#ifndef Magnum_Math_DualQuaternion_h
#define Magnum_Math_DualQuaternion_h
/*
    This file is part of Magnum.

    Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018
              Vladimír Vondruš <mosra@centrum.cz>

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    Software is furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included
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    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
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*/

/** @file
 * @brief Class @ref Magnum::Math::DualQuaternion
 */

#include <cmath>

#include "Magnum/Math/Dual.h"
#include "Magnum/Math/Functions.h"
#include "Magnum/Math/Matrix4.h"
#include "Magnum/Math/Quaternion.h"

namespace Magnum { namespace Math {

namespace Implementation {
    template<class, class> struct DualQuaternionConverter;
}

/** @relatesalso DualQuaternion
@brief Screw linear interpolation of two dual quaternions
@param normalizedA  First dual quaternion
@param normalizedB  Second dual quaternion
@param t            Interpolation phase (from range @f$ [0; 1] @f$)

Expects that both dual quaternions are normalized. @f[
\begin{array}{rcl}
    l + \epsilon m & = & \hat q_A^* \hat q_B \\
    \frac{\hat a} 2 & = & acos \left( l_S \right) - \epsilon m_S \frac 1 {|l_V|} \\
    \hat {\boldsymbol n} & = & \boldsymbol n_0 + \epsilon \boldsymbol n_\epsilon
    ~~~~~~~~ \boldsymbol n_0 = l_V \frac 1 {|l_V|}
    ~~~~~~~~ \boldsymbol n_\epsilon = \left( m_V - {\boldsymbol n}_0 \frac {a_\epsilon} 2 l_S \right)\frac 1 {|l_V|} \\
    {\hat q}_{ScLERP} & = & \hat q_A (\hat q_A^* \hat q_B)^t =
        \hat q_A \left[ \hat {\boldsymbol n} sin \left( t \frac {\hat a} 2 \right), cos \left( t \frac {\hat a} 2 \right) \right] \\
\end{array}
@f]
@see @ref DualQuaternion::isNormalized(),
    @ref slerp(const Quaternion<T>&, const Quaternion<T>&, T),
    @ref lerp(const T&, const T&, U)
*/
template<class T> inline DualQuaternion<T> sclerp(const DualQuaternion<T>& normalizedA, const DualQuaternion<T>& normalizedB, const T t) {
    CORRADE_ASSERT(normalizedA.isNormalized() && normalizedB.isNormalized(),
        "Math::sclerp(): dual quaternions must be normalized", {});
    const T dotResult = dot(normalizedA.real().vector(), normalizedB.real().vector());

    /* Avoid division by zero */
    const T cosHalfAngle = dotResult + normalizedA.real().scalar()*normalizedB.real().scalar();
    if(std::abs(cosHalfAngle) >= T(1))
        return {normalizedA.real(), {Implementation::lerp(normalizedA.dual().vector(), normalizedB.dual().vector(), t), T(0)}};

    /* l + εm = q_A^**q_B, multiplying with -1 ensures shortest path when dot < 0 */
    const DualQuaternion<T> diff = normalizedA.quaternionConjugated()*(dotResult < T(0) ? -normalizedB : normalizedB);
    const Quaternion<T>& l = diff.real();
    const Quaternion<T>& m = diff.dual();

    /* a/2 = acos(l_S) - εm_S/|l_V| */
    const T invr = l.vector().lengthInverted();
    const Dual<T> aHalf{std::acos(l.scalar()), -m.scalar()*invr};

    /* direction = n_0 = l_V/|l_V|
       moment = n_ε = (m_V - n_0*(a_ε/2)*l_S)/|l_V| */
    const Vector3<T> direction = l.vector()*invr;
    const Vector3<T> moment = (m.vector() - direction*(aHalf.dual()*l.scalar()))*invr;
    const Dual<Vector3<T>> n{direction, moment};

    /* q_ScLERP = q_A*(cos(t*a/2) + n*sin(t*a/2)) */
    Dual<T> sin, cos;
    std::tie(sin, cos) = Math::sincos(t*Dual<Rad<T>>(aHalf));
    return normalizedA*DualQuaternion<T>{n*sin, cos};
}

/**
@brief Dual quaternion
@tparam T   Underlying data type

Represents 3D rotation and translation. See @ref transformations for brief
introduction.
@see @ref Magnum::DualQuaternion, @ref Magnum::DualQuaterniond, @ref Dual,
    @ref Quaternion, @ref Matrix4
*/
template<class T> class DualQuaternion: public Dual<Quaternion<T>> {
    public:
        typedef T Type;                         /**< @brief Underlying data type */

        /**
         * @brief Rotation dual quaternion
         * @param angle             Rotation angle (counterclockwise)
         * @param normalizedAxis    Normalized rotation axis
         *
         * Expects that the rotation axis is normalized. @f[
         *      \hat q = [\boldsymbol a \cdot sin \frac \theta 2, cos \frac \theta 2] + \epsilon [\boldsymbol 0, 0]
         * @f]
         * @see @ref rotation() const, @ref Quaternion::rotation(),
         *      @ref Matrix4::rotation(), @ref DualComplex::rotation(),
         *      @ref Vector3::xAxis(), @ref Vector3::yAxis(),
         *      @ref Vector3::zAxis(), @ref Vector::isNormalized()
         */
        static DualQuaternion<T> rotation(Rad<T> angle, const Vector3<T>& normalizedAxis) {
            return {Quaternion<T>::rotation(angle, normalizedAxis), {{}, T(0)}};
        }

        /** @todo Rotation about axis with arbitrary origin, screw motion */

        /**
         * @brief Translation dual quaternion
         * @param vector    Translation vector
         *
         * @f[
         *      \hat q = [\boldsymbol 0, 1] + \epsilon [\frac{\boldsymbol v}{2}, 0]
         * @f]
         * @see @ref translation() const,
         *      @ref Matrix4::translation(const Vector3<T>&),
         *      @ref DualComplex::translation(), @ref Vector3::xAxis(),
         *      @ref Vector3::yAxis(), @ref Vector3::zAxis()
         */
        static DualQuaternion<T> translation(const Vector3<T>& vector) {
            return {{}, {vector/T(2), T(0)}};
        }

        /**
         * @brief Create dual quaternion from transformation matrix
         *
         * Expects that the matrix represents rigid transformation.
         * @see @ref toMatrix(), @ref Quaternion::fromMatrix(),
         *      @ref Matrix4::isRigidTransformation()
         */
        static DualQuaternion<T> fromMatrix(const Matrix4<T>& matrix) {
            CORRADE_ASSERT(matrix.isRigidTransformation(),
                "Math::DualQuaternion::fromMatrix(): the matrix doesn't represent rigid transformation", {});

            Quaternion<T> q = Implementation::quaternionFromMatrix(matrix.rotationScaling());
            return {q, Quaternion<T>(matrix.translation()/2)*q};
        }

        /**
         * @brief Default constructor
         *
         * Creates unit dual quaternion. @f[
         *      \hat q = [\boldsymbol 0, 1] + \epsilon [\boldsymbol 0, 0]
         * @f]
         */
        constexpr /*implicit*/ DualQuaternion(IdentityInitT = IdentityInit) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : Dual<Quaternion<T>>{{}, {{}, T(0)}}
            #endif
            {}

        /** @brief Construct zero-initialized dual quaternion */
        constexpr explicit DualQuaternion(ZeroInitT) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : Dual<Quaternion<T>>{Quaternion<T>{ZeroInit}, Quaternion<T>{ZeroInit}}
            #endif
            {}

        /** @brief Construct without initializing the contents */
        explicit DualQuaternion(NoInitT) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : Dual<Quaternion<T>>{NoInit}
            #endif
            {}

        /**
         * @brief Construct dual quaternion from real and dual part
         *
         * @f[
         *      \hat q = q_0 + \epsilon q_\epsilon
         * @f]
         */
        constexpr /*implicit*/ DualQuaternion(const Quaternion<T>& real, const Quaternion<T>& dual = Quaternion<T>({}, T(0))) noexcept: Dual<Quaternion<T>>(real, dual) {}

        /**
         * @brief Construct dual quaternion from dual vector and scalar parts
         *
         * @f[
         *      \hat q = [\hat{\boldsymbol v}, \hat s] = [\boldsymbol v_0, s_0] + \epsilon [\boldsymbol v_\epsilon, s_\epsilon]
         * @f]
         */
        constexpr /*implicit*/ DualQuaternion(const Dual<Vector3<T>>& vector, const Dual<T>& scalar) noexcept
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : Dual<Quaternion<T>>{{vector.real(), scalar.real()}, {vector.dual(), scalar.dual()}}
            #endif
            {}

        /**
         * @brief Construct dual quaternion from vector
         *
         * To be used in transformations later. @f[
         *      \hat q = [\boldsymbol 0, 1] + \epsilon [\boldsymbol v, 0]
         * @f]
         * @see @ref transformPointNormalized()
         */
        #ifdef DOXYGEN_GENERATING_OUTPUT
        constexpr explicit DualQuaternion(const Vector3<T>& vector) noexcept;
        #else
        constexpr explicit DualQuaternion(const Vector3<T>& vector) noexcept: Dual<Quaternion<T>>({}, {vector, T(0)}) {}
        #endif

        /**
         * @brief Construct dual quaternion from another of different type
         *
         * Performs only default casting on the values, no rounding or anything
         * else.
         */
        template<class U> constexpr explicit DualQuaternion(const DualQuaternion<U>& other) noexcept: Dual<Quaternion<T>>(other) {}

        /** @brief Construct dual quaternion from external representation */
        template<class U, class V = decltype(Implementation::DualQuaternionConverter<T, U>::from(std::declval<U>()))> constexpr explicit DualQuaternion(const U& other): DualQuaternion{Implementation::DualQuaternionConverter<T, U>::from(other)} {}

        /** @brief Copy constructor */
        constexpr /*implicit*/ DualQuaternion(const Dual<Quaternion<T>>& other) noexcept: Dual<Quaternion<T>>(other) {}

        /** @brief Convert dual quaternion to external representation */
        template<class U, class V = decltype(Implementation::DualQuaternionConverter<T, U>::to(std::declval<DualQuaternion<T>>()))> constexpr explicit operator U() const {
            return Implementation::DualQuaternionConverter<T, U>::to(*this);
        }

        /**
         * @brief Whether the dual quaternion is normalized
         *
         * Dual quaternion is normalized if it has unit length: @f[
         *      |\hat q|^2 = |\hat q| = 1 + \epsilon 0
         * @f]
         * @see @ref lengthSquared(), @ref normalized()
         * @todoc Improve the equation as in Quaternion::isNormalized()
         */
        bool isNormalized() const {
            /* Comparing dual part to zero considering the magnitude of the
               translation -- the epsilon be much larger for large values. */
            Dual<T> a = lengthSquared();
            return Implementation::isNormalizedSquared(a.real()) &&
                   TypeTraits<T>::equalsZero(a.dual(), Math::max(Math::abs(Math::Dual<Quaternion<T>>::dual().vector()).max(), Math::abs(Math::Dual<Quaternion<T>>::dual().scalar())));
        }

        /**
         * @brief Rotation part of unit dual quaternion
         *
         * @see @ref Quaternion::angle(), @ref Quaternion::axis()
         */
        constexpr Quaternion<T> rotation() const {
            return Dual<Quaternion<T>>::real();
        }

        /**
         * @brief Translation part of unit dual quaternion
         *
         * @f[
         *      \boldsymbol a = 2 (q_\epsilon q_0^*)_V
         * @f]
         * @see @ref translation(const Vector3<T>&)
         */
        Vector3<T> translation() const {
            return (Dual<Quaternion<T>>::dual()*Dual<Quaternion<T>>::real().conjugated()).vector()*T(2);
        }

        /**
         * @brief Convert dual quaternion to transformation matrix
         *
         * @see @ref fromMatrix(), @ref Quaternion::toMatrix()
         */
        Matrix4<T> toMatrix() const {
            return Matrix4<T>::from(Dual<Quaternion<T>>::real().toMatrix(), translation());
        }

        /**
         * @brief Quaternion-conjugated dual quaternion
         *
         * @f[
         *      \hat q^* = q_0^* + q_\epsilon^*
         * @f]
         * @see @ref dualConjugated(), @ref conjugated(),
         *      @ref Quaternion::conjugated()
         */
        DualQuaternion<T> quaternionConjugated() const {
            return {Dual<Quaternion<T>>::real().conjugated(), Dual<Quaternion<T>>::dual().conjugated()};
        }

        /**
         * @brief Dual-conjugated dual quaternion
         *
         * @f[
         *      \overline{\hat q} = q_0 - \epsilon q_\epsilon
         * @f]
         * @see @ref quaternionConjugated(), @ref conjugated(),
         *      @ref Dual::conjugated()
         */
        DualQuaternion<T> dualConjugated() const {
            return Dual<Quaternion<T>>::conjugated();
        }

        /**
         * @brief Conjugated dual quaternion
         *
         * Both quaternion and dual conjugation. @f[
         *      \overline{\hat q^*} = q_0^* - \epsilon q_\epsilon^* = q_0^* + \epsilon [\boldsymbol q_{V \epsilon}, -q_{S \epsilon}]
         * @f]
         * @see @ref quaternionConjugated(), @ref dualConjugated(),
         *      @ref Quaternion::conjugated(), @ref Dual::conjugated()
         */
        DualQuaternion<T> conjugated() const {
            return {Dual<Quaternion<T>>::real().conjugated(), {Dual<Quaternion<T>>::dual().vector(), -Dual<Quaternion<T>>::dual().scalar()}};
        }

        /**
         * @brief Dual quaternion length squared
         *
         * Should be used instead of @ref length() for comparing dual
         * quaternion length with other values, because it doesn't compute the
         * square root. @f[
         *      |\hat q|^2 = \sqrt{\hat q^* \hat q}^2 = q_0 \cdot q_0 + \epsilon 2 (q_0 \cdot q_\epsilon)
         * @f]
         */
        Dual<T> lengthSquared() const {
            return {Dual<Quaternion<T>>::real().dot(), T(2)*dot(Dual<Quaternion<T>>::real(), Dual<Quaternion<T>>::dual())};
        }

        /**
         * @brief Dual quaternion length
         *
         * See @ref lengthSquared() which is faster for comparing length with other
         * values. @f[
         *      |\hat q| = \sqrt{\hat q^* \hat q} = |q_0| + \epsilon \frac{q_0 \cdot q_\epsilon}{|q_0|}
         * @f]
         */
        Dual<T> length() const {
            return Math::sqrt(lengthSquared());
        }

        /**
         * @brief Normalized dual quaternion (of unit length)
         *
         * @see @ref isNormalized()
         */
        DualQuaternion<T> normalized() const {
            return (*this)/length();
        }

        /**
         * @brief Inverted dual quaternion
         *
         * See @ref invertedNormalized() which is faster for normalized dual
         * quaternions. @f[
         *      \hat q^{-1} = \frac{\hat q^*}{|\hat q|^2}
         * @f]
         */
        DualQuaternion<T> inverted() const {
            return quaternionConjugated()/lengthSquared();
        }

        /**
         * @brief Inverted normalized dual quaternion
         *
         * Equivalent to @ref quaternionConjugated(). Expects that the
         * quaternion is normalized. @f[
         *      \hat q^{-1} = \frac{\hat q^*}{|\hat q|^2} = \hat q^*
         * @f]
         * @see @ref isNormalized(), @ref inverted()
         */
        DualQuaternion<T> invertedNormalized() const {
            CORRADE_ASSERT(isNormalized(),
                           "Math::DualQuaternion::invertedNormalized(): dual quaternion must be normalized", {});
            return quaternionConjugated();
        }

        /**
         * @brief Rotate and translate point with dual quaternion
         *
         * See @ref transformPointNormalized(), which is faster for normalized
         * dual quaternions. @f[
         *      v' = \hat q v \overline{\hat q^{-1}} = \hat q ([\boldsymbol 0, 1] + \epsilon [\boldsymbol v, 0]) \overline{\hat q^{-1}}
         * @f]
         * @see @ref DualQuaternion(const Vector3<T>&), @ref dual(),
         *      @ref Matrix4::transformPoint(),
         *      @ref Quaternion::transformVector(),
         *      @ref DualComplex::transformPoint()
         */
        Vector3<T> transformPoint(const Vector3<T>& vector) const {
            return ((*this)*DualQuaternion<T>(vector)*inverted().dualConjugated()).dual().vector();
        }

        /**
         * @brief Rotate and translate point with normalized dual quaternion
         *
         * Faster alternative to @ref transformPoint(), expects that the dual
         * quaternion is normalized. @f[
         *      v' = \hat q v \overline{\hat q^{-1}} = \hat q v \overline{\hat q^*} = \hat q ([\boldsymbol 0, 1] + \epsilon [\boldsymbol v, 0]) \overline{\hat q^*}
         * @f]
         * @see @ref isNormalized(), @ref DualQuaternion(const Vector3<T>&),
         *      @ref dual(), @ref Matrix4::transformPoint(),
         *      @ref Quaternion::transformVectorNormalized(),
         *      @ref DualComplex::transformPoint()
         */
        Vector3<T> transformPointNormalized(const Vector3<T>& vector) const {
            CORRADE_ASSERT(isNormalized(),
                           "Math::DualQuaternion::transformPointNormalized(): dual quaternion must be normalized", {});
            return ((*this)*DualQuaternion<T>(vector)*conjugated()).dual().vector();
        }

        MAGNUM_DUAL_SUBCLASS_IMPLEMENTATION(DualQuaternion, Quaternion, T)
        MAGNUM_DUAL_SUBCLASS_MULTIPLICATION_IMPLEMENTATION(DualQuaternion, Quaternion)
};

MAGNUM_DUAL_OPERATOR_IMPLEMENTATION(DualQuaternion, Quaternion, T)

/** @debugoperator{Magnum::Math::DualQuaternion} */
template<class T> Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug& debug, const DualQuaternion<T>& value) {
    return debug << "DualQuaternion({{" << Corrade::Utility::Debug::nospace
        << value.real().vector().x() << Corrade::Utility::Debug::nospace << ","
        << value.real().vector().y() << Corrade::Utility::Debug::nospace << ","
        << value.real().vector().z() << Corrade::Utility::Debug::nospace << "},"
        << value.real().scalar() << Corrade::Utility::Debug::nospace << "}, {{"
        << Corrade::Utility::Debug::nospace
        << value.dual().vector().x() << Corrade::Utility::Debug::nospace << ","
        << value.dual().vector().y() << Corrade::Utility::Debug::nospace << ","
        << value.dual().vector().z() << Corrade::Utility::Debug::nospace << "},"
        << value.dual().scalar() << Corrade::Utility::Debug::nospace << "})";
}

/* Explicit instantiation for commonly used types */
#ifndef DOXYGEN_GENERATING_OUTPUT
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const DualQuaternion<Float>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const DualQuaternion<Double>&);
#endif

}}

#endif