magnum/src/Magnum/Math/CubicHermite.h

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

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

    Permission is hereby granted, free of charge, to any person obtaining a
    copy of this software and associated documentation files (the "Software"),
    to deal in the Software without restriction, including without limitation
    the rights to use, copy, modify, merge, publish, distribute, sublicense,
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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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*/

/** @file
 * @brief Class @ref Magnum::Math::CubicHermite, alias @ref Magnum::Math::CubicHermite2D, @ref Magnum::Math::CubicHermite3D, function @ref Magnum::Math::select(), @ref Magnum::Math::lerp(), @ref Magnum::Math::splerp()
 */

#include "Magnum/Math/Complex.h"
#include "Magnum/Math/Quaternion.h"

namespace Magnum { namespace Math {

/**
@brief Cubic Hermite spline point

Represents a point on a [cubic Hermite spline](https://en.wikipedia.org/wiki/Cubic_Hermite_spline).

Unlike @ref Bezier, which describes a curve segment, this structure describes
a spline point @f$ \boldsymbol{p} @f$, with in-tangent @f$ \boldsymbol{m} @f$
and out-tangent @f$ \boldsymbol{n} @f$. This form is more suitable for
animation keyframe representation. The structure assumes the in/out tangents
to be in their final form, i.e. already normalized by length of their adjacent
segments.

Cubic Hermite splines are fully interchangeable with cubic Bézier curves, use
@ref fromBezier() and @ref Bezier::fromCubicHermite() for the conversion.

@see @ref CubicHermite2D, @ref CubicHermite3D,
    @ref Magnum::CubicHermite2D, @ref Magnum::CubicHermite2Dd,
    @ref Magnum::CubicHermite3D, @ref Magnum::CubicHermite3Dd,
    @ref CubicBezier
@experimental
*/
template<class T> class CubicHermite {
    public:
        typedef T Type;             /**< @brief Underlying data type */

        /**
         * @brief Create cubic Hermite spline point from adjacent Bézier curve segments
         *
         * Given two adjacent cubic Bézier curve segments defined by points
         * @f$ \boldsymbol{a}_i @f$ and @f$ \boldsymbol{b}_i @f$,
         * @f$ i \in \{ 0, 1, 2, 3 \} @f$, the corresponding cubic Hermite
         * spline point @f$ \boldsymbol{p} @f$, in-tangent @f$ \boldsymbol{m} @f$
         * and out-tangent @f$ \boldsymbol{n} @f$ is defined as: @f[
         *      \begin{array}{rcl}
         *          \boldsymbol{m} & = & 3 (\boldsymbol{a}_3 - \boldsymbol{a}_2)
         *                           =   3 (\boldsymbol{b}_0 - \boldsymbol{a}_2) \\
         *          \boldsymbol{p} & = & \boldsymbol{a}_3 = \boldsymbol{b}_0 \\
         *          \boldsymbol{n} & = & 3 (\boldsymbol{b}_1 - \boldsymbol{a}_3)
         *                           =   3 (\boldsymbol{b}_1 - \boldsymbol{b}_0)
         *      \end{array}
         * @f]
         *
         * Expects that the two segments are adjacent (i.e., the endpoint of
         * first segment is the start point of the second). If you need to
         * create a cubic Hermite spline point that's at the beginning or at
         * the end of a curve, simply pass a dummy Bézier segment that
         * satisfies this constraint as the other parameter:
         *
         * @snippet MagnumMath.cpp CubicHermite-fromBezier
         *
         * Enabled only on vector underlying types. See
         * @ref Bezier::fromCubicHermite() for the inverse operation.
         */
        template<UnsignedInt dimensions, class U> static
        #ifdef DOXYGEN_GENERATING_OUTPUT
        CubicHermite<T>
        #else
        typename std::enable_if<std::is_base_of<Vector<dimensions, U>, T>::value, CubicHermite<T>>::type
        #endif
        fromBezier(const CubicBezier<dimensions, U>& a, const CubicBezier<dimensions, U>& b) {
            return CORRADE_CONSTEXPR_ASSERT(a[3] == b[0],
                "Math::CubicHermite::fromBezier(): segments are not adjacent"),
                CubicHermite<T>{3*(a[3] - a[2]), a[3], 3*(b[1] - a[3])};
        }

        /**
         * @brief Default constructor
         *
         * Equivalent to @ref CubicHermite(ZeroInitT) for vector types and to
         * @ref CubicHermite(IdentityInitT) for complex and quaternion types.
         */
        constexpr /*implicit*/ CubicHermite() noexcept: CubicHermite{typename std::conditional<std::is_constructible<T, IdentityInitT>::value, IdentityInitT, ZeroInitT>::type{typename std::conditional<std::is_constructible<T, IdentityInitT>::value, IdentityInitT, ZeroInitT>::type::Init{}}} {}

        /**
         * @brief Default constructor
         *
         * Construct cubic Hermite spline point with all control points being
         * zero.
         */
        constexpr explicit CubicHermite(ZeroInitT) noexcept: CubicHermite{ZeroInit, typename std::conditional<std::is_constructible<T, ZeroInitT>::value, ZeroInitT*, void*>::type{}} {}

        /**
         * @brief Identity constructor
         *
         * The @ref point() is constructed as identity in order to have
         * interpolation working correctly; @ref inTangent() and
         * @ref outTangent() is constructed as zero. Enabled only for complex
         * and quaternion types.
         */
        template<class U = T, class = typename std::enable_if<std::is_constructible<U, IdentityInitT>::value>::type> constexpr explicit CubicHermite(IdentityInitT) noexcept: _inTangent{ZeroInit}, _point{IdentityInit}, _outTangent{ZeroInit} {}

        /** @brief Construct cubic Hermite spline point without initializing its contents */
        explicit CubicHermite(NoInitT) noexcept: CubicHermite{NoInit, typename std::conditional<std::is_constructible<T, NoInitT>::value, NoInitT*, void*>::type{}} {}

        /**
         * @brief Construct cubic Hermite spline point with given control points
         * @param inTangent     In-tangent @f$ \boldsymbol{m} @f$
         * @param point         Point @f$ \boldsymbol{p} @f$
         * @param outTangent    Out-tangent @f$ \boldsymbol{n} @f$
         */
        constexpr /*implicit*/ CubicHermite(const T& inTangent, const T& point, const T& outTangent) noexcept: _inTangent{inTangent}, _point{point}, _outTangent{outTangent} {}

        /**
         * @brief Construct subic Hermite spline point from another of different type
         *
         * Performs only default casting on the values, no rounding or
         * anything else.
         */
        template<class U> constexpr explicit CubicHermite(const CubicHermite<U>& other) noexcept: _inTangent{T(other._inTangent)}, _point{T(other._point)}, _outTangent{T(other._outTangent)} {}

        /**
         * @brief Raw data
         * @return One-dimensional array of three elements
         *
         * @see @ref inTangent(), @ref point(), @ref outTangent()
         */
        T* data() { return &_inTangent; }
        constexpr const T* data() const { return &_inTangent; } /**< @overload */

        /** @brief Equality comparison */
        bool operator==(const CubicHermite<T>& other) const;

        /** @brief Non-equality comparison */
        bool operator!=(const CubicHermite<T>& other) const {
            return !operator==(other);
        }

        /** @brief In-tangent @f$ \boldsymbol{m} @f$ */
        T& inTangent() { return _inTangent; }
        /* returns const& so [] operations are also constexpr */
        constexpr const T& inTangent() const { return _inTangent; } /**< @overload */

        /** @brief Point @f$ \boldsymbol{p} @f$ */
        T& point() { return _point; }
        /* returns const& so [] operations are also constexpr */
        constexpr const T& point() const { return _point; } /**< @overload */

        /** @brief Out-tangent @f$ \boldsymbol{n} @f$ */
        T& outTangent() { return _outTangent; }
        /* returns const& so [] operations are also constexpr */
        constexpr const T& outTangent() const { return _outTangent; } /**< @overload */

    private:
        template<class> friend class CubicHermite;

        /* Called from CubicHermite(ZeroInit), either using the ZeroInit
           constructor (if available) or passing zero directly (for scalar
           types) */
        constexpr explicit CubicHermite(ZeroInitT, ZeroInitT*) noexcept: _inTangent{ZeroInit}, _point{ZeroInit}, _outTangent{ZeroInit} {}
        constexpr explicit CubicHermite(ZeroInitT, void*) noexcept: _inTangent{T(0)}, _point{T(0)}, _outTangent{T(0)} {}

        /* Called from CubicHermite(NoInit), either using the NoInit
           constructor (if available) or not doing anything */
        explicit CubicHermite(NoInitT, NoInitT*) noexcept: _inTangent{NoInit}, _point{NoInit}, _outTangent{NoInit} {}
        explicit CubicHermite(NoInitT, void*) noexcept {}

        T _inTangent;
        T _point;
        T _outTangent;
};

/**
@brief Scalar cubic Hermite spline point

Convenience alternative to @cpp CubicHermite<T> @ce. See @ref CubicHermite for
more information.
@see @ref CubicHermite2D, @ref CubicHermite3D, @ref CubicHermiteComplex,
    @ref CubicHermiteQuaternion, @ref Magnum::CubicHermite1D,
    @ref Magnum::CubicHermite1Dd
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using CubicHermite1D = CubicHermite<T>;
#endif

/**
@brief Two-dimensional cubic Hermite spline point

Convenience alternative to @cpp CubicHermite<Vector2<T>> @ce. See
@ref CubicHermite for more information.
@see @ref CubicHermite1D, @ref CubicHermite3D, @ref CubicHermiteComplex,
    @ref CubicHermiteQuaternion, @ref Magnum::CubicHermite2D,
    @ref Magnum::CubicHermite2Dd
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using CubicHermite2D = CubicHermite<Vector2<T>>;
#endif

/**
@brief Three-dimensional cubic Hermite spline point

Convenience alternative to @cpp CubicHermite<Vector2<T>> @ce. See
@ref CubicHermite for more information.
@see @ref CubicHermite1D, @ref CubicHermite2D, @ref CubicHermiteComplex,
    @ref CubicHermiteQuaternion, @ref Magnum::CubicHermite3D,
    @ref Magnum::CubicHermite3Dd
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using CubicHermite3D = CubicHermite<Vector3<T>>;
#endif

/**
@brief Cubic Hermite spline complex number

Convenience alternative to @cpp CubicHermite<Complex<T>> @ce. See
@ref CubicHermite for more information.
@see @ref CubicHermite1D, @ref CubicHermite2D, @ref CubicHermite3D,
    @ref CubicHermiteQuaternion, @ref Magnum::CubicHermiteComplex,
    @ref Magnum::CubicHermiteComplexd
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using CubicHermiteComplex = CubicHermite<Complex<T>>;
#endif

/**
@brief Cubic Hermite spline quaternion

Convenience alternative to @cpp CubicHermite<Quaternion<T>> @ce. See
@ref CubicHermite for more information.
@see @ref CubicHermite1D, @ref CubicHermite2D, @ref CubicHermite3D,
    @ref CubicHermiteComplex, @ref Magnum::CubicHermiteQuaternion,
    @ref Magnum::CubicHermiteQuaterniond
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using CubicHermiteQuaternion = CubicHermite<Quaternion<T>>;
#endif

/** @debugoperator{CubicHermite} */
template<class T> Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug& debug, const CubicHermite<T>& value) {
    return debug << "CubicHermite(" << Corrade::Utility::Debug::nospace
        << value.inTangent() << Corrade::Utility::Debug::nospace << ","
        << value.point() << Corrade::Utility::Debug::nospace << ","
        << value.outTangent() << 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 CubicHermite<Float>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Double>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Vector2<Float>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Vector3<Float>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Vector4<Float>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Vector2<Double>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Vector3<Double>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Vector4<Double>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Complex<Float>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Complex<Double>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Quaternion<Float>>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const CubicHermite<Quaternion<Double>>&);
#endif

/** @relatesalso CubicHermite
@brief Constant interpolation of two cubic Hermite spline points
@param a     First spline point
@param b     Second spline point
@param t     Interpolation phase (from range @f$ [0; 1] @f$)

Given segment points @f$ \boldsymbol{p}_i @f$, in-tangents @f$ \boldsymbol{m}_i @f$
and out-tangents @f$ \boldsymbol{n}_i @f$, the interpolated value @f$ \boldsymbol{p} @f$
at phase @f$ t @f$ is: @f[
    \boldsymbol{p}(t) = \begin{cases}
        \boldsymbol{p}_a, & t < 1 \\
        \boldsymbol{p}_b, & t \ge 1
    \end{cases}
@f]

Equivalent to calling @ref select(const T&, const T&, U) on
@ref CubicHermite::point() extracted from both @p a and @p b.
@see @ref lerp(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref splerp(const CubicHermite<T>&, const CubicHermite<T>&, U)
*/
template<class T, class U> T select(const CubicHermite<T>& a, const CubicHermite<T>& b, U t) {
    /* Not using select() from Functions.h to avoid the header dependency */
    return t < U(1) ? a.point() : b.point();
}

/** @relatesalso CubicHermite
@brief Linear interpolation of two cubic Hermite points
@param a     First spline point
@param b     Second spline point
@param t     Interpolation phase (from range @f$ [0; 1] @f$)

Given segment points @f$ \boldsymbol{p}_i @f$, in-tangents @f$ \boldsymbol{m}_i @f$
and out-tangents @f$ \boldsymbol{n}_i @f$, the interpolated value @f$ \boldsymbol{p} @f$
at phase @f$ t @f$ is: @f[
    \boldsymbol{p}(t) = (1 - t) \boldsymbol{p}_a + t \boldsymbol{p}_b
@f]

Equivalent to calling @ref lerp(const T&, const T&, U) on
@ref CubicHermite::point() extracted from both @p a and @p b.
@see @ref lerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref lerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref lerpShortestPath(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref slerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref slerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref slerpShortestPath(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref splerp(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref splerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref splerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
*/
template<class T, class U> T lerp(const CubicHermite<T>& a, const CubicHermite<T>& b, U t) {
    /* To avoid dependency on Functions.h */
    return Implementation::lerp(a.point(), b.point(), t);
}

/** @relatesalso CubicHermite
@brief Linear interpolation of two cubic Hermite complex numbers

Equivalent to calling @ref lerp(const Complex<T>&, const Complex<T>&, T) on
@ref CubicHermite::point() extracted from @p a and @p b. Compared to
@ref lerp(const CubicHermite<T>&, const CubicHermite<T>&, U) this adds a
normalization step after. Expects that @ref CubicHermite::point() is a
normalized complex number in both @p a and @p b.
@see @ref Complex::isNormalized(),
    @ref lerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref slerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref splerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T)
*/
template<class T> Complex<T> lerp(const CubicHermiteComplex<T>& a, const CubicHermiteComplex<T>& b, T t) {
    return lerp(a.point(), b.point(), t);
}

/** @relatesalso CubicHermite
@brief Linear interpolation of two cubic Hermite quaternions

Equivalent to calling @ref lerp(const Quaternion<T>&, const Quaternion<T>&, T)
on @ref CubicHermite::point() extracted from @p a and @p b. Compared to
@ref lerp(const CubicHermite<T>&, const CubicHermite<T>&, U) this adds a
normalization step after. Expects that @ref CubicHermite::point() is a
normalized quaternion in both @p a and @p b.
@see @ref Quaternion::isNormalized(),
    @ref lerpShortestPath(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref lerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref slerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref splerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
*/
template<class T> Quaternion<T> lerp(const CubicHermiteQuaternion<T>& a, const CubicHermiteQuaternion<T>& b, T t) {
    return lerp(a.point(), b.point(), t);
}

/** @relatesalso CubicHermite
@brief Linear shortest-path interpolation of two cubic Hermite quaternions

Equivalent to calling @ref lerpShortestPath(const Quaternion<T>&, const Quaternion<T>&, T)
on @ref CubicHermite::point() extracted from @p a and @p b. Expects that
@ref CubicHermite::point() is a normalized quaternion in both @p a and @p b.

Note that rotations interpolated with this function may go along a completely
different path compared to @ref splerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T).
Use @ref lerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
for behavior that is consistent with spline interpolation.
@see @ref Quaternion::isNormalized(),
    @ref slerpShortestPath(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
*/
template<class T> Quaternion<T> lerpShortestPath(const CubicHermiteQuaternion<T>& a, const CubicHermiteQuaternion<T>& b, T t) {
    return lerpShortestPath(a.point(), b.point(), t);
}

/** @relatesalso CubicHermite
@brief Spherical linear interpolation of two cubic Hermite complex numbers

Equivalent to calling @ref slerp(const Complex<T>&, const Complex<T>&, T) on
@ref CubicHermite::point() extracted from @p a and @p b. Expects that
@ref CubicHermite::point() is a normalized complex number in both @p a and @p b.
@see @ref Complex::isNormalized(),
    @ref slerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref lerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref splerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T)
 */
template<class T> inline Complex<T> slerp(const CubicHermiteComplex<T>& a, const CubicHermiteComplex<T>& b, T t) {
    return slerp(a.point(), b.point(), t);
}

/** @relatesalso CubicHermite
@brief Spherical linear interpolation of two cubic Hermite quaternions

Equivalent to calling @ref slerp(const Quaternion<T>&, const Quaternion<T>&, T)
on @ref CubicHermite::point() extracted from @p a and @p b. Expects that
@ref CubicHermite::point() is a normalized complex number in both @p a and @p b.
@see @ref Quaternion::isNormalized(),
    @ref slerpShortestPath(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref slerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref lerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref splerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
 */
template<class T> inline Quaternion<T> slerp(const CubicHermiteQuaternion<T>& a, const CubicHermiteQuaternion<T>& b, T t) {
    return slerp(a.point(), b.point(), t);
}

/** @relatesalso CubicHermite
@brief Spherical linear shortest-path interpolation of two cubic Hermite quaternions

Equivalent to calling @ref slerpShortestPath(const Quaternion<T>&, const Quaternion<T>&, T)
on @ref CubicHermite::point() extracted from @p a and @p b. Expects that
@ref CubicHermite::point() is a normalized quaternion in both @p a and @p b.

Note that rotations interpolated with this function may go along a completely
different path compared to @ref splerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T).
Use @ref slerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
for spherical linear interpolation with behavior that is consistent with spline
interpolation.
@see @ref Quaternion::isNormalized(),
    @ref lerpShortestPath(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
*/
template<class T> Quaternion<T> slerpShortestPath(const CubicHermiteQuaternion<T>& a, const CubicHermiteQuaternion<T>& b, T t) {
    return slerpShortestPath(a.point(), b.point(), t);
}

/** @relatesalso CubicHermite
@brief Spline interpolation of two cubic Hermite points
@param a     First spline point
@param b     Second spline point
@param t     Interpolation phase

Given segment points @f$ \boldsymbol{p}_i @f$, in-tangents @f$ \boldsymbol{m}_i @f$
and out-tangents @f$ \boldsymbol{n}_i @f$, the interpolated value @f$ \boldsymbol{p} @f$
at phase @f$ t @f$ is: @f[
    \boldsymbol{p}(t) = (2 t^3 - 3 t^2 + 1) \boldsymbol{p}_a + (t^3 - 2 t^2 + t) \boldsymbol{n}_a + (-2 t^3 + 3 t^2) \boldsymbol{p}_b + (t^3 - t^2)\boldsymbol{m}_b
@f]

@see @ref splerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref splerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref lerp(const CubicHermite<T>&, const CubicHermite<T>&, U)
*/
template<class T, class U> T splerp(const CubicHermite<T>& a, const CubicHermite<T>& b, U t) {
    return (U(2)*t*t*t - U(3)*t*t + U(1))*a.point() +
        (t*t*t - U(2)*t*t + t)*a.outTangent() +
        (U(-2)*t*t*t + U(3)*t*t)*b.point() +
        (t*t*t - t*t)*b.inTangent();
}

/** @relatesalso CubicHermite
@brief Spline interpolation of two cubic Hermite complex numbers

Unlike @ref splerp(const CubicHermite<T>&, const CubicHermite<T>&, U) this adds
a normalization step after. Given segment points @f$ \boldsymbol{p}_i @f$,
in-tangents @f$ \boldsymbol{m}_i @f$ and out-tangents @f$ \boldsymbol{n}_i @f$,
the interpolated value @f$ \boldsymbol{p} @f$ at phase @f$ t @f$ is: @f[
    \begin{array}{rcl}
        \boldsymbol{p'}(t) & = & (2 t^3 - 3 t^2 + 1) \boldsymbol{p}_a + (t^3 - 2 t^2 + t) \boldsymbol{n}_a + (-2 t^3 + 3 t^2) \boldsymbol{p}_b + (t^3 - t^2)\boldsymbol{m}_b \\
        \boldsymbol{p}(t) & = & \cfrac{\boldsymbol{p'}(t)}{|\boldsymbol{p'}(t)|}
    \end{array}
@f]

Expects that @ref CubicHermite::point() is a normalized complex number in both
@p a and @p b.
@see @ref Complex::isNormalized(),
    @ref splerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref lerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref slerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T)
*/
template<class T> Complex<T> splerp(const CubicHermiteComplex<T>& a, const CubicHermiteComplex<T>& b, T t) {
    CORRADE_ASSERT(a.point().isNormalized() && b.point().isNormalized(),
        "Math::splerp(): complex spline points" << a.point() << "and" << b.point() << "are not normalized", {});
    return ((T(2)*t*t*t - T(3)*t*t + T(1))*a.point() +
        (t*t*t - T(2)*t*t + t)*a.outTangent() +
        (T(-2)*t*t*t + T(3)*t*t)*b.point() +
        (t*t*t - t*t)*b.inTangent()).normalized();
}

/** @relatesalso CubicHermite
@brief Spline interpolation of two cubic Hermite quaternions

Unlike @ref splerp(const CubicHermite<T>&, const CubicHermite<T>&, U) this adds
a normalization step after. Given segment points @f$ \boldsymbol{p}_i @f$,
in-tangents @f$ \boldsymbol{m}_i @f$ and out-tangents @f$ \boldsymbol{n}_i @f$,
the interpolated value @f$ \boldsymbol{p} @f$ at phase @f$ t @f$ is: @f[
    \begin{array}{rcl}
        \boldsymbol{p'}(t) & = & (2 t^3 - 3 t^2 + 1) \boldsymbol{p}_a + (t^3 - 2 t^2 + t) \boldsymbol{n}_a + (-2 t^3 + 3 t^2) \boldsymbol{p}_b + (t^3 - t^2)\boldsymbol{m}_b \\
        \boldsymbol{p}(t) & = & \cfrac{\boldsymbol{p'}(t)}{|\boldsymbol{p'}(t)|}
    \end{array}
@f]

Expects that @ref CubicHermite::point() is a normalized quaternion in both @p a
and @p b.
@see @ref Quaternion::isNormalized(),
    @ref splerp(const CubicHermiteComplex<T>&, const CubicHermiteComplex<T>&, T),
    @ref select(const CubicHermite<T>&, const CubicHermite<T>&, U),
    @ref lerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T),
    @ref slerp(const CubicHermiteQuaternion<T>&, const CubicHermiteQuaternion<T>&, T)
*/
template<class T> Quaternion<T> splerp(const CubicHermiteQuaternion<T>& a, const CubicHermiteQuaternion<T>& b, T t) {
    CORRADE_ASSERT(a.point().isNormalized() && b.point().isNormalized(),
        "Math::splerp(): quaternion spline points" << a.point() << "and" << b.point() << "are not normalized", {});
    return ((T(2)*t*t*t - T(3)*t*t + T(1))*a.point() +
        (t*t*t - T(2)*t*t + t)*a.outTangent() +
        (T(-2)*t*t*t + T(3)*t*t)*b.point() +
        (t*t*t - t*t)*b.inTangent()).normalized();
}

template<class T> inline bool CubicHermite<T>::operator==(const CubicHermite<T>& other) const {
    /* For non-scalar types default implementation of TypeTraits would be used,
       which is just operator== */
    return TypeTraits<T>::equals(_inTangent, other._inTangent) &&
        TypeTraits<T>::equals(_point, other._point) &&
        TypeTraits<T>::equals(_outTangent, other._outTangent);
}

namespace Implementation {

template<class T> struct StrictWeakOrdering<CubicHermite<T>> {
    bool operator()(const CubicHermite<T>& a, const CubicHermite<T>& b) const {
        StrictWeakOrdering<T> o;
        if(o(a.inTangent(), b.inTangent()))
            return true;
        if(o(b.inTangent(), a.inTangent()))
            return false;
        if(o(a.point(), b.point()))
            return true;
        if(o(b.point(), a.point()))
            return false;
        return o(a.outTangent(), b.outTangent());
    }
};

}

}}

#endif