magnum/src/Magnum/Math/Matrix.h

#ifndef Magnum_Math_Matrix_h
#define Magnum_Math_Matrix_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,
    and/or sell copies of the Software, and to permit persons to whom the
    Software is furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included
    in all copies or substantial portions of the Software.

    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
    THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
    FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
    DEALINGS IN THE SOFTWARE.
*/

/** @file
 * @brief Class @ref Magnum::Math::Matrix, alias @ref Magnum::Math::Matrix2x2, @ref Magnum::Math::Matrix3x3, @ref Magnum::Math::Matrix4x4
 */

#include "Magnum/Math/RectangularMatrix.h"

namespace Magnum { namespace Math {

namespace Implementation {
    template<std::size_t, class> struct MatrixDeterminant;

    template<std::size_t size, std::size_t col, std::size_t otherSize, class T, std::size_t ...row> constexpr Vector<size, T> valueOrIdentityVector(Sequence<row...>, const RectangularMatrix<otherSize, otherSize, T>& other) {
        return {(col < otherSize && row < otherSize ? other[col][row] :
            col == row ? T{1} : T{0})...};
    }

    template<std::size_t size, std::size_t col, std::size_t otherSize, class T> constexpr Vector<size, T> valueOrIdentityVector(const RectangularMatrix<otherSize, otherSize, T>& other) {
        return valueOrIdentityVector<size, col>(typename Implementation::GenerateSequence<size>::Type(), other);
    }
}

/**
@brief Square matrix
@tparam size    Matrix size
@tparam T       Data type

See @ref matrix-vector for brief introduction.

@configurationvalueref{Magnum::Math::Matrix}
@see @ref Matrix2x2, @ref Matrix3x3, @ref Matrix4x4
 */
template<std::size_t size, class T> class Matrix: public RectangularMatrix<size, size, T> {
    public:
        enum: std::size_t {
            Size = size         /**< Matrix size */
        };

        /**
         * @brief Default constructor
         *
         * Creates identity matrix. @p value allows you to specify value on
         * diagonal.
         */
        constexpr /*implicit*/ Matrix(IdentityInitT = IdentityInit, T value = T(1)) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : RectangularMatrix<size, size, T>{typename Implementation::GenerateSequence<size>::Type(), Vector<size, T>(value)}
            #endif
            {}

        /** @copydoc RectangularMatrix::RectangularMatrix(ZeroInitT) */
        constexpr explicit Matrix(ZeroInitT) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : RectangularMatrix<size, size, T>{ZeroInit}
            #endif
            {}

        /** @copydoc RectangularMatrix::RectangularMatrix(NoInitT) */
        constexpr explicit Matrix(NoInitT) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : RectangularMatrix<size, size, T>{NoInit}
            #endif
            {}

        /** @brief Construct matrix from column vectors */
        template<class ...U> constexpr /*implicit*/ Matrix(const Vector<size, T>& first, const U&... next) noexcept: RectangularMatrix<size, size, T>(first, next...) {}

        /** @brief Construct matrix with one value for all elements */
        constexpr explicit Matrix(T value) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : RectangularMatrix<size, size, T>{typename Implementation::GenerateSequence<size>::Type(), value}
            #endif
            {}

        /**
         * @brief Construct matrix from another of different type
         *
         * Performs only default casting on the values, no rounding or
         * anything else. Example usage:
         *
         * @code{.cpp}
         * Matrix2x2<Float> floatingPoint({1.3f, 2.7f},
         *                                {-15.0f, 7.0f});
         * Matrix2x2<Byte> integral(floatingPoint);
         * // integral == {{1, 2}, {-15, 7}}
         * @endcode
         */
        template<class U> constexpr explicit Matrix(const RectangularMatrix<size, size, U>& other) noexcept: RectangularMatrix<size, size, T>(other) {}

        /** @brief Construct matrix from external representation */
        template<class U, class V = decltype(Implementation::RectangularMatrixConverter<size, size, T, U>::from(std::declval<U>()))> constexpr explicit Matrix(const U& other): RectangularMatrix<size, size, T>(Implementation::RectangularMatrixConverter<size, size, T, U>::from(other)) {}

        /**
         * @brief Construct matrix by slicing or expanding another of a different size
         *
         * If the other matrix is larger, takes only the first @cpp size @ce
         * columns and rows from it; if the other matrix is smaller, it's
         * expanded to an identity (ones on diagonal, zeros elsewhere).
         */
        template<std::size_t otherSize> constexpr explicit Matrix(const RectangularMatrix<otherSize, otherSize, T>& other) noexcept
            /** @todoc remove workaround when doxygen is sane */
            #ifndef DOXYGEN_GENERATING_OUTPUT
            : Matrix<size, T>{typename Implementation::GenerateSequence<size>::Type(), other}
            #endif
            {}

        /** @brief Copy constructor */
        constexpr /*implicit*/ Matrix(const RectangularMatrix<size, size, T>& other) noexcept: RectangularMatrix<size, size, T>(other) {}

        /**
         * @brief Whether the matrix is orthogonal
         *
         * The matrix is orthogonal if its transpose is equal to its inverse: @f[
         *      Q^T = Q^{-1}
         * @f]
         * @see @ref transposed(), @ref inverted(),
         *      @ref Matrix3::isRigidTransformation(),
         *      @ref Matrix4::isRigidTransformation()
         */
        bool isOrthogonal() const;

        /**
         * @brief Trace of the matrix
         *
         * @f[
         * tr(A) = \sum_{i=1}^n a_{i,i}
         * @f]
         */
        T trace() const { return RectangularMatrix<size, size, T>::diagonal().sum(); }

        /** @brief Matrix without given column and row */
        Matrix<size-1, T> ij(std::size_t skipCol, std::size_t skipRow) const;

        /**
         * @brief Determinant
         *
         * Returns `0` if the matrix is noninvertible and `1` if the matrix is
         * orthogonal. Computed recursively using Laplace's formula: @f[
         *      \det(A) = \sum_{j=1}^n (-1)^{i+j} a_{i,j} \det(A^{i,j})
         * @f] @f$ A^{i, j} @f$ is matrix without i-th row and j-th column, see
         * @ref ij(). The formula is expanded down to 2x2 matrix, where the
         * determinant is computed directly: @f[
         *      \det(A) = a_{0, 0} a_{1, 1} - a_{1, 0} a_{0, 1}
         * @f]
         */
        T determinant() const { return Implementation::MatrixDeterminant<size, T>()(*this); }

        /**
         * @brief Inverted matrix
         *
         * Computed using Cramer's rule: @f[
         *      A^{-1} = \frac{1}{\det(A)} Adj(A)
         * @f]
         * See @ref invertedOrthogonal(), @ref Matrix3::invertedRigid() and
         * @ref Matrix4::invertedRigid() which are faster alternatives for
         * particular matrix types.
         * @see @ref Algorithms::gaussJordanInverted()
         * @m_keyword{inverse(),GLSL inverse(),}
         */
        Matrix<size, T> inverted() const;

        /**
         * @brief Inverted orthogonal matrix
         *
         * Equivalent to @ref transposed(), expects that the matrix is
         * orthogonal. @f[
         *      A^{-1} = A^T
         * @f]
         * @see @ref inverted(), @ref isOrthogonal(),
         *      @ref Matrix3::invertedRigid(),
         *      @ref Matrix4::invertedRigid()
         */
        Matrix<size, T> invertedOrthogonal() const {
            CORRADE_ASSERT(isOrthogonal(),
                "Math::Matrix::invertedOrthogonal(): the matrix is not orthogonal:" << Corrade::Utility::Debug::Debug::newline << *this, {});
            return RectangularMatrix<size, size, T>::transposed();
        }

        #ifndef DOXYGEN_GENERATING_OUTPUT
        /* Reimplementation of functions to return correct type */
        Matrix<size, T> operator*(const Matrix<size, T>& other) const {
            return RectangularMatrix<size, size, T>::operator*(other);
        }
        template<std::size_t otherCols> RectangularMatrix<otherCols, size, T> operator*(const RectangularMatrix<otherCols, size, T>& other) const {
            return RectangularMatrix<size, size, T>::operator*(other);
        }
        Vector<size, T> operator*(const Vector<size, T>& other) const {
            return RectangularMatrix<size, size, T>::operator*(other);
        }
        Matrix<size, T> transposed() const {
            return RectangularMatrix<size, size, T>::transposed();
        }
        MAGNUM_RECTANGULARMATRIX_SUBCLASS_IMPLEMENTATION(size, size, Matrix<size, T>)
        #endif

    private:
        /* Implementation for RectangularMatrix<cols, rows, T>::RectangularMatrix(const RectangularMatrix<cols, rows, U>&) */
        template<std::size_t otherSize, std::size_t ...col> constexpr explicit Matrix(Implementation::Sequence<col...>, const RectangularMatrix<otherSize, otherSize, T>& other) noexcept: RectangularMatrix<size, size, T>{Implementation::valueOrIdentityVector<size, col>(other)...} {}
};

/**
@brief 2x2 matrix

Convenience alternative to `Matrix<2, T>`. See @ref Matrix for more
information.
@see @ref Magnum::Matrix2x2, @ref Magnum::Matrix2x2d
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix2x2 = Matrix<2, T>;
#endif

/**
@brief 3x3 matrix

Convenience alternative to `Matrix<3, T>`. See @ref Matrix for more
information. Note that this is different from @ref Matrix3, which contains
additional functions for transformations in 2D.
@see @ref Magnum::Matrix3x3, @ref Magnum::Matrix3x3d
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix3x3 = Matrix<3, T>;
#endif

/**
@brief 4x4 matrix

Convenience alternative to `Matrix<4, T>`. See @ref Matrix for more
information. Note that this is different from @ref Matrix4, which contains
additional functions for transformations in 3D.
@see @ref Magnum::Matrix4x4, @ref Magnum::Matrix4x4d
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix4x4 = Matrix<4, T>;
#endif

MAGNUM_MATRIX_OPERATOR_IMPLEMENTATION(Matrix<size, T>)

#ifndef DOXYGEN_GENERATING_OUTPUT
#define MAGNUM_MATRIX_SUBCLASS_IMPLEMENTATION(size, Type, VectorType)       \
    VectorType<T>& operator[](std::size_t col) {                            \
        return static_cast<VectorType<T>&>(Matrix<size, T>::operator[](col)); \
    }                                                                       \
    constexpr const VectorType<T> operator[](std::size_t col) const {       \
        return VectorType<T>(Matrix<size, T>::operator[](col));             \
    }                                                                       \
    VectorType<T> row(std::size_t row) const {                              \
        return VectorType<T>(Matrix<size, T>::row(row));                    \
    }                                                                       \
                                                                            \
    Type<T> operator*(const Matrix<size, T>& other) const {                 \
        return Matrix<size, T>::operator*(other);                           \
    }                                                                       \
    template<std::size_t otherCols> RectangularMatrix<otherCols, size, T> operator*(const RectangularMatrix<otherCols, size, T>& other) const { \
        return Matrix<size, T>::operator*(other);                           \
    }                                                                       \
    VectorType<T> operator*(const Vector<size, T>& other) const {           \
        return Matrix<size, T>::operator*(other);                           \
    }                                                                       \
                                                                            \
    Type<T> transposed() const { return Matrix<size, T>::transposed(); }    \
    constexpr VectorType<T> diagonal() const { return Matrix<size, T>::diagonal(); } \
    Type<T> inverted() const { return Matrix<size, T>::inverted(); }        \
    Type<T> invertedOrthogonal() const {                                    \
        return Matrix<size, T>::invertedOrthogonal();                       \
    }

namespace Implementation {

template<std::size_t size, class T> struct MatrixDeterminant {
    T operator()(const Matrix<size, T>& m);
};

template<std::size_t size, class T> T MatrixDeterminant<size, T>::operator()(const Matrix<size, T>& m) {
    T out(0);

    for(std::size_t col = 0; col != size; ++col)
        out += ((col & 1) ? -1 : 1)*m[col][0]*m.ij(col, 0).determinant();

    return out;
}

template<class T> struct MatrixDeterminant<2, T> {
    constexpr T operator()(const Matrix<2, T>& m) const {
        return m[0][0]*m[1][1] - m[1][0]*m[0][1];
    }
};

template<class T> struct MatrixDeterminant<1, T> {
    constexpr T operator()(const Matrix<1, T>& m) const {
        return m[0][0];
    }
};

template<std::size_t size, class T> struct StrictWeakOrdering<Matrix<size, T>>: public StrictWeakOrdering<RectangularMatrix<size, size, T>> {};

}
#endif

template<std::size_t size, class T> bool Matrix<size, T>::isOrthogonal() const {
    /* Normality */
    for(std::size_t i = 0; i != size; ++i)
        if(!(*this)[i].isNormalized()) return false;

    /* Orthogonality */
    for(std::size_t i = 0; i != size-1; ++i)
        for(std::size_t j = i+1; j != size; ++j)
            if(dot((*this)[i], (*this)[j]) > TypeTraits<T>::epsilon())
                return false;

    return true;
}

template<std::size_t size, class T> Matrix<size-1, T> Matrix<size, T>::ij(const std::size_t skipCol, const std::size_t skipRow) const {
    Matrix<size-1, T> out{NoInit};

    for(std::size_t col = 0; col != size-1; ++col)
        for(std::size_t row = 0; row != size-1; ++row)
            out[col][row] = (*this)[col + (col >= skipCol)]
                                    [row + (row >= skipRow)];

    return out;
}

template<std::size_t size, class T> Matrix<size, T> Matrix<size, T>::inverted() const {
    Matrix<size, T> out{NoInit};

    const T _determinant = determinant();

    for(std::size_t col = 0; col != size; ++col)
        for(std::size_t row = 0; row != size; ++row)
            out[col][row] = (((row+col) & 1) ? -1 : 1)*ij(row, col).determinant()/_determinant;

    return out;
}

}}

namespace Corrade { namespace Utility {
    /** @configurationvalue{Magnum::Math::Matrix} */
    template<std::size_t size, class T> struct ConfigurationValue<Magnum::Math::Matrix<size, T>>: public ConfigurationValue<Magnum::Math::RectangularMatrix<size, size, T>> {};
}}

#endif