5, 0.5), "incorrect half size");
        assert_eq!(translation, Vec3::Z * 0.33333334, "incorrect translation");
    }

    #[test]
    fn infinite_plane_math() {
        let (plane, origin) = InfinitePlane3d::from_points(Vec3::X, Vec3::Z, Vec3::NEG_X);
        assert_eq!(*plane.normal, Vec3::NEG_Y, "incorrect normal");
        assert_eq!(origin, Vec3::Z * 0.33333334, "incorrect translation");

        let point_in_plane = Vec3::X + Vec3::Z;
        assert_eq!(
            plane.signed_distance(origin, point_in_plane),
            0.0,
            "incorrect distance"
        );
        assert_eq!(
            plane.project_point(origin, point_in_plane),
            point_in_plane,
            "incorrect point"
        );

        let point_outside = Vec3::Y;
        assert_eq!(
            plane.signed_distance(origin, point_outside),
            -1.0,
            "incorrect distance"
        );
        assert_eq!(
            plane.project_point(origin, point_outside),
            Vec3::ZERO,
            "incorrect point"
        );

        let point_outside = Vec3::NEG_Y;
        assert_eq!(
            plane.signed_distance(origin, point_outside),
            1.0,
            "incorrect distance"
        );
        assert_eq!(
            plane.project_point(origin, point_outside),
            Vec3::ZERO,
            "incorrect point"
        );

        let area_f = |[a, b, c]: [Vec3; 3]| (a - b).cross(a - c).length() * 0.5;
        let (proj, inj) = plane.isometries_xy(origin);

        let triangle = [Vec3::X, Vec3::Y, Vec3::ZERO];
        assert_eq!(area_f(triangle), 0.5, "incorrect area");

        let triangle_proj = triangle.map(|vec3| proj * vec3);
        assert_relative_eq!(area_f(triangle_proj), 0.5);

        let triangle_proj_inj = triangle_proj.map(|vec3| inj * vec3);
        assert_relative_eq!(area_f(triangle_proj_inj), 0.5);
    }

    #[test]
    fn cuboid_math() {
        let cuboid = Cuboid::new(3.0, 7.0, 2.0);
        assert_eq!(
            cuboid,
            Cuboid::from_corners(Vec3::new(-1.5, -3.5, -1.0), Vec3::new(1.5, 3.5, 1.0)),
            "incorrect dimensions when created from corners"
        );
        assert_eq!(cuboid.area(), 82.0, "incorrect area");
        assert_eq!(cuboid.volume(), 42.0, "incorrect volume");
    }

    #[test]
    fn cylinder_math() {
        let cylinder = Cylinder::new(2.0, 9.0);
        assert_eq!(
            cylinder.base(),
            Circle { radius: 2.0 },
            "base produces incorrect circle"
        );
        assert_eq!(
            cylinder.lateral_area(),
            113.097336,
            "incorrect lateral area"
        );
        assert_eq!(cylinder.base_area(), 12.566371, "incorrect base area");
        assert_relative_eq!(cylinder.area(), 138.23007);
        assert_eq!(cylinder.volume(), 113.097336, "incorrect volume");
    }

    #[test]
    fn capsule_math() {
        let capsule = Capsule3d::new(2.0, 9.0);
        assert_eq!(
            capsule.to_cylinder(),
            Cylinder::new(2.0, 9.0),
            "cylinder wasn't created correctly from a capsule"
        );
        assert_eq!(capsule.area(), 163.36282, "incorrect area");
        assert_relative_eq!(capsule.volume(), 146.60765);
    }

    #[test]
    fn cone_math() {
        let cone = Cone {
            radius: 2.0,
            height: 9.0,
        };
        assert_eq!(
            cone.base(),
            Circle { radius: 2.0 },
            "base produces incorrect circle"
        );
        assert_eq!(cone.slant_height(), 9.219544, "incorrect slant height");
        assert_eq!(cone.lateral_area(), 57.92811, "incorrect lateral area");
        assert_eq!(cone.base_area(), 12.566371, "incorrect base area");
        assert_relative_eq!(cone.area(), 70.49447);
        assert_eq!(cone.volume(), 37.699111, "incorrect volume");
    }

    #[test]
    fn torus_math() {
        let torus = Torus {
            minor_radius: 0.3,
            major_radius: 2.8,
        };
        assert_eq!(torus.inner_radius(), 2.5, "incorrect inner radius");
        assert_eq!(torus.outer_radius(), 3.1, "incorrect outer radius");
        assert_eq!(torus.kind(), TorusKind::Ring, "incorrect torus kind");
        assert_eq!(
            Torus::new(0.0, 1.0).kind(),
            TorusKind::Horn,
            "incorrect torus kind"
        );
        assert_eq!(
            Torus::new(-0.5, 1.0).kind(),
            TorusKind::Spindle,
            "incorrect torus kind"
        );
        assert_eq!(
            Torus::new(1.5, 1.0).kind(),
            TorusKind::Invalid,
            "torus should be invalid"
        );
        assert_relative_eq!(torus.area(), 33.16187);
        assert_relative_eq!(torus.volume(), 4.97428, epsilon = 0.00001);
    }

    #[test]
    fn tetrahedron_math() {
        let tetrahedron = Tetrahedron {
            vertices: [
                Vec3::new(0.3, 1.0, 1.7),
                Vec3::new(-2.0, -1.0, 0.0),
                Vec3::new(1.8, 0.5, 1.0),
                Vec3::new(-1.0, -2.0, 3.5),
            ],
        };
        assert_eq!(tetrahedron.area(), 19.251068, "incorrect area");
        assert_eq!(tetrahedron.volume(), 3.2058334, "incorrect volume");
        assert_eq!(
            tetrahedron.signed_volume(),
            3.2058334,
            "incorrect signed volume"
        );
        assert_relative_eq!(tetrahedron.centroid(), Vec3::new(-0.225, -0.375, 1.55));

        assert_eq!(Tetrahedron::default().area(), 3.4641016, "incorrect area");
        assert_eq!(
            Tetrahedron::default().volume(),
            0.33333334,
            "incorrect volume"
        );
        assert_eq!(
            Tetrahedron::default().signed_volume(),
            -0.33333334,
            "incorrect signed volume"
        );
        assert_relative_eq!(Tetrahedron::default().centroid(), Vec3::ZERO);
    }

    #[test]
    fn extrusion_math() {
        let circle = Circle::new(0.75);
        let cylinder = Extrusion::new(circle, 2.5);
        assert_eq!(cylinder.area(), 15.315264, "incorrect surface area");
        assert_eq!(cylinder.volume(), 4.417865, "incorrect volume");

        let annulus = crate::primitives::Annulus::new(0.25, 1.375);
        let tube = Extrusion::new(annulus, 0.333);
        assert_eq!(tube.area(), 14.886437, "incorrect surface area");
        assert_eq!(tube.volume(), 1.9124937, "incorrect volume");

        let polygon = crate::primitives::RegularPolygon::new(3.8, 7);
        let regular_prism = Extrusion::new(polygon, 1.25);
        assert_eq!(regular_prism.area(), 107.8808, "incorrect surface area");
        assert_eq!(regular_prism.volume(), 49.392204, "incorrect volume");
    }

    #[test]
    fn triangle_math() {
        // Default triangle tests
        let mut default_triangle = Triangle3d::default();
        let reverse_default_triangle = Triangle3d::new(
            Vec3::new(0.5, -0.5, 0.0),
            Vec3::new(-0.5, -0.5, 0.0),
            Vec3::new(0.0, 0.5, 0.0),
        );
        assert_eq!(default_triangle.area(), 0.5, "incorrect area");
        assert_relative_eq!(
            default_triangle.perimeter(),
            1.0 + 2.0 * 1.25_f32.sqrt(),
            epsilon = 10e-9
        );
        assert_eq!(default_triangle.normal(), Ok(Dir3::Z), "incorrect normal");
        assert!(
            !default_triangle.is_degenerate(),
            "incorrect degenerate check"
        );
        assert_eq!(
            default_triangle.centroid(),
            Vec3::new(0.0, -0.16666667, 0.0),
            "incorrect centroid"
        );
        assert_eq!(
            default_triangle.largest_side(),
            (Vec3::new(0.0, 0.5, 0.0), Vec3::new(-0.5, -0.5, 0.0))
        );
        default_triangle.reverse();
        assert_eq!(
            default_triangle, reverse_default_triangle,
            "incorrect reverse"
        );
        assert_eq!(
            default_triangle.circumcenter(),
            Vec3::new(0.0, -0.125, 0.0),
            "incorrect circumcenter"
        );

        // Custom triangle tests
        let right_triangle = Triangle3d::new(Vec3::ZERO, Vec3::X, Vec3::Y);
        let obtuse_triangle = Triangle3d::new(Vec3::NEG_X, Vec3::X, Vec3::new(0.0, 0.1, 0.0));
        let acute_triangle = Triangle3d::new(Vec3::ZERO, Vec3::X, Vec3::new(0.5, 5.0, 0.0));

        assert_eq!(
            right_triangle.circumcenter(),
            Vec3::new(0.5, 0.5, 0.0),
            "incorrect circumcenter"
        );
        assert_eq!(
            obtuse_triangle.circumcenter(),
            Vec3::new(0.0, -4.95, 0.0),
            "incorrect circumcenter"
        );
        assert_eq!(
            acute_triangle.circumcenter(),
            Vec3::new(0.5, 2.475, 0.0),
            "incorrect circumcenter"
        );

        assert!(acute_triangle.is_acute());
        assert!(!acute_triangle.is_obtuse());
        assert!(!obtuse_triangle.is_acute());
        assert!(obtuse_triangle.is_obtuse());

        // Arbitrary triangle tests
        let [a, b, c] = [Vec3::ZERO, Vec3::new(1., 1., 0.5), Vec3::new(-3., 2.5, 1.)];
        let triangle = Triangle3d::new(a, b, c);

        assert!(!triangle.is_degenerate(), "incorrectly found degenerate");
        assert_eq!(triangle.area(), 3.0233467, "incorrect area");
        assert_eq!(triangle.perimeter(), 9.832292, "incorrect perimeter");
        assert_eq!(
            triangle.circumcenter(),
            Vec3::new(-1., 1.75, 0.75),
            "incorrect circumcenter"
        );
        assert_eq!(
            triangle.normal(),
            Ok(Dir3::new_unchecked(Vec3::new(
                -0.04134491,
                -0.4134491,
                0.90958804
            ))),
            "incorrect normal"
        );

        // Degenerate triangle tests
        let zero_degenerate_triangle = Triangle3d::new(Vec3::ZERO, Vec3::ZERO, Vec3::ZERO);
        assert!(
            zero_degenerate_triangle.is_degenerate(),
            "incorrect degenerate check"
        );
        assert_eq!(
            zero_degenerate_triangle.normal(),
            Err(InvalidDirectionError::Zero),
            "incorrect normal"
        );
        assert_eq!(
            zero_degenerate_triangle.largest_side(),
            (Vec3::ZERO, Vec3::ZERO),
            "incorrect largest side"
        );

        let dup_degenerate_triangle = Triangle3d::new(Vec3::ZERO, Vec3::X, Vec3::X);
        assert!(
            dup_degenerate_triangle.is_degenerate(),
            "incorrect degenerate check"
        );
        assert_eq!(
            dup_degenerate_triangle.normal(),
            Err(InvalidDirectionError::Zero),
            "incorrect normal"
        );
        assert_eq!(
            dup_degenerate_triangle.largest_side(),
            (Vec3::ZERO, Vec3::X),
            "incorrect largest side"
        );

        let collinear_degenerate_triangle = Triangle3d::new(Vec3::NEG_X, Vec3::ZERO, Vec3::X);
        assert!(
            collinear_degenerate_triangle.is_degenerate(),
            "incorrect degenerate check"
        );
        assert_eq!(
            collinear_degenerate_triangle.normal(),
            Err(InvalidDirectionError::Zero),
            "incorrect normal"
        );
        assert_eq!(
            collinear_degenerate_triangle.largest_side(),
            (Vec3::NEG_X, Vec3::X),
            "incorrect largest side"
        );
    }
}