magnum/src/Magnum/Math/DualComplex.h

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

    Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016
              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::DualComplex
 */

#include "Magnum/Math/Complex.h"
#include "Magnum/Math/Dual.h"
#include "Magnum/Math/Matrix3.h"

namespace Magnum { namespace Math {

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

/**
@brief Dual complex number
@tparam T   Underlying data type

Represents 2D rotation and translation. See @ref transformations for brief
introduction.
@see @ref Magnum::DualComplex, @ref Magnum::DualComplexd, @ref Dual,
    @ref Complex, @ref Matrix3
@todo Can this be done similarly as in dual quaternions? It sort of works, but
    the math beneath is weird.
*/
template<class T> class DualComplex: public Dual<Complex<T>> {
    public:
        typedef T Type;                         /**< @brief Underlying data type */

        /**
         * @brief Rotation dual complex number
         * @param angle         Rotation angle (counterclockwise)
         *
         * @f[
         *      \hat c = (cos \theta + i sin \theta) + \epsilon (0 + i0)
         * @f]
         * @see @ref Complex::rotation(), @ref Matrix3::rotation(),
         *      @ref DualQuaternion::rotation()
         */
        static DualComplex<T> rotation(Rad<T> angle) {
            return {Complex<T>::rotation(angle), {{}, {}}};
        }

        /**
         * @brief Translation dual complex number
         * @param vector    Translation vector
         *
         * @f[
         *      \hat c = (0 + i1) + \epsilon (v_x + iv_y)
         * @f]
         * @see @ref translation() const, @ref Matrix3::translation(const Vector2<T>&),
         *      @ref DualQuaternion::translation(), @ref Vector2::xAxis(),
         *      @ref Vector2::yAxis()
         */
        static DualComplex<T> translation(const Vector2<T>& vector) {
            return {{}, {vector.x(), vector.y()}};
        }

        /**
         * @brief Create dual complex number from rotation matrix
         *
         * Expects that the matrix represents rigid transformation.
         * @see @ref toMatrix(), @ref Complex::fromMatrix(),
         *      @ref Matrix3::isRigidTransformation()
         */
        static DualComplex<T> fromMatrix(const Matrix3<T>& matrix) {
            CORRADE_ASSERT(matrix.isRigidTransformation(),
                "Math::DualComplex::fromMatrix(): the matrix doesn't represent rigid transformation", {});
            return {Implementation::complexFromMatrix(matrix.rotationScaling()), Complex<T>(matrix.translation())};
        }

        /**
         * @brief Default constructor
         *
         * Creates unit dual complex number. @f[
         *      \hat c = (0 + i1) + \epsilon (0 + i0)
         * @f]
         */
        #ifdef DOXYGEN_GENERATING_OUTPUT
        constexpr /*implicit*/ DualComplex(IdentityInitT = IdentityInit) noexcept;
        #else
        constexpr /*implicit*/ DualComplex(IdentityInitT = IdentityInit) noexcept: Dual<Complex<T>>({}, {T(0), T(0)}) {}
        #endif

        /** @brief Construct zero-initialized dual complex number */
        constexpr explicit DualComplex(ZeroInitT) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            /* MSVC 2015 can't handle {} here */
            : Dual<Complex<T>>(Complex<T>{ZeroInit}, Complex<T>{ZeroInit})
            #endif
            {}

        /** @brief Construct without initializing the contents */
        explicit DualComplex(NoInitT) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            /* MSVC 2015 can't handle {} here */
            : Dual<Complex<T>>(NoInit)
            #endif
            {}

        /**
         * @brief Construct dual complex number from real and dual part
         *
         * @f[
         *      \hat c = c_0 + \epsilon c_\epsilon
         * @f]
         */
        constexpr /*implicit*/ DualComplex(const Complex<T>& real, const Complex<T>& dual = Complex<T>(T(0), T(0))) noexcept: Dual<Complex<T>>(real, dual) {}

        /* No constructor from a pair of Dual values because that would be
           ambiguous with the above */

        /**
         * @brief Construct dual complex number from vector
         *
         * To be used in transformations later. @f[
         *      \hat c = (0 + i1) + \epsilon(v_x + iv_y)
         * @f]
         */
        #ifdef DOXYGEN_GENERATING_OUTPUT
        constexpr explicit DualComplex(const Vector2<T>& vector) noexcept;
        #else
        constexpr explicit DualComplex(const Vector2<T>& vector) noexcept: Dual<Complex<T>>({}, Complex<T>(vector)) {}
        #endif

        /**
         * @brief Construct dual complex number from another of different type
         *
         * Performs only default casting on the values, no rounding or anything
         * else.
         */
        template<class U> constexpr explicit DualComplex(const DualComplex<U>& other) noexcept
            #ifndef DOXYGEN_GENERATING_OUTPUT
            /* MSVC 2015 can't handle {} here */
            : Dual<Complex<T>>(other)
            #endif
            {}

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

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

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

        /**
         * @brief Whether the dual complex number is normalized
         *
         * Dual complex number is normalized if its real part has unit length: @f[
         *      |c_0|^2 = |c_0| = 1
         * @f]
         * @see @ref Complex::dot(), @ref normalized()
         * @todoc Improve the equation as in Complex::isNormalized()
         */
        bool isNormalized() const {
            return Implementation::isNormalizedSquared(lengthSquared());
        }

        /**
         * @brief Rotation part of dual complex number
         *
         * @see @ref Complex::angle()
         */
        constexpr Complex<T> rotation() const {
            return Dual<Complex<T>>::real();
        }

        /**
         * @brief Translation part of dual complex number
         *
         * @f[
         *      \boldsymbol a = (c_\epsilon c_0^*)
         * @f]
         * @see @ref translation(const Vector2<T>&)
         */
        Vector2<T> translation() const {
            return Vector2<T>(Dual<Complex<T>>::dual());
        }

        /**
         * @brief Convert dual complex number to transformation matrix
         *
         * @see @ref fromMatrix(), @ref Complex::toMatrix()
         */
        Matrix3<T> toMatrix() const {
            return Matrix3<T>::from(Dual<Complex<T>>::real().toMatrix(), translation());
        }

        /**
         * @brief Multipy with dual complex number
         *
         * @f[
         *      \hat a \hat b = a_0 b_0 + \epsilon (a_0 b_\epsilon + a_\epsilon)
         * @f]
         * @todo can this be done similarly to dual quaternions?
         */
        DualComplex<T> operator*(const DualComplex<T>& other) const {
            return {Dual<Complex<T>>::real()*other.real(), Dual<Complex<T>>::real()*other.dual() + Dual<Complex<T>>::dual()};
        }

        /**
         * @brief Complex-conjugated dual complex number
         *
         * @f[
         *      \hat c^* = c^*_0 + c^*_\epsilon
         * @f]
         * @see @ref dualConjugated(), @ref conjugated(),
         *      @ref Complex::conjugated()
         */
        DualComplex<T> complexConjugated() const {
            return {Dual<Complex<T>>::real().conjugated(), Dual<Complex<T>>::dual().conjugated()};
        }

        /**
         * @brief Dual-conjugated dual complex number
         *
         * @f[
         *      \overline{\hat c} = c_0 - \epsilon c_\epsilon
         * @f]
         * @see @ref complexConjugated(), @ref conjugated(),
         *      @ref Dual::conjugated()
         */
        DualComplex<T> dualConjugated() const {
            return Dual<Complex<T>>::conjugated();
        }

        /**
         * @brief Conjugated dual complex number
         *
         * Both complex and dual conjugation. @f[
         *      \overline{\hat c^*} = c^*_0 - \epsilon c^*_\epsilon = c^*_0 + \epsilon(-a_\epsilon + ib_\epsilon)
         * @f]
         * @see @ref complexConjugated(), @ref dualConjugated(),
         *      @ref Complex::conjugated(), @ref Dual::conjugated()
         */
        DualComplex<T> conjugated() const {
            return {Dual<Complex<T>>::real().conjugated(), {-Dual<Complex<T>>::dual().real(), Dual<Complex<T>>::dual().imaginary()}};
        }

        /**
         * @brief Complex number length squared
         *
         * Should be used instead of length() for comparing complex number
         * length with other values, because it doesn't compute the square root. @f[
         *      |\hat c|^2 = c_0 \cdot c_0 = |c_0|^2
         * @f]
         * @todo Can this be done similarly to dual quaternins?
         */
        T lengthSquared() const {
            return Dual<Complex<T>>::real().dot();
        }

        /**
         * @brief Dual quaternion length
         *
         * See lengthSquared() which is faster for comparing length with other
         * values. @f[
         *      |\hat c| = \sqrt{c_0 \cdot c_0} = |c_0|
         * @f]
         * @todo can this be done similarly to dual quaternions?
         */
        T length() const {
            return Dual<Complex<T>>::real().length();
        }

        /**
         * @brief Normalized dual complex number (of unit length)
         *
         * @f[
         *      c' = \frac{c_0}{|c_0|}
         * @f]
         * @see @ref isNormalized()
         * @todo can this be done similarly to dual quaternions?
         */
        DualComplex<T> normalized() const {
            return {Dual<Complex<T>>::real()/length(), Dual<Complex<T>>::dual()};
        }

        /**
         * @brief Inverted dual complex number
         *
         * See invertedNormalized() which is faster for normalized dual complex
         * numbers. @f[
         *      \hat c^{-1} = c_0^{-1} - \epsilon c_\epsilon
         * @f]
         * @todo can this be done similarly to dual quaternions?
         */
        DualComplex<T> inverted() const {
            return DualComplex<T>(Dual<Complex<T>>::real().inverted(), {{}, {}})*DualComplex<T>({}, -Dual<Complex<T>>::dual());
        }

        /**
         * @brief Inverted normalized dual complex number
         *
         * Expects that the complex number is normalized. @f[
         *      \hat c^{-1} = c_0^{-1} - \epsilon c_\epsilon = c_0^* - \epsilon c_\epsilon
         * @f]
         * @see @ref isNormalized(), @ref inverted()
         * @todo can this be done similarly to dual quaternions?
         */
        DualComplex<T> invertedNormalized() const {
            return DualComplex<T>(Dual<Complex<T>>::real().invertedNormalized(), {{}, {}})*DualComplex<T>({}, -Dual<Complex<T>>::dual());
        }

        /**
         * @brief Rotate and translate point with dual complex number
         *
         * See transformPointNormalized(), which is faster for normalized dual
         * complex number. @f[
         *      v' = \hat c v = \hat c ((0 + i) + \epsilon(v_x + iv_y))
         * @f]
         * @see @ref DualComplex(const Vector2<T>&), @ref dual(),
         *      @ref Matrix3::transformPoint(), @ref Complex::transformVector(),
         *      @ref DualQuaternion::transformPoint()
         */
        Vector2<T> transformPoint(const Vector2<T>& vector) const {
            return Vector2<T>(((*this)*DualComplex<T>(vector)).dual());
        }

        MAGNUM_DUAL_SUBCLASS_IMPLEMENTATION(DualComplex, Vector2, T)
        /* Not using MAGNUM_DUAL_SUBCLASS_MULTIPLICATION_IMPLEMENTATION(), as
           we have special multiplication/division implementation */
};

MAGNUM_DUAL_OPERATOR_IMPLEMENTATION(DualComplex, Vector2, T)

/** @debugoperator{Magnum::Math::DualQuaternion} */
template<class T> Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug& debug, const DualComplex<T>& value) {
    return debug << "DualComplex({" << Corrade::Utility::Debug::nospace
          << value.real().real() << Corrade::Utility::Debug::nospace << ","
          << value.real().imaginary() << Corrade::Utility::Debug::nospace << "}, {"
          << Corrade::Utility::Debug::nospace
          << value.dual().real() << Corrade::Utility::Debug::nospace << ","
          << value.dual().imaginary() << 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 DualComplex<Float>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const DualComplex<Double>&);
#endif

}}

#endif