quaternion.rs - source
Description: Source of the Rust file `/Users/erlendbasso/.cargo/registry/src/github.com-1ecc6299db9ec823/nalgebra-0.32.1/src/geometry/quaternion.rs`.
Keywords: rust, rustlang, rust-lang

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    use approx::{AbsDiffEq, RelativeEq, UlpsEq};
    use num::Zero;
    use std::fmt;
    use std::hash::{Hash, Hasher};

    #[cfg(feature = "serde-serialize-no-std")]
    use crate::base::storage::Owned;
    #[cfg(feature = "serde-serialize-no-std")]
    use serde::{Deserialize, Deserializer, Serialize, Serializer};

    use simba::scalar::{ClosedNeg, RealField};
    use simba::simd::{SimdBool, SimdOption, SimdRealField};

    use crate::base::dimension::{U1, U3, U4};
    use crate::base::storage::{CStride, RStride};
    use crate::base::{
        Matrix3, Matrix4, MatrixView, MatrixViewMut, Normed, Scalar, Unit, Vector3, Vector4,
    };

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

    /// A quaternion. See the type alias `UnitQuaternion = Unit<Quaternion>` for a quaternion
    /// that may be used as a rotation.
    #[repr(C)]
    #[derive(Copy, Clone)]
    #[cfg_attr(
        feature = "rkyv-serialize-no-std",
        derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize),
        archive(
            as = "Quaternion<T::Archived>",
            bound(archive = "
                T: rkyv::Archive,
                Vector4<T>: rkyv::Archive<Archived = Vector4<T::Archived>>
            ")
        )
    )]
    #[cfg_attr(feature = "rkyv-serialize", derive(bytecheck::CheckBytes))]
    #[cfg_attr(feature = "cuda", derive(cust_core::DeviceCopy))]
    pub struct Quaternion<T> {
        /// This quaternion as a 4D vector of coordinates in the `[ x, y, z, w ]` storage order.
        pub coords: Vector4<T>,
    }

    impl<T: fmt::Debug> fmt::Debug for Quaternion<T> {
        fn fmt(&self, formatter: &mut fmt::Formatter<'_>) -> Result<(), fmt::Error> {
            self.coords.as_slice().fmt(formatter)
        }
    }

    impl<T: Scalar + Hash> Hash for Quaternion<T> {
        fn hash<H: Hasher>(&self, state: &mut H) {
            self.coords.hash(state)
        }
    }

    impl<T: Scalar + Eq> Eq for Quaternion<T> {}

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

    impl<T: Scalar + Zero> Default for Quaternion<T> {
        fn default() -> Self {
            Quaternion {
                coords: Vector4::zeros(),
            }
        }
    }

    #[cfg(feature = "bytemuck")]
    unsafe impl<T: Scalar> bytemuck::Zeroable for Quaternion<T> where Vector4<T>: bytemuck::Zeroable {}

    #[cfg(feature = "bytemuck")]
    unsafe impl<T: Scalar> bytemuck::Pod for Quaternion<T>
    where
        Vector4<T>: bytemuck::Pod,
        T: Copy,
    {
    }

    #[cfg(feature = "serde-serialize-no-std")]
    impl<T: Scalar> Serialize for Quaternion<T>
    where
        Owned<T, U4>: Serialize,
    {
        fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
        where
            S: Serializer,
        {
            self.coords.serialize(serializer)
        }
    }

    #[cfg(feature = "serde-serialize-no-std")]
    impl<'a, T: Scalar> Deserialize<'a> for Quaternion<T>
    where
        Owned<T, U4>: Deserialize<'a>,
    {
        fn deserialize<Des>(deserializer: Des) -> Result<Self, Des::Error>
        where
            Des: Deserializer<'a>,
        {
            let coords = Vector4::<T>::deserialize(deserializer)?;

            Ok(Self::from(coords))
        }
    }

    impl<T: SimdRealField> Quaternion<T>
    where
        T::Element: SimdRealField,
    {
        /// Moves this unit quaternion into one that owns its data.
        #[inline]
        #[deprecated(note = "This method is a no-op and will be removed in a future release.")]
        pub fn into_owned(self) -> Self {
            self
        }

        /// Clones this unit quaternion into one that owns its data.
        #[inline]
        #[deprecated(note = "This method is a no-op and will be removed in a future release.")]
        pub fn clone_owned(&self) -> Self {
            Self::from(self.coords.clone_owned())
        }

        /// Normalizes this quaternion.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let q_normalized = q.normalize();
        /// relative_eq!(q_normalized.norm(), 1.0);
        /// ```
        #[inline]
        #[must_use = "Did you mean to use normalize_mut()?"]
        pub fn normalize(&self) -> Self {
            Self::from(self.coords.normalize())
        }

        /// The imaginary part of this quaternion.
        #[inline]
        #[must_use]
        pub fn imag(&self) -> Vector3<T> {
            self.coords.xyz()
        }

        /// The conjugate of this quaternion.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let conj = q.conjugate();
        /// assert!(conj.i == -2.0 && conj.j == -3.0 && conj.k == -4.0 && conj.w == 1.0);
        /// ```
        #[inline]
        #[must_use = "Did you mean to use conjugate_mut()?"]
        pub fn conjugate(&self) -> Self {
            Self::from_parts(self.w.clone(), -self.imag())
        }

        /// Linear interpolation between two quaternion.
        ///
        /// Computes `self * (1 - t) + other * t`.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let q2 = Quaternion::new(10.0, 20.0, 30.0, 40.0);
        ///
        /// assert_eq!(q1.lerp(&q2, 0.1), Quaternion::new(1.9, 3.8, 5.7, 7.6));
        /// ```
        #[inline]
        #[must_use]
        pub fn lerp(&self, other: &Self, t: T) -> Self {
            self * (T::one() - t.clone()) + other * t
        }

        /// The vector part `(i, j, k)` of this quaternion.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// assert_eq!(q.vector()[0], 2.0);
        /// assert_eq!(q.vector()[1], 3.0);
        /// assert_eq!(q.vector()[2], 4.0);
        /// ```
        #[inline]
        #[must_use]
        pub fn vector(&self) -> MatrixView<'_, T, U3, U1, RStride<T, U4, U1>, CStride<T, U4, U1>> {
            self.coords.fixed_rows::<3>(0)
        }

        /// The scalar part `w` of this quaternion.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// assert_eq!(q.scalar(), 1.0);
        /// ```
        #[inline]
        #[must_use]
        pub fn scalar(&self) -> T {
            self.coords[3].clone()
        }

        /// Reinterprets this quaternion as a 4D vector.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{Vector4, Quaternion};
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// // Recall that the quaternion is stored internally as (i, j, k, w)
        /// // while the crate::new constructor takes the arguments as (w, i, j, k).
        /// assert_eq!(*q.as_vector(), Vector4::new(2.0, 3.0, 4.0, 1.0));
        /// ```
        #[inline]
        #[must_use]
        pub fn as_vector(&self) -> &Vector4<T> {
            &self.coords
        }

        /// The norm of this quaternion.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// assert_relative_eq!(q.norm(), 5.47722557, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn norm(&self) -> T {
            self.coords.norm()
        }

        /// A synonym for the norm of this quaternion.
        ///
        /// Aka the length.
        /// This is the same as `.norm()`
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// assert_relative_eq!(q.magnitude(), 5.47722557, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn magnitude(&self) -> T {
            self.norm()
        }

        /// The squared norm of this quaternion.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// assert_eq!(q.magnitude_squared(), 30.0);
        /// ```
        #[inline]
        #[must_use]
        pub fn norm_squared(&self) -> T {
            self.coords.norm_squared()
        }

        /// A synonym for the squared norm of this quaternion.
        ///
        /// Aka the squared length.
        /// This is the same as `.norm_squared()`
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// assert_eq!(q.magnitude_squared(), 30.0);
        /// ```
        #[inline]
        #[must_use]
        pub fn magnitude_squared(&self) -> T {
            self.norm_squared()
        }

        /// The dot product of two quaternions.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let q2 = Quaternion::new(5.0, 6.0, 7.0, 8.0);
        /// assert_eq!(q1.dot(&q2), 70.0);
        /// ```
        #[inline]
        #[must_use]
        pub fn dot(&self, rhs: &Self) -> T {
            self.coords.dot(&rhs.coords)
        }
    }

    impl<T: SimdRealField> Quaternion<T>
    where
        T::Element: SimdRealField,
    {
        /// Inverts this quaternion if it is not zero.
        ///
        /// This method also does not works with SIMD components (see `simd_try_inverse` instead).
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let inv_q = q.try_inverse();
        ///
        /// assert!(inv_q.is_some());
        /// assert_relative_eq!(inv_q.unwrap() * q, Quaternion::identity());
        ///
        /// //Non-invertible case
        /// let q = Quaternion::new(0.0, 0.0, 0.0, 0.0);
        /// let inv_q = q.try_inverse();
        ///
        /// assert!(inv_q.is_none());
        /// ```
        #[inline]
        #[must_use = "Did you mean to use try_inverse_mut()?"]
        pub fn try_inverse(&self) -> Option<Self>
        where
            T: RealField,
        {
            let mut res = self.clone();

            if res.try_inverse_mut() {
                Some(res)
            } else {
                None
            }
        }

        /// Attempt to inverse this quaternion.
        ///
        /// This method also works with SIMD components.
        #[inline]
        #[must_use = "Did you mean to use try_inverse_mut()?"]
        pub fn simd_try_inverse(&self) -> SimdOption<Self> {
            let norm_squared = self.norm_squared();
            let ge = norm_squared.clone().simd_ge(T::simd_default_epsilon());
            SimdOption::new(self.conjugate() / norm_squared, ge)
        }

        /// Calculates the inner product (also known as the dot product).
        /// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
        /// Formula 4.89.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
        /// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
        /// let expected = Quaternion::new(-20.0, 0.0, 0.0, 0.0);
        /// let result = a.inner(&b);
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
        /// ```
        #[inline]
        #[must_use]
        pub fn inner(&self, other: &Self) -> Self {
            (self * other + other * self).half()
        }

        /// Calculates the outer product (also known as the wedge product).
        /// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
        /// Formula 4.89.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
        /// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
        /// let expected = Quaternion::new(0.0, -5.0, 18.0, -11.0);
        /// let result = a.outer(&b);
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
        /// ```
        #[inline]
        #[must_use]
        pub fn outer(&self, other: &Self) -> Self {
            #[allow(clippy::eq_op)]
            (self * other - other * self).half()
        }

        /// Calculates the projection of `self` onto `other` (also known as the parallel).
        /// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
        /// Formula 4.94.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
        /// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
        /// let expected = Quaternion::new(0.0, 3.333333333333333, 1.3333333333333333, 0.6666666666666666);
        /// let result = a.project(&b).unwrap();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
        /// ```
        #[inline]
        #[must_use]
        pub fn project(&self, other: &Self) -> Option<Self>
        where
            T: RealField,
        {
            self.inner(other).right_div(other)
        }

        /// Calculates the rejection of `self` from `other` (also known as the perpendicular).
        /// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
        /// Formula 4.94.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
        /// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
        /// let expected = Quaternion::new(0.0, -1.3333333333333333, 1.6666666666666665, 3.3333333333333335);
        /// let result = a.reject(&b).unwrap();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
        /// ```
        #[inline]
        #[must_use]
        pub fn reject(&self, other: &Self) -> Option<Self>
        where
            T: RealField,
        {
            self.outer(other).right_div(other)
        }

        /// The polar decomposition of this quaternion.
        ///
        /// Returns, from left to right: the quaternion norm, the half rotation angle, the rotation
        /// axis. If the rotation angle is zero, the rotation axis is set to `None`.
        ///
        /// # Example
        /// ```
        /// # use std::f32;
        /// # use nalgebra::{Vector3, Quaternion};
        /// let q = Quaternion::new(0.0, 5.0, 0.0, 0.0);
        /// let (norm, half_ang, axis) = q.polar_decomposition();
        /// assert_eq!(norm, 5.0);
        /// assert_eq!(half_ang, f32::consts::FRAC_PI_2);
        /// assert_eq!(axis, Some(Vector3::x_axis()));
        /// ```
        #[must_use]
        pub fn polar_decomposition(&self) -> (T, T, Option<Unit<Vector3<T>>>)
        where
            T: RealField,
        {
            if let Some((q, n)) = Unit::try_new_and_get(self.clone(), T::zero()) {
                if let Some(axis) = Unit::try_new(self.vector().clone_owned(), T::zero()) {
                    let angle = q.angle() / crate::convert(2.0f64);

                    (n, angle, Some(axis))
                } else {
                    (n, T::zero(), None)
                }
            } else {
                (T::zero(), T::zero(), None)
            }
        }

        /// Compute the natural logarithm of a quaternion.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(2.0, 5.0, 0.0, 0.0);
        /// assert_relative_eq!(q.ln(), Quaternion::new(1.683647, 1.190289, 0.0, 0.0), epsilon = 1.0e-6)
        /// ```
        #[inline]
        #[must_use]
        pub fn ln(&self) -> Self {
            let n = self.norm();
            let v = self.vector();
            let s = self.scalar();

            Self::from_parts(n.clone().simd_ln(), v.normalize() * (s / n).simd_acos())
        }

        /// Compute the exponential of a quaternion.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);
        /// assert_relative_eq!(q.exp(), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5)
        /// ```
        #[inline]
        #[must_use]
        pub fn exp(&self) -> Self {
            self.exp_eps(T::simd_default_epsilon())
        }

        /// Compute the exponential of a quaternion. Returns the identity if the vector part of this quaternion
        /// has a norm smaller than `eps`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);
        /// assert_relative_eq!(q.exp_eps(1.0e-6), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5);
        ///
        /// // Singular case.
        /// let q = Quaternion::new(0.0000001, 0.0, 0.0, 0.0);
        /// assert_eq!(q.exp_eps(1.0e-6), Quaternion::identity());
        /// ```
        #[inline]
        #[must_use]
        pub fn exp_eps(&self, eps: T) -> Self {
            let v = self.vector();
            let nn = v.norm_squared();
            let le = nn.clone().simd_le(eps.clone() * eps);
            le.if_else(Self::identity, || {
                let w_exp = self.scalar().simd_exp();
                let n = nn.simd_sqrt();
                let nv = v * (w_exp.clone() * n.clone().simd_sin() / n.clone());

                Self::from_parts(w_exp * n.simd_cos(), nv)
            })
        }

        /// Raise the quaternion to a given floating power.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// assert_relative_eq!(q.powf(1.5), Quaternion::new( -6.2576659, 4.1549037, 6.2323556, 8.3098075), epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn powf(&self, n: T) -> Self {
            (self.ln() * n).exp()
        }

        /// Transforms this quaternion into its 4D vector form (Vector part, Scalar part).
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{Quaternion, Vector4};
        /// let mut q = Quaternion::identity();
        /// *q.as_vector_mut() = Vector4::new(1.0, 2.0, 3.0, 4.0);
        /// assert!(q.i == 1.0 && q.j == 2.0 && q.k == 3.0 && q.w == 4.0);
        /// ```
        #[inline]
        pub fn as_vector_mut(&mut self) -> &mut Vector4<T> {
            &mut self.coords
        }

        /// The mutable vector part `(i, j, k)` of this quaternion.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{Quaternion, Vector4};
        /// let mut q = Quaternion::identity();
        /// {
        ///     let mut v = q.vector_mut();
        ///     v[0] = 2.0;
        ///     v[1] = 3.0;
        ///     v[2] = 4.0;
        /// }
        /// assert!(q.i == 2.0 && q.j == 3.0 && q.k == 4.0 && q.w == 1.0);
        /// ```
        #[inline]
        pub fn vector_mut(
            &mut self,
        ) -> MatrixViewMut<'_, T, U3, U1, RStride<T, U4, U1>, CStride<T, U4, U1>> {
            self.coords.fixed_rows_mut::<3>(0)
        }

        /// Replaces this quaternion by its conjugate.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::Quaternion;
        /// let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// q.conjugate_mut();
        /// assert!(q.i == -2.0 && q.j == -3.0 && q.k == -4.0 && q.w == 1.0);
        /// ```
        #[inline]
        pub fn conjugate_mut(&mut self) {
            self.coords[0] = -self.coords[0].clone();
            self.coords[1] = -self.coords[1].clone();
            self.coords[2] = -self.coords[2].clone();
        }

        /// Inverts this quaternion in-place if it is not zero.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let mut q = Quaternion::new(1.0f32, 2.0, 3.0, 4.0);
        ///
        /// assert!(q.try_inverse_mut());
        /// assert_relative_eq!(q * Quaternion::new(1.0, 2.0, 3.0, 4.0), Quaternion::identity());
        ///
        /// //Non-invertible case
        /// let mut q = Quaternion::new(0.0f32, 0.0, 0.0, 0.0);
        /// assert!(!q.try_inverse_mut());
        /// ```
        #[inline]
        pub fn try_inverse_mut(&mut self) -> T::SimdBool {
            let norm_squared = self.norm_squared();
            let ge = norm_squared.clone().simd_ge(T::simd_default_epsilon());
            *self = ge.if_else(|| self.conjugate() / norm_squared, || self.clone());
            ge
        }

        /// Normalizes this quaternion.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// q.normalize_mut();
        /// assert_relative_eq!(q.norm(), 1.0);
        /// ```
        #[inline]
        pub fn normalize_mut(&mut self) -> T {
            self.coords.normalize_mut()
        }

        /// Calculates square of a quaternion.
        #[inline]
        #[must_use]
        pub fn squared(&self) -> Self {
            self * self
        }

        /// Divides quaternion into two.
        #[inline]
        #[must_use]
        pub fn half(&self) -> Self {
            self / crate::convert(2.0f64)
        }

        /// Calculates square root.
        #[inline]
        #[must_use]
        pub fn sqrt(&self) -> Self {
            self.powf(crate::convert(0.5))
        }

        /// Check if the quaternion is pure.
        ///
        /// A quaternion is pure if it has no real part (`self.w == 0.0`).
        #[inline]
        #[must_use]
        pub fn is_pure(&self) -> bool {
            self.w.is_zero()
        }

        /// Convert quaternion to pure quaternion.
        #[inline]
        #[must_use]
        pub fn pure(&self) -> Self {
            Self::from_imag(self.imag())
        }

        /// Left quaternionic division.
        ///
        /// Calculates B<sup>-1</sup> * A where A = self, B = other.
        #[inline]
        #[must_use]
        pub fn left_div(&self, other: &Self) -> Option<Self>
        where
            T: RealField,
        {
            other.try_inverse().map(|inv| inv * self)
        }

        /// Right quaternionic division.
        ///
        /// Calculates A * B<sup>-1</sup> where A = self, B = other.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let a = Quaternion::new(0.0, 1.0, 2.0, 3.0);
        /// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
        /// let result = a.right_div(&b).unwrap();
        /// let expected = Quaternion::new(0.4, 0.13333333333333336, -0.4666666666666667, 0.26666666666666666);
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn right_div(&self, other: &Self) -> Option<Self>
        where
            T: RealField,
        {
            other.try_inverse().map(|inv| self * inv)
        }

        /// Calculates the quaternionic cosinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(58.93364616794395, -34.086183690465596, -51.1292755356984, -68.17236738093119);
        /// let result = input.cos();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn cos(&self) -> Self {
            let z = self.imag().magnitude();
            let w = -self.w.clone().simd_sin() * z.clone().simd_sinhc();
            Self::from_parts(self.w.clone().simd_cos() * z.simd_cosh(), self.imag() * w)
        }

        /// Calculates the quaternionic arccosinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let result = input.cos().acos();
        /// assert_relative_eq!(input, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn acos(&self) -> Self {
            let u = Self::from_imag(self.imag().normalize());
            let identity = Self::identity();

            let z = (self + (self.squared() - identity).sqrt()).ln();

            -(u * z)
        }

        /// Calculates the quaternionic sinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(91.78371578403467, 21.886486853029176, 32.82973027954377, 43.77297370605835);
        /// let result = input.sin();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn sin(&self) -> Self {
            let z = self.imag().magnitude();
            let w = self.w.clone().simd_cos() * z.clone().simd_sinhc();
            Self::from_parts(self.w.clone().simd_sin() * z.simd_cosh(), self.imag() * w)
        }

        /// Calculates the quaternionic arcsinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let result = input.sin().asin();
        /// assert_relative_eq!(input, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn asin(&self) -> Self {
            let u = Self::from_imag(self.imag().normalize());
            let identity = Self::identity();

            let z = ((u.clone() * self) + (identity - self.squared()).sqrt()).ln();

            -(u * z)
        }

        /// Calculates the quaternionic tangent.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(0.00003821631725009489, 0.3713971716439371, 0.5570957574659058, 0.7427943432878743);
        /// let result = input.tan();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn tan(&self) -> Self
        where
            T: RealField,
        {
            self.sin().right_div(&self.cos()).unwrap()
        }

        /// Calculates the quaternionic arctangent.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let result = input.tan().atan();
        /// assert_relative_eq!(input, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn atan(&self) -> Self
        where
            T: RealField,
        {
            let u = Self::from_imag(self.imag().normalize());
            let num = u.clone() + self;
            let den = u.clone() - self;
            let fr = num.right_div(&den).unwrap();
            let ln = fr.ln();
            (u.half()) * ln
        }

        /// Calculates the hyperbolic quaternionic sinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(0.7323376060463428, -0.4482074499805421, -0.6723111749708133, -0.8964148999610843);
        /// let result = input.sinh();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn sinh(&self) -> Self {
            (self.exp() - (-self).exp()).half()
        }

        /// Calculates the hyperbolic quaternionic arcsinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(2.385889902585242, 0.514052600662788, 0.7710789009941821, 1.028105201325576);
        /// let result = input.asinh();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn asinh(&self) -> Self {
            let identity = Self::identity();
            (self + (identity + self.squared()).sqrt()).ln()
        }

        /// Calculates the hyperbolic quaternionic cosinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(0.9615851176369566, -0.3413521745610167, -0.5120282618415251, -0.6827043491220334);
        /// let result = input.cosh();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn cosh(&self) -> Self {
            (self.exp() + (-self).exp()).half()
        }

        /// Calculates the hyperbolic quaternionic arccosinus.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(2.4014472020074007, 0.5162761016176176, 0.7744141524264264, 1.0325522032352352);
        /// let result = input.acosh();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn acosh(&self) -> Self {
            let identity = Self::identity();
            (self + (self + identity.clone()).sqrt() * (self - identity).sqrt()).ln()
        }

        /// Calculates the hyperbolic quaternionic tangent.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(1.0248695360556623, -0.10229568178876419, -0.1534435226831464, -0.20459136357752844);
        /// let result = input.tanh();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn tanh(&self) -> Self
        where
            T: RealField,
        {
            self.sinh().right_div(&self.cosh()).unwrap()
        }

        /// Calculates the hyperbolic quaternionic arctangent.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::Quaternion;
        /// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
        /// let expected = Quaternion::new(0.03230293287000163, 0.5173453683196951, 0.7760180524795426, 1.0346907366393903);
        /// let result = input.atanh();
        /// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
        /// ```
        #[inline]
        #[must_use]
        pub fn atanh(&self) -> Self {
            let identity = Self::identity();
            ((identity.clone() + self).ln() - (identity - self).ln()).half()
        }
    }

    impl<T: RealField + AbsDiffEq<Epsilon = T>> AbsDiffEq for Quaternion<T> {
        type Epsilon = T;

        #[inline]
        fn default_epsilon() -> Self::Epsilon {
            T::default_epsilon()
        }

        #[inline]
        fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool {
            self.as_vector().abs_diff_eq(other.as_vector(), epsilon.clone()) ||
            // Account for the double-covering of S², i.e. q = -q
            self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.abs_diff_eq(&-b.clone(), epsilon.clone()))
        }
    }

    impl<T: RealField + RelativeEq<Epsilon = T>> RelativeEq for Quaternion<T> {
        #[inline]
        fn default_max_relative() -> Self::Epsilon {
            T::default_max_relative()
        }

        #[inline]
        fn relative_eq(
            &self,
            other: &Self,
            epsilon: Self::Epsilon,
            max_relative: Self::Epsilon,
        ) -> bool {
            self.as_vector().relative_eq(other.as_vector(), epsilon.clone(), max_relative.clone()) ||
            // Account for the double-covering of S², i.e. q = -q
            self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.relative_eq(&-b.clone(), epsilon.clone(), max_relative.clone()))
        }
    }

    impl<T: RealField + UlpsEq<Epsilon = T>> UlpsEq for Quaternion<T> {
        #[inline]
        fn default_max_ulps() -> u32 {
            T::default_max_ulps()
        }

        #[inline]
        fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
            self.as_vector().ulps_eq(other.as_vector(), epsilon.clone(), max_ulps) ||
            // Account for the double-covering of S², i.e. q = -q.
            self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.ulps_eq(&-b.clone(), epsilon.clone(), max_ulps))
        }
    }

    impl<T: RealField + fmt::Display> fmt::Display for Quaternion<T> {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(
                f,
                "Quaternion {} − ({}, {}, {})",
                self[3], self[0], self[1], self[2]
            )
        }
    }

    /// A unit quaternions. May be used to represent a rotation.
    pub type UnitQuaternion<T> = Unit<Quaternion<T>>;

    #[cfg(feature = "cuda")]
    unsafe impl<T: cust_core::DeviceCopy> cust_core::DeviceCopy for UnitQuaternion<T> {}

    impl<T: Scalar + ClosedNeg + PartialEq> PartialEq for UnitQuaternion<T> {
        #[inline]
        fn eq(&self, rhs: &Self) -> bool {
            self.coords == rhs.coords ||
            // Account for the double-covering of S², i.e. q = -q
            self.coords.iter().zip(rhs.coords.iter()).all(|(a, b)| *a == -b.clone())
        }
    }

    impl<T: Scalar + ClosedNeg + Eq> Eq for UnitQuaternion<T> {}

    impl<T: SimdRealField> Normed for Quaternion<T> {
        type Norm = T::SimdRealField;

        #[inline]
        fn norm(&self) -> T::SimdRealField {
            self.coords.norm()
        }

        #[inline]
        fn norm_squared(&self) -> T::SimdRealField {
            self.coords.norm_squared()
        }

        #[inline]
        fn scale_mut(&mut self, n: Self::Norm) {
            self.coords.scale_mut(n)
        }

        #[inline]
        fn unscale_mut(&mut self, n: Self::Norm) {
            self.coords.unscale_mut(n)
        }
    }

    impl<T: SimdRealField> UnitQuaternion<T>
    where
        T::Element: SimdRealField,
    {
        /// The rotation angle in [0; pi] of this unit quaternion.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{Unit, UnitQuaternion, Vector3};
        /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
        /// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);
        /// assert_eq!(rot.angle(), 1.78);
        /// ```
        #[inline]
        #[must_use]
        pub fn angle(&self) -> T {
            let w = self.quaternion().scalar().simd_abs();
            self.quaternion().imag().norm().simd_atan2(w) * crate::convert(2.0f64)
        }

        /// The underlying quaternion.
        ///
        /// Same as `self.as_ref()`.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{UnitQuaternion, Quaternion};
        /// let axis = UnitQuaternion::identity();
        /// assert_eq!(*axis.quaternion(), Quaternion::new(1.0, 0.0, 0.0, 0.0));
        /// ```
        #[inline]
        #[must_use]
        pub fn quaternion(&self) -> &Quaternion<T> {
            self.as_ref()
        }

        /// Compute the conjugate of this unit quaternion.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{Unit, UnitQuaternion, Vector3};
        /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
        /// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);
        /// let conj = rot.conjugate();
        /// assert_eq!(conj, UnitQuaternion::from_axis_angle(&-axis, 1.78));
        /// ```
        #[inline]
        #[must_use = "Did you mean to use conjugate_mut()?"]
        pub fn conjugate(&self) -> Self {
            Self::new_unchecked(self.as_ref().conjugate())
        }

        /// Inverts this quaternion if it is not zero.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{Unit, UnitQuaternion, Vector3};
        /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
        /// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);
        /// let inv = rot.inverse();
        /// assert_eq!(rot * inv, UnitQuaternion::identity());
        /// assert_eq!(inv * rot, UnitQuaternion::identity());
        /// ```
        #[inline]
        #[must_use = "Did you mean to use inverse_mut()?"]
        pub fn inverse(&self) -> Self {
            self.conjugate()
        }

        /// The rotation angle needed to make `self` and `other` coincide.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitQuaternion, Vector3};
        /// let rot1 = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), 1.0);
        /// let rot2 = UnitQuaternion::from_axis_angle(&Vector3::x_axis(), 0.1);
        /// assert_relative_eq!(rot1.angle_to(&rot2), 1.0045657, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn angle_to(&self, other: &Self) -> T {
            let delta = self.rotation_to(other);
            delta.angle()
        }

        /// The unit quaternion needed to make `self` and `other` coincide.
        ///
        /// The result is such that: `self.rotation_to(other) * self == other`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitQuaternion, Vector3};
        /// let rot1 = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), 1.0);
        /// let rot2 = UnitQuaternion::from_axis_angle(&Vector3::x_axis(), 0.1);
        /// let rot_to = rot1.rotation_to(&rot2);
        /// assert_relative_eq!(rot_to * rot1, rot2, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn rotation_to(&self, other: &Self) -> Self {
            other / self
        }

        /// Linear interpolation between two unit quaternions.
        ///
        /// The result is not normalized.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{UnitQuaternion, Quaternion};
        /// let q1 = UnitQuaternion::new_normalize(Quaternion::new(1.0, 0.0, 0.0, 0.0));
        /// let q2 = UnitQuaternion::new_normalize(Quaternion::new(0.0, 1.0, 0.0, 0.0));
        /// assert_eq!(q1.lerp(&q2, 0.1), Quaternion::new(0.9, 0.1, 0.0, 0.0));
        /// ```
        #[inline]
        #[must_use]
        pub fn lerp(&self, other: &Self, t: T) -> Quaternion<T> {
            self.as_ref().lerp(other.as_ref(), t)
        }

        /// Normalized linear interpolation between two unit quaternions.
        ///
        /// This is the same as `self.lerp` except that the result is normalized.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{UnitQuaternion, Quaternion};
        /// let q1 = UnitQuaternion::new_normalize(Quaternion::new(1.0, 0.0, 0.0, 0.0));
        /// let q2 = UnitQuaternion::new_normalize(Quaternion::new(0.0, 1.0, 0.0, 0.0));
        /// assert_eq!(q1.nlerp(&q2, 0.1), UnitQuaternion::new_normalize(Quaternion::new(0.9, 0.1, 0.0, 0.0)));
        /// ```
        #[inline]
        #[must_use]
        pub fn nlerp(&self, other: &Self, t: T) -> Self {
            let mut res = self.lerp(other, t);
            let _ = res.normalize_mut();

            Self::new_unchecked(res)
        }

        /// Spherical linear interpolation between two unit quaternions.
        ///
        /// Panics if the angle between both quaternion is 180 degrees (in which case the interpolation
        /// is not well-defined). Use `.try_slerp` instead to avoid the panic.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::geometry::UnitQuaternion;
        ///
        /// let q1 = UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_4, 0.0, 0.0);
        /// let q2 = UnitQuaternion::from_euler_angles(-std::f32::consts::PI, 0.0, 0.0);
        ///
        /// let q = q1.slerp(&q2, 1.0 / 3.0);
        ///
        /// assert_eq!(q.euler_angles(), (std::f32::consts::FRAC_PI_2, 0.0, 0.0));
        /// ```
        #[inline]
        #[must_use]
        pub fn slerp(&self, other: &Self, t: T) -> Self
        where
            T: RealField,
        {
            self.try_slerp(other, t, T::default_epsilon())
                .expect("Quaternion slerp: ambiguous configuration.")
        }

        /// Computes the spherical linear interpolation between two unit quaternions or returns `None`
        /// if both quaternions are approximately 180 degrees apart (in which case the interpolation is
        /// not well-defined).
        ///
        /// # Arguments
        /// * `self`: the first quaternion to interpolate from.
        /// * `other`: the second quaternion to interpolate toward.
        /// * `t`: the interpolation parameter. Should be between 0 and 1.
        /// * `epsilon`: the value below which the sinus of the angle separating both quaternion
        /// must be to return `None`.
        #[inline]
        #[must_use]
        pub fn try_slerp(&self, other: &Self, t: T, epsilon: T) -> Option<Self>
        where
            T: RealField,
        {
            let coords = if self.coords.dot(&other.coords) < T::zero() {
                Unit::new_unchecked(self.coords.clone()).try_slerp(
                    &Unit::new_unchecked(-other.coords.clone()),
                    t,
                    epsilon,
                )
            } else {
                Unit::new_unchecked(self.coords.clone()).try_slerp(
                    &Unit::new_unchecked(other.coords.clone()),
                    t,
                    epsilon,
                )
            };

            coords.map(|q| Unit::new_unchecked(Quaternion::from(q.into_inner())))
        }

        /// Compute the conjugate of this unit quaternion in-place.
        #[inline]
        pub fn conjugate_mut(&mut self) {
            self.as_mut_unchecked().conjugate_mut()
        }

        /// Inverts this quaternion if it is not zero.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitQuaternion, Vector3, Unit};
        /// let axisangle = Vector3::new(0.1, 0.2, 0.3);
        /// let mut rot = UnitQuaternion::new(axisangle);
        /// rot.inverse_mut();
        /// assert_relative_eq!(rot * UnitQuaternion::new(axisangle), UnitQuaternion::identity());
        /// assert_relative_eq!(UnitQuaternion::new(axisangle) * rot, UnitQuaternion::identity());
        /// ```
        #[inline]
        pub fn inverse_mut(&mut self) {
            self.as_mut_unchecked().conjugate_mut()
        }

        /// The rotation axis of this unit quaternion or `None` if the rotation is zero.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{UnitQuaternion, Vector3, Unit};
        /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
        /// let angle = 1.2;
        /// let rot = UnitQuaternion::from_axis_angle(&axis, angle);
        /// assert_eq!(rot.axis(), Some(axis));
        ///
        /// // Case with a zero angle.
        /// let rot = UnitQuaternion::from_axis_angle(&axis, 0.0);
        /// assert!(rot.axis().is_none());
        /// ```
        #[inline]
        #[must_use]
        pub fn axis(&self) -> Option<Unit<Vector3<T>>>
        where
            T: RealField,
        {
            let v = if self.quaternion().scalar() >= T::zero() {
                self.as_ref().vector().clone_owned()
            } else {
                -self.as_ref().vector()
            };

            Unit::try_new(v, T::zero())
        }

        /// The rotation axis of this unit quaternion multiplied by the rotation angle.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitQuaternion, Vector3, Unit};
        /// let axisangle = Vector3::new(0.1, 0.2, 0.3);
        /// let rot = UnitQuaternion::new(axisangle);
        /// assert_relative_eq!(rot.scaled_axis(), axisangle, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn scaled_axis(&self) -> Vector3<T>
        where
            T: RealField,
        {
            if let Some(axis) = self.axis() {
                axis.into_inner() * self.angle()
            } else {
                Vector3::zero()
            }
        }

        /// The rotation axis and angle in (0, pi] of this unit quaternion.
        ///
        /// Returns `None` if the angle is zero.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{UnitQuaternion, Vector3, Unit};
        /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
        /// let angle = 1.2;
        /// let rot = UnitQuaternion::from_axis_angle(&axis, angle);
        /// assert_eq!(rot.axis_angle(), Some((axis, angle)));
        ///
        /// // Case with a zero angle.
        /// let rot = UnitQuaternion::from_axis_angle(&axis, 0.0);
        /// assert!(rot.axis_angle().is_none());
        /// ```
        #[inline]
        #[must_use]
        pub fn axis_angle(&self) -> Option<(Unit<Vector3<T>>, T)>
        where
            T: RealField,
        {
            self.axis().map(|axis| (axis, self.angle()))
        }

        /// Compute the exponential of a quaternion.
        ///
        /// Note that this function yields a `Quaternion<T>` because it loses the unit property.
        #[inline]
        #[must_use]
        pub fn exp(&self) -> Quaternion<T> {
            self.as_ref().exp()
        }

        /// Compute the natural logarithm of a quaternion.
        ///
        /// Note that this function yields a `Quaternion<T>` because it loses the unit property.
        /// The vector part of the return value corresponds to the axis-angle representation (divided
        /// by 2.0) of this unit quaternion.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{Vector3, UnitQuaternion};
        /// let axisangle = Vector3::new(0.1, 0.2, 0.3);
        /// let q = UnitQuaternion::new(axisangle);
        /// assert_relative_eq!(q.ln().vector().into_owned(), axisangle, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn ln(&self) -> Quaternion<T>
        where
            T: RealField,
        {
            if let Some(v) = self.axis() {
                Quaternion::from_imag(v.into_inner() * self.angle())
            } else {
                Quaternion::zero()
            }
        }

        /// Raise the quaternion to a given floating power.
        ///
        /// This returns the unit quaternion that identifies a rotation with axis `self.axis()` and
        /// angle `self.angle() × n`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitQuaternion, Vector3, Unit};
        /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
        /// let angle = 1.2;
        /// let rot = UnitQuaternion::from_axis_angle(&axis, angle);
        /// let pow = rot.powf(2.0);
        /// assert_relative_eq!(pow.axis().unwrap(), axis, epsilon = 1.0e-6);
        /// assert_eq!(pow.angle(), 2.4);
        /// ```
        #[inline]
        #[must_use]
        pub fn powf(&self, n: T) -> Self
        where
            T: RealField,
        {
            if let Some(v) = self.axis() {
                Self::from_axis_angle(&v, self.angle() * n)
            } else {
                Self::identity()
            }
        }

        /// Builds a rotation matrix from this unit quaternion.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use std::f32;
        /// # use nalgebra::{UnitQuaternion, Vector3, Matrix3};
        /// let q = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);
        /// let rot = q.to_rotation_matrix();
        /// let expected = Matrix3::new(0.8660254, -0.5,      0.0,
        ///                             0.5,       0.8660254, 0.0,
        ///                             0.0,       0.0,       1.0);
        ///
        /// assert_relative_eq!(*rot.matrix(), expected, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn to_rotation_matrix(self) -> Rotation<T, 3> {
            let i = self.as_ref()[0].clone();
            let j = self.as_ref()[1].clone();
            let k = self.as_ref()[2].clone();
            let w = self.as_ref()[3].clone();

            let ww = w.clone() * w.clone();
            let ii = i.clone() * i.clone();
            let jj = j.clone() * j.clone();
            let kk = k.clone() * k.clone();
            let ij = i.clone() * j.clone() * crate::convert(2.0f64);
            let wk = w.clone() * k.clone() * crate::convert(2.0f64);
            let wj = w.clone() * j.clone() * crate::convert(2.0f64);
            let ik = i.clone() * k.clone() * crate::convert(2.0f64);
            let jk = j * k * crate::convert(2.0f64);
            let wi = w * i * crate::convert(2.0f64);

            Rotation::from_matrix_unchecked(Matrix3::new(
                ww.clone() + ii.clone() - jj.clone() - kk.clone(),
                ij.clone() - wk.clone(),
                wj.clone() + ik.clone(),
                wk + ij,
                ww.clone() - ii.clone() + jj.clone() - kk.clone(),
                jk.clone() - wi.clone(),
                ik - wj,
                wi + jk,
                ww - ii - jj + kk,
            ))
        }

        /// Converts this unit quaternion into its equivalent Euler angles.
        ///
        /// The angles are produced in the form (roll, pitch, yaw).
        #[inline]
        #[deprecated(note = "This is renamed to use `.euler_angles()`.")]
        pub fn to_euler_angles(self) -> (T, T, T)
        where
            T: RealField,
        {
            self.euler_angles()
        }

        /// Retrieves the euler angles corresponding to this unit quaternion.
        ///
        /// The angles are produced in the form (roll, pitch, yaw).
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::UnitQuaternion;
        /// let rot = UnitQuaternion::from_euler_angles(0.1, 0.2, 0.3);
        /// let euler = rot.euler_angles();
        /// assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
        /// assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
        /// assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn euler_angles(&self) -> (T, T, T)
        where
            T: RealField,
        {
            self.clone().to_rotation_matrix().euler_angles()
        }

        /// Converts this unit quaternion into its equivalent homogeneous transformation matrix.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use std::f32;
        /// # use nalgebra::{UnitQuaternion, Vector3, Matrix4};
        /// let rot = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);
        /// let expected = Matrix4::new(0.8660254, -0.5,      0.0, 0.0,
        ///                             0.5,       0.8660254, 0.0, 0.0,
        ///                             0.0,       0.0,       1.0, 0.0,
        ///                             0.0,       0.0,       0.0, 1.0);
        ///
        /// assert_relative_eq!(rot.to_homogeneous(), expected, epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn to_homogeneous(self) -> Matrix4<T> {
            self.to_rotation_matrix().to_homogeneous()
        }

        /// Rotate a point by this unit quaternion.
        ///
        /// This is the same as the multiplication `self * pt`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use std::f32;
        /// # use nalgebra::{UnitQuaternion, Vector3, Point3};
        /// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
        /// let transformed_point = rot.transform_point(&Point3::new(1.0, 2.0, 3.0));
        ///
        /// assert_relative_eq!(transformed_point, Point3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn transform_point(&self, pt: &Point3<T>) -> Point3<T> {
            self * pt
        }

        /// Rotate a vector by this unit quaternion.
        ///
        /// This is the same as the multiplication `self * v`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use std::f32;
        /// # use nalgebra::{UnitQuaternion, Vector3};
        /// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
        /// let transformed_vector = rot.transform_vector(&Vector3::new(1.0, 2.0, 3.0));
        ///
        /// assert_relative_eq!(transformed_vector, Vector3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn transform_vector(&self, v: &Vector3<T>) -> Vector3<T> {
            self * v
        }

        /// Rotate a point by the inverse of this unit quaternion. This may be
        /// cheaper than inverting the unit quaternion and transforming the
        /// point.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use std::f32;
        /// # use nalgebra::{UnitQuaternion, Vector3, Point3};
        /// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
        /// let transformed_point = rot.inverse_transform_point(&Point3::new(1.0, 2.0, 3.0));
        ///
        /// assert_relative_eq!(transformed_point, Point3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn inverse_transform_point(&self, pt: &Point3<T>) -> Point3<T> {
            // TODO: would it be useful performancewise not to call inverse explicitly (i-e. implement
            // the inverse transformation explicitly here) ?
            self.inverse() * pt
        }

        /// Rotate a vector by the inverse of this unit quaternion. This may be
        /// cheaper than inverting the unit quaternion and transforming the
        /// vector.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use std::f32;
        /// # use nalgebra::{UnitQuaternion, Vector3};
        /// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
        /// let transformed_vector = rot.inverse_transform_vector(&Vector3::new(1.0, 2.0, 3.0));
        ///
        /// assert_relative_eq!(transformed_vector, Vector3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn inverse_transform_vector(&self, v: &Vector3<T>) -> Vector3<T> {
            self.inverse() * v
        }

        /// Rotate a vector by the inverse of this unit quaternion. This may be
        /// cheaper than inverting the unit quaternion and transforming the
        /// vector.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use std::f32;
        /// # use nalgebra::{UnitQuaternion, Vector3};
        /// let rot = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_2);
        /// let transformed_vector = rot.inverse_transform_unit_vector(&Vector3::x_axis());
        ///
        /// assert_relative_eq!(transformed_vector, -Vector3::y_axis(), epsilon = 1.0e-6);
        /// ```
        #[inline]
        #[must_use]
        pub fn inverse_transform_unit_vector(&self, v: &Unit<Vector3<T>>) -> Unit<Vector3<T>> {
            self.inverse() * v
        }

        /// Appends to `self` a rotation given in the axis-angle form, using a linearized formulation.
        ///
        /// This is faster, but approximate, way to compute `UnitQuaternion::new(axisangle) * self`.
        #[inline]
        #[must_use]
        pub fn append_axisangle_linearized(&self, axisangle: &Vector3<T>) -> Self {
            let half: T = crate::convert(0.5);
            let q1 = self.clone().into_inner();
            let q2 = Quaternion::from_imag(axisangle * half);
            Unit::new_normalize(&q1 + q2 * &q1)
        }
    }

    impl<T: RealField> Default for UnitQuaternion<T> {
        fn default() -> Self {
            Self::identity()
        }
    }

    impl<T: RealField + fmt::Display> fmt::Display for UnitQuaternion<T> {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            if let Some(axis) = self.axis() {
                let axis = axis.into_inner();
                write!(
                    f,
                    "UnitQuaternion angle: {} − axis: ({}, {}, {})",
                    self.angle(),
                    axis[0],
                    axis[1],
                    axis[2]
                )
            } else {
                write!(
                    f,
                    "UnitQuaternion angle: {} − axis: (undefined)",
                    self.angle()
                )
            }
        }
    }

    impl<T: RealField + AbsDiffEq<Epsilon = T>> AbsDiffEq for UnitQuaternion<T> {
        type Epsilon = T;

        #[inline]
        fn default_epsilon() -> Self::Epsilon {
            T::default_epsilon()
        }

        #[inline]
        fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool {
            self.as_ref().abs_diff_eq(other.as_ref(), epsilon)
        }
    }

    impl<T: RealField + RelativeEq<Epsilon = T>> RelativeEq for UnitQuaternion<T> {
        #[inline]
        fn default_max_relative() -> Self::Epsilon {
            T::default_max_relative()
        }

        #[inline]
        fn relative_eq(
            &self,
            other: &Self,
            epsilon: Self::Epsilon,
            max_relative: Self::Epsilon,
        ) -> bool {
            self.as_ref()
                .relative_eq(other.as_ref(), epsilon, max_relative)
        }
    }

    impl<T: RealField + UlpsEq<Epsilon = T>> UlpsEq for UnitQuaternion<T> {
        #[inline]
        fn default_max_ulps() -> u32 {
            T::default_max_ulps()
        }

        #[inline]
        fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
            self.as_ref().ulps_eq(other.as_ref(), epsilon, max_ulps)
        }
    }