nalgebra/src/geometry/orthographic.rs

#[cfg(feature = "arbitrary")]
use quickcheck::{Arbitrary, Gen};
#[cfg(feature = "rand-no-std")]
use rand::{
    distributions::{Distribution, Standard},
    Rng,
};
#[cfg(feature = "serde-serialize-no-std")]
use serde::{Deserialize, Deserializer, Serialize, Serializer};
use std::fmt;

use simba::scalar::RealField;

use crate::base::dimension::U3;
use crate::base::storage::Storage;
use crate::base::{Matrix4, Vector, Vector3};

use crate::geometry::{Point3, Projective3};

/// A 3D orthographic projection stored as a homogeneous 4x4 matrix.
#[repr(C)]
#[cfg_attr(feature = "rkyv-serialize", derive(bytecheck::CheckBytes))]
#[cfg_attr(
    feature = "rkyv-serialize-no-std",
    derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize),
    archive(as = "Orthographic3<T::Archived>", bound(archive = "
        T: rkyv::Archive,
        Matrix4<T>: rkyv::Archive<Archived = Matrix4<T::Archived>>
    "))
)]
#[cfg_attr(feature = "cuda", derive(cust_core::DeviceCopy))]
#[derive(Copy, Clone)]
pub struct Orthographic3<T> {
    matrix: Matrix4<T>,
}

impl<T: RealField> fmt::Debug for Orthographic3<T> {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> Result<(), fmt::Error> {
        self.matrix.fmt(f)
    }
}

impl<T: RealField> PartialEq for Orthographic3<T> {
    #[inline]
    fn eq(&self, right: &Self) -> bool {
        self.matrix == right.matrix
    }
}

#[cfg(feature = "bytemuck")]
unsafe impl<T> bytemuck::Zeroable for Orthographic3<T>
where
    T: RealField + bytemuck::Zeroable,
    Matrix4<T>: bytemuck::Zeroable,
{
}

#[cfg(feature = "bytemuck")]
unsafe impl<T> bytemuck::Pod for Orthographic3<T>
where
    T: RealField + bytemuck::Pod,
    Matrix4<T>: bytemuck::Pod,
{
}

#[cfg(feature = "serde-serialize-no-std")]
impl<T: RealField + Serialize> Serialize for Orthographic3<T> {
    fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
    where
        S: Serializer,
    {
        self.matrix.serialize(serializer)
    }
}

#[cfg(feature = "serde-serialize-no-std")]
impl<'a, T: RealField + Deserialize<'a>> Deserialize<'a> for Orthographic3<T> {
    fn deserialize<Des>(deserializer: Des) -> Result<Self, Des::Error>
    where
        Des: Deserializer<'a>,
    {
        let matrix = Matrix4::<T>::deserialize(deserializer)?;

        Ok(Self::from_matrix_unchecked(matrix))
    }
}

impl<T> Orthographic3<T> {
    /// Wraps the given matrix to interpret it as a 3D orthographic matrix.
    ///
    /// It is not checked whether or not the given matrix actually represents an orthographic
    /// projection.
    ///
    /// # Example
    /// ```
    /// # use nalgebra::{Orthographic3, Point3, Matrix4};
    /// let mat = Matrix4::new(
    ///     2.0 / 9.0, 0.0,        0.0,         -11.0 / 9.0,
    ///     0.0,       2.0 / 18.0, 0.0,         -22.0 / 18.0,
    ///     0.0,       0.0,       -2.0 / 999.9, -1000.1 / 999.9,
    ///     0.0,       0.0,        0.0,         1.0
    /// );
    /// let proj = Orthographic3::from_matrix_unchecked(mat);
    /// assert_eq!(proj, Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0));
    /// ```
    #[inline]
    pub const fn from_matrix_unchecked(matrix: Matrix4<T>) -> Self {
        Self { matrix }
    }
}

impl<T: RealField> Orthographic3<T> {
    /// Creates a new orthographic projection matrix.
    ///
    /// This follows the OpenGL convention, so this will flip the `z` axis.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Orthographic3, Point3};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// // Check this projection actually transforms the view cuboid into the double-unit cube.
    /// // See https://www.nalgebra.org/docs/user_guide/projections#orthographic-projection for more details.
    /// let p1 = Point3::new(1.0, 2.0, -0.1);
    /// let p2 = Point3::new(1.0, 2.0, -1000.0);
    /// let p3 = Point3::new(1.0, 20.0, -0.1);
    /// let p4 = Point3::new(1.0, 20.0, -1000.0);
    /// let p5 = Point3::new(10.0, 2.0, -0.1);
    /// let p6 = Point3::new(10.0, 2.0, -1000.0);
    /// let p7 = Point3::new(10.0, 20.0, -0.1);
    /// let p8 = Point3::new(10.0, 20.0, -1000.0);
    ///
    /// assert_relative_eq!(proj.project_point(&p1), Point3::new(-1.0, -1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p2), Point3::new(-1.0, -1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p3), Point3::new(-1.0,  1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p4), Point3::new(-1.0,  1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p5), Point3::new( 1.0, -1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p6), Point3::new( 1.0, -1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p7), Point3::new( 1.0,  1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p8), Point3::new( 1.0,  1.0,  1.0));
    ///
    /// // This also works with flipped axis. In other words, we allow that
    /// // `left > right`, `bottom > top`, and/or `znear > zfar`.
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    ///
    /// assert_relative_eq!(proj.project_point(&p1), Point3::new( 1.0,  1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p2), Point3::new( 1.0,  1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p3), Point3::new( 1.0, -1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p4), Point3::new( 1.0, -1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p5), Point3::new(-1.0,  1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p6), Point3::new(-1.0,  1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p7), Point3::new(-1.0, -1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p8), Point3::new(-1.0, -1.0, -1.0));
    /// ```
    #[inline]
    pub fn new(left: T, right: T, bottom: T, top: T, znear: T, zfar: T) -> Self {
        let matrix = Matrix4::<T>::identity();
        let mut res = Self::from_matrix_unchecked(matrix);

        res.set_left_and_right(left, right);
        res.set_bottom_and_top(bottom, top);
        res.set_znear_and_zfar(znear, zfar);

        res
    }

    /// Creates a new orthographic projection matrix from an aspect ratio and the vertical field of view.
    #[inline]
    pub fn from_fov(aspect: T, vfov: T, znear: T, zfar: T) -> Self {
        assert!(
            znear != zfar,
            "The far plane must not be equal to the near plane."
        );
        assert!(
            !relative_eq!(aspect, T::zero()),
            "The aspect ratio must not be zero."
        );

        let half: T = crate::convert(0.5);
        let width = zfar.clone() * (vfov * half.clone()).tan();
        let height = width.clone() / aspect;

        Self::new(
            -width.clone() * half.clone(),
            width * half.clone(),
            -height.clone() * half.clone(),
            height * half,
            znear,
            zfar,
        )
    }

    /// Retrieves the inverse of the underlying homogeneous matrix.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Orthographic3, Point3, Matrix4};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// let inv = proj.inverse();
    ///
    /// assert_relative_eq!(inv * proj.as_matrix(), Matrix4::identity());
    /// assert_relative_eq!(proj.as_matrix() * inv, Matrix4::identity());
    ///
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    /// let inv = proj.inverse();
    /// assert_relative_eq!(inv * proj.as_matrix(), Matrix4::identity());
    /// assert_relative_eq!(proj.as_matrix() * inv, Matrix4::identity());
    /// ```
    #[inline]
    #[must_use]
    pub fn inverse(&self) -> Matrix4<T> {
        let mut res = self.clone().to_homogeneous();

        let inv_m11 = T::one() / self.matrix[(0, 0)].clone();
        let inv_m22 = T::one() / self.matrix[(1, 1)].clone();
        let inv_m33 = T::one() / self.matrix[(2, 2)].clone();

        res[(0, 0)] = inv_m11.clone();
        res[(1, 1)] = inv_m22.clone();
        res[(2, 2)] = inv_m33.clone();

        res[(0, 3)] = -self.matrix[(0, 3)].clone() * inv_m11;
        res[(1, 3)] = -self.matrix[(1, 3)].clone() * inv_m22;
        res[(2, 3)] = -self.matrix[(2, 3)].clone() * inv_m33;

        res
    }

    /// Computes the corresponding homogeneous matrix.
    ///
    /// # Example
    /// ```
    /// # use nalgebra::{Orthographic3, Point3, Matrix4};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// let expected = Matrix4::new(
    ///     2.0 / 9.0, 0.0,        0.0,         -11.0 / 9.0,
    ///     0.0,       2.0 / 18.0, 0.0,         -22.0 / 18.0,
    ///     0.0,       0.0,       -2.0 / 999.9, -1000.1 / 999.9,
    ///     0.0,       0.0,        0.0,         1.0
    /// );
    /// assert_eq!(proj.to_homogeneous(), expected);
    /// ```
    #[inline]
    #[must_use]
    pub fn to_homogeneous(self) -> Matrix4<T> {
        self.matrix
    }

    /// A reference to the underlying homogeneous transformation matrix.
    ///
    /// # Example
    /// ```
    /// # use nalgebra::{Orthographic3, Point3, Matrix4};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// let expected = Matrix4::new(
    ///     2.0 / 9.0, 0.0,        0.0,         -11.0 / 9.0,
    ///     0.0,       2.0 / 18.0, 0.0,         -22.0 / 18.0,
    ///     0.0,       0.0,       -2.0 / 999.9, -1000.1 / 999.9,
    ///     0.0,       0.0,        0.0,         1.0
    /// );
    /// assert_eq!(*proj.as_matrix(), expected);
    /// ```
    #[inline]
    #[must_use]
    pub fn as_matrix(&self) -> &Matrix4<T> {
        &self.matrix
    }

    /// A reference to this transformation seen as a `Projective3`.
    ///
    /// # Example
    /// ```
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_eq!(proj.as_projective().to_homogeneous(), proj.to_homogeneous());
    /// ```
    #[inline]
    #[must_use]
    pub fn as_projective(&self) -> &Projective3<T> {
        unsafe { &*(self as *const Orthographic3<T> as *const Projective3<T>) }
    }

    /// This transformation seen as a `Projective3`.
    ///
    /// # Example
    /// ```
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_eq!(proj.to_projective().to_homogeneous(), proj.to_homogeneous());
    /// ```
    #[inline]
    #[must_use]
    pub fn to_projective(self) -> Projective3<T> {
        Projective3::from_matrix_unchecked(self.matrix)
    }

    /// Retrieves the underlying homogeneous matrix.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Orthographic3, Point3, Matrix4};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// let expected = Matrix4::new(
    ///     2.0 / 9.0, 0.0,        0.0,         -11.0 / 9.0,
    ///     0.0,       2.0 / 18.0, 0.0,         -22.0 / 18.0,
    ///     0.0,       0.0,       -2.0 / 999.9, -1000.1 / 999.9,
    ///     0.0,       0.0,        0.0,         1.0
    /// );
    /// assert_eq!(proj.into_inner(), expected);
    /// ```
    #[inline]
    pub fn into_inner(self) -> Matrix4<T> {
        self.matrix
    }

    /// Retrieves the underlying homogeneous matrix.
    /// Deprecated: Use [`Orthographic3::into_inner`] instead.
    #[deprecated(note = "use `.into_inner()` instead")]
    #[inline]
    pub fn unwrap(self) -> Matrix4<T> {
        self.matrix
    }

    /// The left offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_relative_eq!(proj.left(), 1.0, epsilon = 1.0e-6);
    ///
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    /// assert_relative_eq!(proj.left(), 10.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    #[must_use]
    pub fn left(&self) -> T {
        (-T::one() - self.matrix[(0, 3)].clone()) / self.matrix[(0, 0)].clone()
    }

    /// The right offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_relative_eq!(proj.right(), 10.0, epsilon = 1.0e-6);
    ///
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    /// assert_relative_eq!(proj.right(), 1.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    #[must_use]
    pub fn right(&self) -> T {
        (T::one() - self.matrix[(0, 3)].clone()) / self.matrix[(0, 0)].clone()
    }

    /// The bottom offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_relative_eq!(proj.bottom(), 2.0, epsilon = 1.0e-6);
    ///
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    /// assert_relative_eq!(proj.bottom(), 20.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    #[must_use]
    pub fn bottom(&self) -> T {
        (-T::one() - self.matrix[(1, 3)].clone()) / self.matrix[(1, 1)].clone()
    }

    /// The top offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_relative_eq!(proj.top(), 20.0, epsilon = 1.0e-6);
    ///
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    /// assert_relative_eq!(proj.top(), 2.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    #[must_use]
    pub fn top(&self) -> T {
        (T::one() - self.matrix[(1, 3)].clone()) / self.matrix[(1, 1)].clone()
    }

    /// The near plane offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_relative_eq!(proj.znear(), 0.1, epsilon = 1.0e-6);
    ///
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    /// assert_relative_eq!(proj.znear(), 1000.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    #[must_use]
    pub fn znear(&self) -> T {
        (T::one() + self.matrix[(2, 3)].clone()) / self.matrix[(2, 2)].clone()
    }

    /// The far plane offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// assert_relative_eq!(proj.zfar(), 1000.0, epsilon = 1.0e-6);
    ///
    /// let proj = Orthographic3::new(10.0, 1.0, 20.0, 2.0, 1000.0, 0.1);
    /// assert_relative_eq!(proj.zfar(), 0.1, epsilon = 1.0e-6);
    /// ```
    #[inline]
    #[must_use]
    pub fn zfar(&self) -> T {
        (-T::one() + self.matrix[(2, 3)].clone()) / self.matrix[(2, 2)].clone()
    }

    // TODO: when we get specialization, specialize the Mul impl instead.
    /// Projects a point. Faster than matrix multiplication.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Orthographic3, Point3};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    ///
    /// let p1 = Point3::new(1.0, 2.0, -0.1);
    /// let p2 = Point3::new(1.0, 2.0, -1000.0);
    /// let p3 = Point3::new(1.0, 20.0, -0.1);
    /// let p4 = Point3::new(1.0, 20.0, -1000.0);
    /// let p5 = Point3::new(10.0, 2.0, -0.1);
    /// let p6 = Point3::new(10.0, 2.0, -1000.0);
    /// let p7 = Point3::new(10.0, 20.0, -0.1);
    /// let p8 = Point3::new(10.0, 20.0, -1000.0);
    ///
    /// assert_relative_eq!(proj.project_point(&p1), Point3::new(-1.0, -1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p2), Point3::new(-1.0, -1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p3), Point3::new(-1.0,  1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p4), Point3::new(-1.0,  1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p5), Point3::new( 1.0, -1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p6), Point3::new( 1.0, -1.0,  1.0));
    /// assert_relative_eq!(proj.project_point(&p7), Point3::new( 1.0,  1.0, -1.0));
    /// assert_relative_eq!(proj.project_point(&p8), Point3::new( 1.0,  1.0,  1.0));
    /// ```
    #[inline]
    #[must_use]
    pub fn project_point(&self, p: &Point3<T>) -> Point3<T> {
        Point3::new(
            self.matrix[(0, 0)].clone() * p[0].clone() + self.matrix[(0, 3)].clone(),
            self.matrix[(1, 1)].clone() * p[1].clone() + self.matrix[(1, 3)].clone(),
            self.matrix[(2, 2)].clone() * p[2].clone() + self.matrix[(2, 3)].clone(),
        )
    }

    /// Un-projects a point. Faster than multiplication by the underlying matrix inverse.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Orthographic3, Point3};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    ///
    /// let p1 = Point3::new(-1.0, -1.0, -1.0);
    /// let p2 = Point3::new(-1.0, -1.0,  1.0);
    /// let p3 = Point3::new(-1.0,  1.0, -1.0);
    /// let p4 = Point3::new(-1.0,  1.0,  1.0);
    /// let p5 = Point3::new( 1.0, -1.0, -1.0);
    /// let p6 = Point3::new( 1.0, -1.0,  1.0);
    /// let p7 = Point3::new( 1.0,  1.0, -1.0);
    /// let p8 = Point3::new( 1.0,  1.0,  1.0);
    ///
    /// assert_relative_eq!(proj.unproject_point(&p1), Point3::new(1.0, 2.0, -0.1), epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.unproject_point(&p2), Point3::new(1.0, 2.0, -1000.0), epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.unproject_point(&p3), Point3::new(1.0, 20.0, -0.1), epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.unproject_point(&p4), Point3::new(1.0, 20.0, -1000.0), epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.unproject_point(&p5), Point3::new(10.0, 2.0, -0.1), epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.unproject_point(&p6), Point3::new(10.0, 2.0, -1000.0), epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.unproject_point(&p7), Point3::new(10.0, 20.0, -0.1), epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.unproject_point(&p8), Point3::new(10.0, 20.0, -1000.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    #[must_use]
    pub fn unproject_point(&self, p: &Point3<T>) -> Point3<T> {
        Point3::new(
            (p[0].clone() - self.matrix[(0, 3)].clone()) / self.matrix[(0, 0)].clone(),
            (p[1].clone() - self.matrix[(1, 3)].clone()) / self.matrix[(1, 1)].clone(),
            (p[2].clone() - self.matrix[(2, 3)].clone()) / self.matrix[(2, 2)].clone(),
        )
    }

    // TODO: when we get specialization, specialize the Mul impl instead.
    /// Projects a vector. Faster than matrix multiplication.
    ///
    /// Vectors are not affected by the translation part of the projection.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Orthographic3, Vector3};
    /// let proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    ///
    /// let v1 = Vector3::x();
    /// let v2 = Vector3::y();
    /// let v3 = Vector3::z();
    ///
    /// assert_relative_eq!(proj.project_vector(&v1), Vector3::x() * 2.0 / 9.0);
    /// assert_relative_eq!(proj.project_vector(&v2), Vector3::y() * 2.0 / 18.0);
    /// assert_relative_eq!(proj.project_vector(&v3), Vector3::z() * -2.0 / 999.9);
    /// ```
    #[inline]
    #[must_use]
    pub fn project_vector<SB>(&self, p: &Vector<T, U3, SB>) -> Vector3<T>
    where
        SB: Storage<T, U3>,
    {
        Vector3::new(
            self.matrix[(0, 0)].clone() * p[0].clone(),
            self.matrix[(1, 1)].clone() * p[1].clone(),
            self.matrix[(2, 2)].clone() * p[2].clone(),
        )
    }

    /// Sets the left offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_left(2.0);
    /// assert_relative_eq!(proj.left(), 2.0, epsilon = 1.0e-6);
    ///
    /// // It is OK to set a left offset greater than the current right offset.
    /// proj.set_left(20.0);
    /// assert_relative_eq!(proj.left(), 20.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_left(&mut self, left: T) {
        let right = self.right();
        self.set_left_and_right(left, right);
    }

    /// Sets the right offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_right(15.0);
    /// assert_relative_eq!(proj.right(), 15.0, epsilon = 1.0e-6);
    ///
    /// // It is OK to set a right offset smaller than the current left offset.
    /// proj.set_right(-3.0);
    /// assert_relative_eq!(proj.right(), -3.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_right(&mut self, right: T) {
        let left = self.left();
        self.set_left_and_right(left, right);
    }

    /// Sets the bottom offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_bottom(8.0);
    /// assert_relative_eq!(proj.bottom(), 8.0, epsilon = 1.0e-6);
    ///
    /// // It is OK to set a bottom offset greater than the current top offset.
    /// proj.set_bottom(50.0);
    /// assert_relative_eq!(proj.bottom(), 50.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_bottom(&mut self, bottom: T) {
        let top = self.top();
        self.set_bottom_and_top(bottom, top);
    }

    /// Sets the top offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_top(15.0);
    /// assert_relative_eq!(proj.top(), 15.0, epsilon = 1.0e-6);
    ///
    /// // It is OK to set a top offset smaller than the current bottom offset.
    /// proj.set_top(-3.0);
    /// assert_relative_eq!(proj.top(), -3.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_top(&mut self, top: T) {
        let bottom = self.bottom();
        self.set_bottom_and_top(bottom, top);
    }

    /// Sets the near plane offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_znear(8.0);
    /// assert_relative_eq!(proj.znear(), 8.0, epsilon = 1.0e-6);
    ///
    /// // It is OK to set a znear greater than the current zfar.
    /// proj.set_znear(5000.0);
    /// assert_relative_eq!(proj.znear(), 5000.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_znear(&mut self, znear: T) {
        let zfar = self.zfar();
        self.set_znear_and_zfar(znear, zfar);
    }

    /// Sets the far plane offset of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_zfar(15.0);
    /// assert_relative_eq!(proj.zfar(), 15.0, epsilon = 1.0e-6);
    ///
    /// // It is OK to set a zfar smaller than the current znear.
    /// proj.set_zfar(-3.0);
    /// assert_relative_eq!(proj.zfar(), -3.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_zfar(&mut self, zfar: T) {
        let znear = self.znear();
        self.set_znear_and_zfar(znear, zfar);
    }

    /// Sets the view cuboid offsets along the `x` axis.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_left_and_right(7.0, 70.0);
    /// assert_relative_eq!(proj.left(), 7.0, epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.right(), 70.0, epsilon = 1.0e-6);
    ///
    /// // It is also OK to have `left > right`.
    /// proj.set_left_and_right(70.0, 7.0);
    /// assert_relative_eq!(proj.left(), 70.0, epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.right(), 7.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_left_and_right(&mut self, left: T, right: T) {
        assert!(
            left != right,
            "The left corner must not be equal to the right corner."
        );
        self.matrix[(0, 0)] = crate::convert::<_, T>(2.0) / (right.clone() - left.clone());
        self.matrix[(0, 3)] = -(right.clone() + left.clone()) / (right - left);
    }

    /// Sets the view cuboid offsets along the `y` axis.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_bottom_and_top(7.0, 70.0);
    /// assert_relative_eq!(proj.bottom(), 7.0, epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.top(), 70.0, epsilon = 1.0e-6);
    ///
    /// // It is also OK to have `bottom > top`.
    /// proj.set_bottom_and_top(70.0, 7.0);
    /// assert_relative_eq!(proj.bottom(), 70.0, epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.top(), 7.0, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_bottom_and_top(&mut self, bottom: T, top: T) {
        assert_ne!(
            bottom, top,
            "The top corner must not be equal to the bottom corner."
        );
        self.matrix[(1, 1)] = crate::convert::<_, T>(2.0) / (top.clone() - bottom.clone());
        self.matrix[(1, 3)] = -(top.clone() + bottom.clone()) / (top - bottom);
    }

    /// Sets the near and far plane offsets of the view cuboid.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Orthographic3;
    /// let mut proj = Orthographic3::new(1.0, 10.0, 2.0, 20.0, 0.1, 1000.0);
    /// proj.set_znear_and_zfar(50.0, 5000.0);
    /// assert_relative_eq!(proj.znear(), 50.0, epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.zfar(), 5000.0, epsilon = 1.0e-6);
    ///
    /// // It is also OK to have `znear > zfar`.
    /// proj.set_znear_and_zfar(5000.0, 0.5);
    /// assert_relative_eq!(proj.znear(), 5000.0, epsilon = 1.0e-6);
    /// assert_relative_eq!(proj.zfar(), 0.5, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn set_znear_and_zfar(&mut self, znear: T, zfar: T) {
        assert!(
            zfar != znear,
            "The near-plane and far-plane must not be superimposed."
        );
        self.matrix[(2, 2)] = -crate::convert::<_, T>(2.0) / (zfar.clone() - znear.clone());
        self.matrix[(2, 3)] = -(zfar.clone() + znear.clone()) / (zfar - znear);
    }
}

#[cfg(feature = "rand-no-std")]
impl<T: RealField> Distribution<Orthographic3<T>> for Standard
where
    Standard: Distribution<T>,
{
    /// Generate an arbitrary random variate for testing purposes.
    fn sample<R: Rng + ?Sized>(&self, r: &mut R) -> Orthographic3<T> {
        use crate::base::helper;
        let left = r.gen();
        let right = helper::reject_rand(r, |x: &T| *x > left);
        let bottom = r.gen();
        let top = helper::reject_rand(r, |x: &T| *x > bottom);
        let znear = r.gen();
        let zfar = helper::reject_rand(r, |x: &T| *x > znear);

        Orthographic3::new(left, right, bottom, top, znear, zfar)
    }
}

#[cfg(feature = "arbitrary")]
impl<T: RealField + Arbitrary> Arbitrary for Orthographic3<T>
where
    Matrix4<T>: Send,
{
    fn arbitrary(g: &mut Gen) -> Self {
        use crate::base::helper;
        let left = Arbitrary::arbitrary(g);
        let right = helper::reject(g, |x: &T| *x > left);
        let bottom = Arbitrary::arbitrary(g);
        let top = helper::reject(g, |x: &T| *x > bottom);
        let znear = Arbitrary::arbitrary(g);
        let zfar = helper::reject(g, |x: &T| *x > znear);

        Self::new(left, right, bottom, top, znear, zfar)
    }
}

impl<T: RealField> From<Orthographic3<T>> for Matrix4<T> {
    #[inline]
    fn from(orth: Orthographic3<T>) -> Self {
        orth.into_inner()
    }
}