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    // Copyright 2016 The CGMath Developers. For a full listing of the authors,
    // refer to the Cargo.toml file at the top-level directory of this distribution.
    //
    // Licensed under the Apache License, Version 2.0 (the "License");
    // you may not use this file except in compliance with the License.
    // You may obtain a copy of the License at
    //
    //     http://www.apache.org/licenses/LICENSE-2.0
    //
    // Unless required by applicable law or agreed to in writing, software
    // distributed under the License is distributed on an "AS IS" BASIS,
    // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    // See the License for the specific language governing permissions and
    // limitations under the License.

    use rand::distributions::{Distribution, Standard};
    use rand::Rng;
    use num_traits::cast;

    use structure::*;

    use angle::Rad;
    use approx;
    use quaternion::Quaternion;
    #[cfg(feature = "mint")]
    use mint;
    use num::BaseFloat;

    /// A set of [Euler angles] representing a rotation in three-dimensional space.
    ///
    /// This type is marked as `#[repr(C)]`.
    ///
    /// The axis rotation sequence is XYZ. That is, the rotation is first around
    /// the X axis, then the Y axis, and lastly the Z axis (using intrinsic
    /// rotations). Since all three rotation axes are used, the angles are
    /// Tait–Bryan angles rather than proper Euler angles.
    ///
    /// # Ranges
    ///
    /// - x: [-pi, pi]
    /// - y: [-pi/2, pi/2]
    /// - z: [-pi, pi]
    ///
    /// # Defining rotations using Euler angles
    ///
    /// Note that while [Euler angles] are intuitive to define, they are prone to
    /// [gimbal lock] and are challenging to interpolate between. Instead we
    /// recommend that you convert them to a more robust representation, such as a
    /// quaternion or a rotation matrix. To this end, `From<Euler<A>>` conversions
    /// are provided for the following types:
    ///
    /// - [`Basis3`](struct.Basis3.html)
    /// - [`Matrix3`](struct.Matrix3.html)
    /// - [`Matrix4`](struct.Matrix4.html)
    /// - [`Quaternion`](struct.Quaternion.html)
    ///
    /// For example, to define a quaternion that applies the following:
    ///
    /// 1. a 90° rotation around the _x_ axis
    /// 2. a 45° rotation around the _y_ axis
    /// 3. a 15° rotation around the _z_ axis
    ///
    /// you can use the following code:
    ///
    /// ```
    /// use cgmath::{Deg, Euler, Quaternion};
    ///
    /// let rotation = Quaternion::from(Euler {
    ///     x: Deg(90.0),
    ///     y: Deg(45.0),
    ///     z: Deg(15.0),
    /// });
    /// ```
    ///
    /// [Euler angles]: https://en.wikipedia.org/wiki/Euler_angles
    /// [gimbal lock]: https://en.wikipedia.org/wiki/Gimbal_lock#Gimbal_lock_in_applied_mathematics
    /// [convert]: #defining-rotations-using-euler-angles
    #[repr(C)]
    #[derive(Copy, Clone, Debug, PartialEq, Eq)]
    #[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
    pub struct Euler<A> {
        /// The angle to apply around the _x_ axis. Also known at the _pitch_.
        pub x: A,
        /// The angle to apply around the _y_ axis. Also known at the _yaw_.
        pub y: A,
        /// The angle to apply around the _z_ axis. Also known at the _roll_.
        pub z: A,
    }

    impl<A> Euler<A> {
        /// Construct a set of euler angles.
        ///
        /// # Arguments
        ///
        /// * `x` - The angle to apply around the _x_ axis. Also known at the _pitch_.
        /// * `y` - The angle to apply around the _y_ axis. Also known at the _yaw_.
        /// * `z` - The angle to apply around the _z_ axis. Also known at the _roll_.
        pub const fn new(x: A, y: A, z: A) -> Euler<A> {
            Euler { x: x, y: y, z: z }
        }
    }

    impl<S: BaseFloat> From<Quaternion<S>> for Euler<Rad<S>> {
        fn from(src: Quaternion<S>) -> Euler<Rad<S>> {
            let sig: S = cast(0.499).unwrap();
            let two: S = cast(2).unwrap();
            let one: S = cast(1).unwrap();

            let (qw, qx, qy, qz) = (src.s, src.v.x, src.v.y, src.v.z);
            let (sqw, sqx, sqy, sqz) = (qw * qw, qx * qx, qy * qy, qz * qz);

            let unit = sqx + sqz + sqy + sqw;
            let test = qx * qz + qy * qw;

            // We set x to zero and z to the value, but the other way would work too.
            if test > sig * unit {
                // x + z = 2 * atan(x / w)
                Euler {
                    x: Rad::zero(),
                    y: Rad::turn_div_4(),
                    z: Rad::atan2(qx, qw) * two,
                }
            } else if test < -sig * unit {
                // x - z = 2 * atan(x / w)
                Euler {
                    x: Rad::zero(),
                    y: -Rad::turn_div_4(),
                    z: -Rad::atan2(qx, qw) * two,
                }
            } else {
                // Using the quat-to-matrix equation from either
                // http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToMatrix/index.htm
                // or equation 15 on page 7 of
                // http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770024290.pdf
                // to fill in the equations on page A-2 of the NASA document gives the below.
                Euler {
                    x: Rad::atan2(two * (-qy * qz + qx * qw), one - two * (sqx + sqy)),
                    y: Rad::asin(two * (qx * qz + qy * qw)),
                    z: Rad::atan2(two * (-qx * qy + qz * qw), one - two * (sqy + sqz)),
                }
            }
        }
    }

    impl<A: Angle> approx::AbsDiffEq for Euler<A> {
        type Epsilon = A::Epsilon;

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

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

    impl<A: Angle> approx::RelativeEq for Euler<A> {
        #[inline]
        fn default_max_relative() -> A::Epsilon {
            A::default_max_relative()
        }

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

    impl<A: Angle> approx::UlpsEq for Euler<A> {
        #[inline]
        fn default_max_ulps() -> u32 {
            A::default_max_ulps()
        }

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

    impl<A> Distribution<Euler<A>> for Standard
        where Standard: Distribution<A>,
            A: Angle {
        fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Euler<A> {
            Euler {
                x: rng.gen(),
                y: rng.gen(),
                z: rng.gen(),
            }
        }
    }

    #[cfg(feature = "mint")]
    type MintEuler<S> = mint::EulerAngles<S, mint::IntraXYZ>;

    #[cfg(feature = "mint")]
    impl<S, A: Angle + From<S>> From<MintEuler<S>> for Euler<A> {
        fn from(mint: MintEuler<S>) -> Self {
            Euler {
                x: mint.a.into(),
                y: mint.b.into(),
                z: mint.c.into(),
            }
        }
    }

    #[cfg(feature = "mint")]
    impl<S: Clone, A: Angle + Into<S>> Into<MintEuler<S>> for Euler<A> {
        fn into(self) -> MintEuler<S> {
            MintEuler::from([self.x.into(), self.y.into(), self.z.into()])
        }
    }

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