gl_mat.hh

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#ifndef GL_MAT_HH
#define GL_MAT_HH
 
// This code implements a mathematical matrix, comparable in functionality
// and syntax to the matrix types in GLSL.
//
// The code was tuned for 4x4 matrixes, but when it didn't complicate the
// code, other sizes (even non-square) are supported as well.
//
// The functionality of this code is focused on openGL(ES). So only the
// operation 'matrix x column-vector' is supported and not e.g. 'row-vector x
// matrix'. The internal layout is also compatible with openGL(ES).
//
// In the past we had (some) manual SSE optimizations in this code. Though for
// the functions that matter (matrix-vector and matrix-matrix multiplication),
// the compiler's auto-vectorization has become as good as the manually
// vectorized code.
 
#include "gl_vec.hh"
#include "xrange.hh"
#include <cassert>
 
namespace gl {
 
// Matrix with M rows and N columns (M by N), elements have type T.
// Internally elements are stored in column-major order. This is the same
// convention as openGL (and Fortran), but different from the typical row-major
// order in C++. This allows to directly pass these matrices to openGL(ES).
template<int M, int N, typename T> class matMxN
{
public:
        // Default copy-constructor and assignment operator.
 
        // Construct identity matrix.
        constexpr matMxN()
        {
                for (auto i : xrange(N)) {
                        for (auto j : xrange(M)) {
                                c[i][j] = (i == j) ? T(1) : T(0);
                        }
                }
        }
 
        // Construct diagonal matrix.
        constexpr explicit matMxN(const vecN<(M < N ? M : N), T>& d)
        {
                for (auto i : xrange(N)) {
                        for (auto j : xrange(M)) {
                                c[i][j] = (i == j) ? d[i] : T(0);
                        }
                }
        }
 
        // Construct from larger matrix (higher order elements are dropped).
        template<int M2, int N2> constexpr explicit matMxN(const matMxN<M2, N2, T>& x)
        {
                static_assert(((M2 > M) && (N2 >= N)) || ((M2 >= M) && (N2 > N)),
                              "matrix must have strictly larger dimensions");
                for (auto i : xrange(N)) c[i] = vecN<M, T>(x[i]);
        }
 
        // Construct matrix from 2 given columns (only valid when N == 2).
        constexpr matMxN(const vecN<M, T>& x, const vecN<M, T>& y)
        {
                static_assert(N == 2, "wrong #constructor arguments");
                c[0] = x; c[1] = y;
        }
 
        // Construct matrix from 3 given columns (only valid when N == 3).
        constexpr matMxN(const vecN<M, T>& x, const vecN<M, T>& y, const vecN<M, T>& z)
        {
                static_assert(N == 3, "wrong #constructor arguments");
                c[0] = x; c[1] = y; c[2] = z;
        }
 
        // Construct matrix from 4 given columns (only valid when N == 4).
        constexpr matMxN(const vecN<M, T>& x, const vecN<M, T>& y,
                         const vecN<M, T>& z, const vecN<M, T>& w)
        {
                static_assert(N == 4, "wrong #constructor arguments");
                c[0] = x; c[1] = y; c[2] = z; c[3] = w;
        }
 
        // Access the i-the column of the matrix.
        // Vectors are also indexable, so 'A[i][j]' returns the element
        // at the i-th column, j-th row.
        [[nodiscard]] constexpr const vecN<M, T>& operator[](unsigned i) const {
                #ifdef DEBUG
                assert(i < N);
                #endif
                return c[i];
        }
        [[nodiscard]] constexpr vecN<M, T>& operator[](unsigned i) {
                #ifdef DEBUG
                assert(i < N);
                #endif
                return c[i];
        }
 
        // Assignment version of the +,-,* operations defined below.
        constexpr matMxN& operator+=(const matMxN& x) { *this = *this + x; return *this; }
        constexpr matMxN& operator-=(const matMxN& x) { *this = *this - x; return *this; }
        constexpr matMxN& operator*=(T             x) { *this = *this * x; return *this; }
        constexpr matMxN& operator*=(const matMxN<N, N, T>& x) { *this = *this * x; return *this; }
 
        [[nodiscard]] constexpr bool operator==(const matMxN&) const = default;
 
private:
        vecN<M, T> c[N];
};
 
 
// Convenience typedefs (same names as used by GLSL).
using mat2 = matMxN<2, 2, float>;
using mat3 = matMxN<3, 3, float>;
using mat4 = matMxN<4, 4, float>;
 
 
// -- Matrix functions --
 
// matrix + matrix
template<int M, int N, typename T>
[[nodiscard]] constexpr matMxN<M, N, T> operator+(const matMxN<M, N, T>& A, const matMxN<M, N, T>& B)
{
        matMxN<M, N, T> R;
        for (auto i : xrange(N)) R[i] = A[i] + B[i];
        return R;
}
 
// matrix - matrix
template<int M, int N, typename T>
[[nodiscard]] constexpr matMxN<M, N, T> operator-(const matMxN<M, N, T>& A, const matMxN<M, N, T>& B)
{
        matMxN<M, N, T> R;
        for (auto i : xrange(N)) R[i] = A[i] - B[i];
        return R;
}
 
// matrix negation
template<int M, int N, typename T>
[[nodiscard]] constexpr matMxN<M, N, T> operator-(const matMxN<M, N, T>& A)
{
        return matMxN<M, N, T>(vecN<(M < N ? M : N), T>()) - A;
}
 
// scalar * matrix
template<int M, int N, typename T>
[[nodiscard]] constexpr matMxN<M, N, T> operator*(T x, const matMxN<M, N, T>& A)
{
        matMxN<M, N, T> R;
        for (auto i : xrange(N)) R[i] = x * A[i];
        return R;
}
 
// matrix * scalar
template<int M, int N, typename T>
[[nodiscard]] constexpr matMxN<M, N, T> operator*(const matMxN<M, N, T>& A, T x)
{
        matMxN<M, N, T> R;
        for (auto i : xrange(N)) R[i] = A[i] * x;
        return R;
}
 
// matrix * column-vector
template<int M, int N, typename T>
[[nodiscard]] constexpr vecN<M, T> operator*(const matMxN<M, N, T>& A, const vecN<N, T>& x)
{
        vecN<M, T> r;
        for (auto i : xrange(N)) r += A[i] * x[i];
        return r;
}
 
// matrix * matrix
template<int M, int N, int O, typename T>
[[nodiscard]] constexpr matMxN<M, O, T> operator*(const matMxN<M, N, T>& A, const matMxN<N, O, T>& B)
{
        matMxN<M, O, T> R;
        for (auto i : xrange(O)) R[i] = A * B[i];
        return R;
}
 
// matrix transpose
template<int M, int N, typename T>
[[nodiscard]] constexpr matMxN<N, M, T> transpose(const matMxN<M, N, T>& A)
{
        matMxN<N, M, T> R;
        for (auto i : xrange(N)) {
                for (auto j : xrange(M)) {
                        R[j][i] = A[i][j];
                }
        }
        return R;
}
 
// determinant of a 2x2 matrix
template<typename T>
[[nodiscard]] constexpr T determinant(const matMxN<2, 2, T>& A)
{
        return A[0][0] * A[1][1] - A[0][1] * A[1][0];
}
 
// determinant of a 3x3 matrix
template<typename T>
[[nodiscard]] constexpr T determinant(const matMxN<3, 3, T>& A)
{
        return A[0][0] * (A[1][1] * A[2][2] - A[1][2] * A[2][1])
             - A[1][0] * (A[0][1] * A[2][2] - A[0][2] * A[2][1])
             + A[2][0] * (A[0][1] * A[1][2] - A[0][2] * A[1][1]);
}
 
// determinant of a 4x4 matrix
template<typename T>
[[nodiscard]] constexpr T determinant(const matMxN<4, 4, T>& A)
{
        // Implementation based on glm:  http://glm.g-truc.net
        T f0 = A[2][2] * A[3][3] - A[3][2] * A[2][3];
        T f1 = A[2][1] * A[3][3] - A[3][1] * A[2][3];
        T f2 = A[2][1] * A[3][2] - A[3][1] * A[2][2];
        T f3 = A[2][0] * A[3][3] - A[3][0] * A[2][3];
        T f4 = A[2][0] * A[3][2] - A[3][0] * A[2][2];
        T f5 = A[2][0] * A[3][1] - A[3][0] * A[2][1];
        vecN<4, T> c((A[1][1] * f0 - A[1][2] * f1 + A[1][3] * f2),
                     (A[1][2] * f3 - A[1][3] * f4 - A[1][0] * f0),
                     (A[1][0] * f1 - A[1][1] * f3 + A[1][3] * f5),
                     (A[1][1] * f4 - A[1][2] * f5 - A[1][0] * f2));
        return dot(A[0], c);
}
 
// inverse of a 2x2 matrix
template<typename T>
[[nodiscard]] constexpr matMxN<2, 2, T> inverse(const matMxN<2, 2, T>& A)
{
        T d = T(1) / determinant(A);
        return d * matMxN<2, 2, T>(vecN<2, T>( A[1][1], -A[0][1]),
                                   vecN<2, T>(-A[1][0],  A[0][0]));
}
 
// inverse of a 3x3 matrix
template<typename T>
[[nodiscard]] constexpr matMxN<3, 3, T> inverse(const matMxN<3, 3, T>& A)
{
        T d = T(1) / determinant(A);
        return d * matMxN<3, 3, T>(
                vecN<3, T>(A[1][1] * A[2][2] - A[1][2] * A[2][1],
                           A[0][2] * A[2][1] - A[0][1] * A[2][2],
                           A[0][1] * A[1][2] - A[0][2] * A[1][1]),
                vecN<3, T>(A[1][2] * A[2][0] - A[1][0] * A[2][2],
                           A[0][0] * A[2][2] - A[0][2] * A[2][0],
                           A[0][2] * A[1][0] - A[0][0] * A[1][2]),
                vecN<3, T>(A[1][0] * A[2][1] - A[1][1] * A[2][0],
                           A[0][1] * A[2][0] - A[0][0] * A[2][1],
                           A[0][0] * A[1][1] - A[0][1] * A[1][0]));
}
 
// inverse of a 4x4 matrix
template<typename T>
[[nodiscard]] constexpr matMxN<4, 4, T> inverse(const matMxN<4, 4, T>& A)
{
        // Implementation based on glm:  http://glm.g-truc.net
 
        T c00 = A[2][2] * A[3][3] - A[3][2] * A[2][3];
        T c02 = A[1][2] * A[3][3] - A[3][2] * A[1][3];
        T c03 = A[1][2] * A[2][3] - A[2][2] * A[1][3];
 
        T c04 = A[2][1] * A[3][3] - A[3][1] * A[2][3];
        T c06 = A[1][1] * A[3][3] - A[3][1] * A[1][3];
        T c07 = A[1][1] * A[2][3] - A[2][1] * A[1][3];
 
        T c08 = A[2][1] * A[3][2] - A[3][1] * A[2][2];
        T c10 = A[1][1] * A[3][2] - A[3][1] * A[1][2];
        T c11 = A[1][1] * A[2][2] - A[2][1] * A[1][2];
 
        T c12 = A[2][0] * A[3][3] - A[3][0] * A[2][3];
        T c14 = A[1][0] * A[3][3] - A[3][0] * A[1][3];
        T c15 = A[1][0] * A[2][3] - A[2][0] * A[1][3];
 
        T c16 = A[2][0] * A[3][2] - A[3][0] * A[2][2];
        T c18 = A[1][0] * A[3][2] - A[3][0] * A[1][2];
        T c19 = A[1][0] * A[2][2] - A[2][0] * A[1][2];
 
        T c20 = A[2][0] * A[3][1] - A[3][0] * A[2][1];
        T c22 = A[1][0] * A[3][1] - A[3][0] * A[1][1];
        T c23 = A[1][0] * A[2][1] - A[2][0] * A[1][1];
 
        vecN<4, T> f0(c00, c00, c02, c03);
        vecN<4, T> f1(c04, c04, c06, c07);
        vecN<4, T> f2(c08, c08, c10, c11);
        vecN<4, T> f3(c12, c12, c14, c15);
        vecN<4, T> f4(c16, c16, c18, c19);
        vecN<4, T> f5(c20, c20, c22, c23);
 
        vecN<4, T> v0(A[1][0], A[0][0], A[0][0], A[0][0]);
        vecN<4, T> v1(A[1][1], A[0][1], A[0][1], A[0][1]);
        vecN<4, T> v2(A[1][2], A[0][2], A[0][2], A[0][2]);
        vecN<4, T> v3(A[1][3], A[0][3], A[0][3], A[0][3]);
 
        vecN<4, T> i0(v1 * f0 - v2 * f1 + v3 * f2);
        vecN<4, T> i1(v0 * f0 - v2 * f3 + v3 * f4);
        vecN<4, T> i2(v0 * f1 - v1 * f3 + v3 * f5);
        vecN<4, T> i3(v0 * f2 - v1 * f4 + v2 * f5);
 
        vecN<4, T> sa(+1, -1, +1, -1);
        vecN<4, T> sb(-1, +1, -1, +1);
        matMxN<4, 4, T> inverse(i0 * sa, i1 * sb, i2 * sa, i3 * sb);
 
        vecN<4, T> row0(inverse[0][0], inverse[1][0], inverse[2][0], inverse[3][0]);
        T OneOverDeterminant = static_cast<T>(1) / dot(A[0], row0);
 
        return inverse * OneOverDeterminant;
}
 
// norm-2 squared
template<int M, int N, typename T>
[[nodiscard]] constexpr T norm2_2(const matMxN<M, N, T>& A)
{
        vecN<M, T> t;
        for (auto i : xrange(N)) t += A[i] * A[i];
        return sum(t);
}
 
// Textual representation. (Only) used to debug unittest.
template<int M, int N, typename T>
std::ostream& operator<<(std::ostream& os, const matMxN<M, N, T>& A)
{
        for (auto j : xrange(M)) {
                for (auto i : xrange(N)) {
                        os << A[i][j] << ' ';
                }
                os << '\n';
        }
        return os;
}
 
} // namespace gl
 
#endif