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    use approx::{AbsDiffEq, RelativeEq, UlpsEq};
    use num_complex::Complex;
    use std::fmt;

    use alga::general::RealField;
    use crate::base::{Matrix2, Matrix3, Unit, Vector1, Vector2};
    use crate::geometry::{Rotation2, Point2};

    /// A complex number with a norm equal to 1.
    pub type UnitComplex<N> = Unit<Complex<N>>;

    impl<N: RealField> UnitComplex<N> {
        /// The rotation angle in `]-pi; pi]` of this unit complex number.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::UnitComplex;
        /// let rot = UnitComplex::new(1.78);
        /// assert_eq!(rot.angle(), 1.78);
        /// ```
        #[inline]
        pub fn angle(&self) -> N {
            self.im.atan2(self.re)
        }

        /// The sine of the rotation angle.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::UnitComplex;
        /// let angle = 1.78f32;
        /// let rot = UnitComplex::new(angle);
        /// assert_eq!(rot.sin_angle(), angle.sin());
        /// ```
        #[inline]
        pub fn sin_angle(&self) -> N {
            self.im
        }

        /// The cosine of the rotation angle.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::UnitComplex;
        /// let angle = 1.78f32;
        /// let rot = UnitComplex::new(angle);
        /// assert_eq!(rot.cos_angle(),angle.cos());
        /// ```
        #[inline]
        pub fn cos_angle(&self) -> N {
            self.re
        }

        /// The rotation angle returned as a 1-dimensional vector.
        ///
        /// This is generally used in the context of generic programming. Using
        /// the `.angle()` method instead is more common.
        #[inline]
        pub fn scaled_axis(&self) -> Vector1<N> {
            Vector1::new(self.angle())
        }

        /// The rotation axis and angle in ]0, pi] of this complex number.
        ///
        /// This is generally used in the context of generic programming. Using
        /// the `.angle()` method instead is more common.
        /// Returns `None` if the angle is zero.
        #[inline]
        pub fn axis_angle(&self) -> Option<(Unit<Vector1<N>>, N)> {
            let ang = self.angle();

            if ang.is_zero() {
                None
            } else if ang.is_sign_negative() {
                Some((Unit::new_unchecked(Vector1::x()), -ang))
            } else {
                Some((Unit::new_unchecked(-Vector1::<N>::x()), ang))
            }
        }

        /// The underlying complex number.
        ///
        /// Same as `self.as_ref()`.
        ///
        /// # Example
        /// ```
        /// # extern crate num_complex;
        /// # use num_complex::Complex;
        /// # use nalgebra::UnitComplex;
        /// let angle = 1.78f32;
        /// let rot = UnitComplex::new(angle);
        /// assert_eq!(*rot.complex(), Complex::new(angle.cos(), angle.sin()));
        /// ```
        #[inline]
        pub fn complex(&self) -> &Complex<N> {
            self.as_ref()
        }

        /// Compute the conjugate of this unit complex number.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::UnitComplex;
        /// let rot = UnitComplex::new(1.78);
        /// let conj = rot.conjugate();
        /// assert_eq!(rot.complex().im, -conj.complex().im);
        /// assert_eq!(rot.complex().re, conj.complex().re);
        /// ```
        #[inline]
        pub fn conjugate(&self) -> Self {
            Self::new_unchecked(self.conj())
        }

        /// Inverts this complex number if it is not zero.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::UnitComplex;
        /// let rot = UnitComplex::new(1.2);
        /// let inv = rot.inverse();
        /// assert_relative_eq!(rot * inv, UnitComplex::identity(), epsilon = 1.0e-6);
        /// assert_relative_eq!(inv * rot, UnitComplex::identity(), epsilon = 1.0e-6);
        /// ```
        #[inline]
        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::UnitComplex;
        /// let rot1 = UnitComplex::new(0.1);
        /// let rot2 = UnitComplex::new(1.7);
        /// assert_relative_eq!(rot1.angle_to(&rot2), 1.6);
        /// ```
        #[inline]
        pub fn angle_to(&self, other: &Self) -> N {
            let delta = self.rotation_to(other);
            delta.angle()
        }

        /// The unit complex number 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::UnitComplex;
        /// let rot1 = UnitComplex::new(0.1);
        /// let rot2 = UnitComplex::new(1.7);
        /// let rot_to = rot1.rotation_to(&rot2);
        ///
        /// assert_relative_eq!(rot_to * rot1, rot2);
        /// assert_relative_eq!(rot_to.inverse() * rot2, rot1);
        /// ```
        #[inline]
        pub fn rotation_to(&self, other: &Self) -> Self {
            other / self
        }

        /// Compute in-place the conjugate of this unit complex number.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::UnitComplex;
        /// let angle = 1.7;
        /// let rot = UnitComplex::new(angle);
        /// let mut conj = UnitComplex::new(angle);
        /// conj.conjugate_mut();
        /// assert_eq!(rot.complex().im, -conj.complex().im);
        /// assert_eq!(rot.complex().re, conj.complex().re);
        /// ```
        #[inline]
        pub fn conjugate_mut(&mut self) {
            let me = self.as_mut_unchecked();
            me.im = -me.im;
        }

        /// Inverts in-place this unit complex number.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::UnitComplex;
        /// let angle = 1.7;
        /// let mut rot = UnitComplex::new(angle);
        /// rot.inverse_mut();
        /// assert_relative_eq!(rot * UnitComplex::new(angle), UnitComplex::identity());
        /// assert_relative_eq!(UnitComplex::new(angle) * rot, UnitComplex::identity());
        /// ```
        #[inline]
        pub fn inverse_mut(&mut self) {
            self.conjugate_mut()
        }

        /// Raise this unit complex number to a given floating power.
        ///
        /// This returns the unit complex number that identifies a rotation angle equal to
        /// `self.angle() × n`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::UnitComplex;
        /// let rot = UnitComplex::new(0.78);
        /// let pow = rot.powf(2.0);
        /// assert_relative_eq!(pow.angle(), 2.0 * 0.78);
        /// ```
        #[inline]
        pub fn powf(&self, n: N) -> Self {
            Self::from_angle(self.angle() * n)
        }

        /// Builds the rotation matrix corresponding to this unit complex number.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{UnitComplex, Rotation2};
        /// # use std::f32;
        /// let rot = UnitComplex::new(f32::consts::FRAC_PI_6);
        /// let expected = Rotation2::new(f32::consts::FRAC_PI_6);
        /// assert_eq!(rot.to_rotation_matrix(), expected);
        /// ```
        #[inline]
        pub fn to_rotation_matrix(&self) -> Rotation2<N> {
            let r = self.re;
            let i = self.im;

            Rotation2::from_matrix_unchecked(Matrix2::new(r, -i, i, r))
        }

        /// Converts this unit complex number into its equivalent homogeneous transformation matrix.
        ///
        /// # Example
        /// ```
        /// # use nalgebra::{UnitComplex, Matrix3};
        /// # use std::f32;
        /// let rot = UnitComplex::new(f32::consts::FRAC_PI_6);
        /// let expected = Matrix3::new(0.8660254, -0.5,      0.0,
        ///                             0.5,       0.8660254, 0.0,
        ///                             0.0,       0.0,       1.0);
        /// assert_eq!(rot.to_homogeneous(), expected);
        /// ```
        #[inline]
        pub fn to_homogeneous(&self) -> Matrix3<N> {
            self.to_rotation_matrix().to_homogeneous()
        }

        /// Rotate the given point by this unit complex number.
        ///
        /// This is the same as the multiplication `self * pt`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitComplex, Point2};
        /// # use std::f32;
        /// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
        /// let transformed_point = rot.transform_point(&Point2::new(1.0, 2.0));
        /// assert_relative_eq!(transformed_point, Point2::new(-2.0, 1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        pub fn transform_point(&self, pt: &Point2<N>) -> Point2<N> {
            self * pt
        }

        /// Rotate the given vector by this unit complex number.
        ///
        /// This is the same as the multiplication `self * v`.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitComplex, Vector2};
        /// # use std::f32;
        /// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
        /// let transformed_vector = rot.transform_vector(&Vector2::new(1.0, 2.0));
        /// assert_relative_eq!(transformed_vector, Vector2::new(-2.0, 1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        pub fn transform_vector(&self, v: &Vector2<N>) -> Vector2<N> {
            self * v
        }

        /// Rotate the given point by the inverse of this unit complex number.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitComplex, Point2};
        /// # use std::f32;
        /// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
        /// let transformed_point = rot.inverse_transform_point(&Point2::new(1.0, 2.0));
        /// assert_relative_eq!(transformed_point, Point2::new(2.0, -1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        pub fn inverse_transform_point(&self, pt: &Point2<N>) -> Point2<N> {
            // FIXME: would it be useful performancewise not to call inverse explicitly (i-e. implement
            // the inverse transformation explicitly here) ?
            self.inverse() * pt
        }

        /// Rotate the given vector by the inverse of this unit complex number.
        ///
        /// # Example
        /// ```
        /// # #[macro_use] extern crate approx;
        /// # use nalgebra::{UnitComplex, Vector2};
        /// # use std::f32;
        /// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
        /// let transformed_vector = rot.inverse_transform_vector(&Vector2::new(1.0, 2.0));
        /// assert_relative_eq!(transformed_vector, Vector2::new(2.0, -1.0), epsilon = 1.0e-6);
        /// ```
        #[inline]
        pub fn inverse_transform_vector(&self, v: &Vector2<N>) -> Vector2<N> {
            self.inverse() * v
        }
    }

    impl<N: RealField + fmt::Display> fmt::Display for UnitComplex<N> {
        fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
            write!(f, "UnitComplex angle: {}", self.angle())
        }
    }

    impl<N: RealField> AbsDiffEq for UnitComplex<N> {
        type Epsilon = N;

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

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

    impl<N: RealField> RelativeEq for UnitComplex<N> {
        #[inline]
        fn default_max_relative() -> Self::Epsilon {
            N::default_max_relative()
        }

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

    impl<N: RealField> UlpsEq for UnitComplex<N> {
        #[inline]
        fn default_max_ulps() -> u32 {
            N::default_max_ulps()
        }

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

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