() const noexcept
{
return size_cast(std::make_index_sequence
{});
}
/// Returns a zero matrix, where every element is equal to zero.
static constexpr matrix zero() noexcept
{
return {};
}
/// @private
template
static constexpr inline matrix one(std::index_sequence) noexcept
{
//return {column_vector_type::one() ...};
// MSVC bug workaround (I must be referenced for parameter pack expansion)
return {(I ? column_vector_type::one() : column_vector_type::one()) ...};
}
/// Returns a matrix of ones, where every element is equal to one.
static constexpr matrix one() noexcept
{
return one(std::make_index_sequence{});
}
/// @private
template
static constexpr inline column_vector_type identity_column(std::size_t i, std::index_sequence) noexcept
{
return {(I == i ? T{1} : T{0}) ...};
}
/// @private
template
static constexpr inline matrix identity(std::index_sequence) noexcept
{
return {identity_column(I, std::make_index_sequence{}) ...};
}
/// Returns an identity matrix, with ones on the main diagonal and zeros elsewhere.
static constexpr matrix identity() noexcept
{
return identity(std::make_index_sequence{});
}
};
/// 2x2 matrix.
template
using matrix2 = matrix;
/// 2x2 matrix.
template
using matrix2x2 = matrix;
/// 3x3 matrix.
template
using matrix3 = matrix;
/// 3x3 matrix.
template
using matrix3x3 = matrix;
/// 4x4 matrix.
template
using matrix4 = matrix;
/// 4x4 matrix.
template
using matrix4x4 = matrix;
/**
* Adds two matrices.
*
* @param a First matrix.
* @param b Second matrix.
*
* @return Sum of the two matrices.
*/
template
constexpr matrix add(const matrix& a, const matrix& b) noexcept;
/**
* Adds a matrix and a scalar.
*
* @param a Matrix.
* @param b scalar.
*
* @return Sum of the matrix and scalar.
*/
template
constexpr matrix add(const matrix& a, T b) noexcept;
/**
* Calculates the determinant of a square matrix.
*
* @param m Matrix of which to take the determinant.
*
* @return Determinant of @p m.
*
* @warning Currently only implemented for 2x2, 3x3, and 4x4 matrices.
*/
template
constexpr T determinant(const matrix& m) noexcept;
/**
* Calculates the inverse of a square matrix.
*
* @param m Matrix of which to take the inverse.
*
* @return Inverse of matrix @p m.
*
* @warning Currently only implemented for 2x2, 3x3, and 4x4 matrices.
*/
template
constexpr matrix inverse(const matrix& m) noexcept;
/**
* Performs a component-wise multiplication of two matrices.
*
* @param x First matrix multiplicand.
* @param y Second matrix multiplicand.
*/
template
constexpr matrix componentwise_mul(const matrix& a, const matrix& b) noexcept;
/**
* Divides a matrix by a matrix.
*
* @param a First matrix.
* @param b Second matrix.
* @return Result of the division.
*/
template
constexpr matrix div(const matrix& a, const matrix& b) noexcept;
/**
* Divides a matrix by a scalar.
*
* @param a Matrix.
* @param b Scalar.
* @return Result of the division.
*/
template
constexpr matrix div(const matrix& a, T b) noexcept;
/**
* Divides a scalar by a matrix.
*
* @param a Scalar.
* @param b Matrix.
* @return Result of the division.
*/
template
constexpr matrix div(T a, const matrix& b) noexcept;
/**
* Creates a viewing transformation matrix.
*
* @param position Position of the view point.
* @param target Position of the target.
* @param up Normalized direction of the up vector.
* @return Viewing transformation matrix.
*/
template
constexpr matrix look_at(const vector& position, const vector& target, vector up);
/**
* Multiplies two matrices
*
* @tparam T Matrix element type.
* @tparam N Number of columns in matrix @p a, and rows in matrix @p b.
* @tparam M Number of rows in matrix @p a.
* @tparam P Number of columns in matrix @p b.
*
* @param a First matrix.
* @param b Second matrix.
*
* @return Product of `a * b`.
*/
template
constexpr matrix mul(const matrix& a, const matrix& b) noexcept;
/**
* Multiplies a matrix by a scalar.
*
* @param a Matrix.
* @param b Scalar.
* @return Product of the matrix and the scalar.
*/
template
constexpr matrix mul(const matrix& a, T b) noexcept;
/**
* Calculates the product of a matrix and a row vector.
*
* @param a Matrix.
* @param b Row vector
*
* @return Product of the matrix and the row vector.
*/
template
constexpr typename matrix::column_vector_type mul(const matrix& a, const typename matrix::row_vector_type& b) noexcept;
/**
* Calculates the product of a column vector and a matrix.
*
* @param a Column vector.
* @param b Matrix.
*
* @return Product of the column vector and the matrix.
*/
template
constexpr typename matrix::row_vector_type mul(const typename matrix::column_vector_type& a, const matrix& b) noexcept;
/**
* Constructs a rotation matrix.
*
* @param angle Angle of rotation, in radians.
* @param axis Axis of rotation
* @return Rotation matrix.
*/
template
matrix rotate(T angle, const vector& axis);
/**
* Produces a matrix which rotates Cartesian coordinates about the x-axis by a given angle.
*
* @param angle Angle of rotation, in radians.
* @return Rotation matrix.
*/
template
matrix3 rotate_x(T angle);
/**
* Produces a matrix which rotates Cartesian coordinates about the y-axis by a given angle.
*
* @param angle Angle of rotation, in radians.
* @return Rotation matrix.
*/
template
matrix3 rotate_y(T angle);
/**
* Produces a matrix which rotates Cartesian coordinates about the z-axis by a given angle.
*
* @param angle Angle of rotation, in radians.
* @return Rotation matrix.
*/
template
matrix3 rotate_z(T angle);
/**
* Scales a matrix.
*
* @param m Matrix to scale.
* @param v Scale vector.
* @return Scaled matrix.
*/
template
constexpr matrix scale(const matrix& m, const vector& v);
/**
* Subtracts a matrix from another matrix.
*
* @param a First matrix.
* @param b Second matrix.
*
* @return Difference between the two matrices.
*/
template
constexpr matrix sub(const matrix& a, const matrix& b) noexcept;
/**
* Subtracts a scalar from matrix.
*
* @param a Matrix.
* @param b Scalar.
*
* @return Difference between the matrix and scalar.
*/
template
constexpr matrix sub(const matrix& a, T b) noexcept;
/**
* Subtracts a matrix from a scalar.
*
* @param a Scalar.
* @param b Matrix.
*
* @return Difference between the scalar and matrix.
*/
template
constexpr matrix sub(T a, const matrix& b) noexcept;
/**
* Translates a matrix.
*
* @param m Matrix to translate.
* @param v Translation vector.
* @return Translated matrix.
*/
template
constexpr matrix translate(const matrix& m, const vector& v);
/**
* Calculates the transpose of a matrix.
*
* @param m Matrix to transpose.
*
* @return Transposed matrix.
*/
template
constexpr matrix transpose(const matrix& m) noexcept;
/// @private
template
constexpr inline matrix add(const matrix& a, const matrix& b, std::index_sequence) noexcept
{
return {(a[I] + b[I]) ...};
}
template
constexpr matrix add(const matrix& a, const matrix& b) noexcept
{
return add(a, b, std::make_index_sequence{});
}
/// @private
template
constexpr inline matrix add(const matrix& a, T b, std::index_sequence) noexcept
{
return {(a[I] + b) ...};
}
template
constexpr matrix add(const matrix& a, T b) noexcept
{
return add(a, b, std::make_index_sequence{});
}
template
constexpr T determinant(const matrix& m) noexcept
{
return
m[0][0] * m[1][1] -
m[0][1] * m[1][0];
}
template
constexpr T determinant(const matrix& m) noexcept
{
return
m[0][0] * m[1][1] * m[2][2] +
m[0][1] * m[1][2] * m[2][0] +
m[0][2] * m[1][0] * m[2][1] -
m[0][0] * m[1][2] * m[2][1] -
m[0][1] * m[1][0] * m[2][2] -
m[0][2] * m[1][1] * m[2][0];
}
template
constexpr T determinant(const matrix& m) noexcept
{
return
m[0][3] * m[1][2] * m[2][1] * m[3][0] - m[0][2] * m[1][3] * m[2][1] * m[3][0] -
m[0][3] * m[1][1] * m[2][2] * m[3][0] + m[0][1] * m[1][3] * m[2][2] * m[3][0] +
m[0][2] * m[1][1] * m[2][3] * m[3][0] - m[0][1] * m[1][2] * m[2][3] * m[3][0] -
m[0][3] * m[1][2] * m[2][0] * m[3][1] + m[0][2] * m[1][3] * m[2][0] * m[3][1] +
m[0][3] * m[1][0] * m[2][2] * m[3][1] - m[0][0] * m[1][3] * m[2][2] * m[3][1] -
m[0][2] * m[1][0] * m[2][3] * m[3][1] + m[0][0] * m[1][2] * m[2][3] * m[3][1] +
m[0][3] * m[1][1] * m[2][0] * m[3][2] - m[0][1] * m[1][3] * m[2][0] * m[3][2] -
m[0][3] * m[1][0] * m[2][1] * m[3][2] + m[0][0] * m[1][3] * m[2][1] * m[3][2] +
m[0][1] * m[1][0] * m[2][3] * m[3][2] - m[0][0] * m[1][1] * m[2][3] * m[3][2] -
m[0][2] * m[1][1] * m[2][0] * m[3][3] + m[0][1] * m[1][2] * m[2][0] * m[3][3] +
m[0][2] * m[1][0] * m[2][1] * m[3][3] - m[0][0] * m[1][2] * m[2][1] * m[3][3] -
m[0][1] * m[1][0] * m[2][2] * m[3][3] + m[0][0] * m[1][1] * m[2][2] * m[3][3];
}
template
constexpr matrix inverse(const matrix& m) noexcept
{
const T inv_det = T{1} / determinant(m);
return
{
m[1][1] * inv_det,
-m[0][1] * inv_det,
-m[1][0] * inv_det,
m[0][0] * inv_det
};
}
template
constexpr matrix inverse(const matrix& m) noexcept
{
const T inv_det = T{1} / determinant(m);
return
{
(m[1][1] * m[2][2] - m[1][2] * m[2][1]) * inv_det,
(m[0][2] * m[2][1] - m[0][1] * m[2][2]) * inv_det,
(m[0][1] * m[1][2] - m[0][2] * m[1][1]) * inv_det,
(m[1][2] * m[2][0] - m[1][0] * m[2][2]) * inv_det,
(m[0][0] * m[2][2] - m[0][2] * m[2][0]) * inv_det,
(m[0][2] * m[1][0] - m[0][0] * m[1][2]) * inv_det,
(m[1][0] * m[2][1] - m[1][1] * m[2][0]) * inv_det,
(m[0][1] * m[2][0] - m[0][0] * m[2][1]) * inv_det,
(m[0][0] * m[1][1] - m[0][1] * m[1][0]) * inv_det
};
}
template
constexpr matrix inverse(const matrix& m) noexcept
{
const T inv_det = T{1} / determinant(m);
return
{
(m[1][2] * m[2][3] * m[3][1] - m[1][3] * m[2][2] * m[3][1] + m[1][3] * m[2][1] * m[3][2] - m[1][1] * m[2][3] * m[3][2] - m[1][2] * m[2][1] * m[3][3] + m[1][1] * m[2][2] * m[3][3]) * inv_det,
(m[0][3] * m[2][2] * m[3][1] - m[0][2] * m[2][3] * m[3][1] - m[0][3] * m[2][1] * m[3][2] + m[0][1] * m[2][3] * m[3][2] + m[0][2] * m[2][1] * m[3][3] - m[0][1] * m[2][2] * m[3][3]) * inv_det,
(m[0][2] * m[1][3] * m[3][1] - m[0][3] * m[1][2] * m[3][1] + m[0][3] * m[1][1] * m[3][2] - m[0][1] * m[1][3] * m[3][2] - m[0][2] * m[1][1] * m[3][3] + m[0][1] * m[1][2] * m[3][3]) * inv_det,
(m[0][3] * m[1][2] * m[2][1] - m[0][2] * m[1][3] * m[2][1] - m[0][3] * m[1][1] * m[2][2] + m[0][1] * m[1][3] * m[2][2] + m[0][2] * m[1][1] * m[2][3] - m[0][1] * m[1][2] * m[2][3]) * inv_det,
(m[1][3] * m[2][2] * m[3][0] - m[1][2] * m[2][3] * m[3][0] - m[1][3] * m[2][0] * m[3][2] + m[1][0] * m[2][3] * m[3][2] + m[1][2] * m[2][0] * m[3][3] - m[1][0] * m[2][2] * m[3][3]) * inv_det,
(m[0][2] * m[2][3] * m[3][0] - m[0][3] * m[2][2] * m[3][0] + m[0][3] * m[2][0] * m[3][2] - m[0][0] * m[2][3] * m[3][2] - m[0][2] * m[2][0] * m[3][3] + m[0][0] * m[2][2] * m[3][3]) * inv_det,
(m[0][3] * m[1][2] * m[3][0] - m[0][2] * m[1][3] * m[3][0] - m[0][3] * m[1][0] * m[3][2] + m[0][0] * m[1][3] * m[3][2] + m[0][2] * m[1][0] * m[3][3] - m[0][0] * m[1][2] * m[3][3]) * inv_det,
(m[0][2] * m[1][3] * m[2][0] - m[0][3] * m[1][2] * m[2][0] + m[0][3] * m[1][0] * m[2][2] - m[0][0] * m[1][3] * m[2][2] - m[0][2] * m[1][0] * m[2][3] + m[0][0] * m[1][2] * m[2][3]) * inv_det,
(m[1][1] * m[2][3] * m[3][0] - m[1][3] * m[2][1] * m[3][0] + m[1][3] * m[2][0] * m[3][1] - m[1][0] * m[2][3] * m[3][1] - m[1][1] * m[2][0] * m[3][3] + m[1][0] * m[2][1] * m[3][3]) * inv_det,
(m[0][3] * m[2][1] * m[3][0] - m[0][1] * m[2][3] * m[3][0] - m[0][3] * m[2][0] * m[3][1] + m[0][0] * m[2][3] * m[3][1] + m[0][1] * m[2][0] * m[3][3] - m[0][0] * m[2][1] * m[3][3]) * inv_det,
(m[0][1] * m[1][3] * m[3][0] - m[0][3] * m[1][1] * m[3][0] + m[0][3] * m[1][0] * m[3][1] - m[0][0] * m[1][3] * m[3][1] - m[0][1] * m[1][0] * m[3][3] + m[0][0] * m[1][1] * m[3][3]) * inv_det,
(m[0][3] * m[1][1] * m[2][0] - m[0][1] * m[1][3] * m[2][0] - m[0][3] * m[1][0] * m[2][1] + m[0][0] * m[1][3] * m[2][1] + m[0][1] * m[1][0] * m[2][3] - m[0][0] * m[1][1] * m[2][3]) * inv_det,
(m[1][2] * m[2][1] * m[3][0] - m[1][1] * m[2][2] * m[3][0] - m[1][2] * m[2][0] * m[3][1] + m[1][0] * m[2][2] * m[3][1] + m[1][1] * m[2][0] * m[3][2] - m[1][0] * m[2][1] * m[3][2]) * inv_det,
(m[0][1] * m[2][2] * m[3][0] - m[0][2] * m[2][1] * m[3][0] + m[0][2] * m[2][0] * m[3][1] - m[0][0] * m[2][2] * m[3][1] - m[0][1] * m[2][0] * m[3][2] + m[0][0] * m[2][1] * m[3][2]) * inv_det,
(m[0][2] * m[1][1] * m[3][0] - m[0][1] * m[1][2] * m[3][0] - m[0][2] * m[1][0] * m[3][1] + m[0][0] * m[1][2] * m[3][1] + m[0][1] * m[1][0] * m[3][2] - m[0][0] * m[1][1] * m[3][2]) * inv_det,
(m[0][1] * m[1][2] * m[2][0] - m[0][2] * m[1][1] * m[2][0] + m[0][2] * m[1][0] * m[2][1] - m[0][0] * m[1][2] * m[2][1] - m[0][1] * m[1][0] * m[2][2] + m[0][0] * m[1][1] * m[2][2]) * inv_det
};
}
/// @private
template
constexpr inline matrix componentwise_mul(const matrix& a, const matrix& b, std::index_sequence) noexcept
{
return {(a[I] * b[I]) ...};
}
template
constexpr matrix componentwise_mul(const matrix& a, const matrix& b) noexcept
{
return componentwise_mul(a, b, std::make_index_sequence{});
}
/// @private
template
constexpr inline matrix div(const matrix& a, const matrix& b, std::index_sequence) noexcept
{
return {(a[I] / b[I]) ...};
}
template
constexpr matrix div(const matrix& a, const matrix& b) noexcept
{
return div(a, b, std::make_index_sequence{});
}
/// @private
template
constexpr inline matrix div(const matrix& a, T b, std::index_sequence) noexcept
{
return {(a[I] / b) ...};
}
template
constexpr matrix div(const matrix& a, T b) noexcept
{
return div(a, b, std::make_index_sequence{});
}
/// @private
template
constexpr inline matrix div(T a, const matrix& b, std::index_sequence) noexcept
{
return {(a / b[I]) ...};
}
template
constexpr matrix div(T a, const matrix& b) noexcept
{
return div(a, b, std::make_index_sequence{});
}
template
constexpr matrix look_at(const vector& position, const vector& target, vector up)
{
vector forward = normalize(sub(target, position));
vector right = normalize(cross(forward, up));
up = cross(right, forward);
matrix m =
{{
{right[0], up[0], -forward[0], T(0)},
{right[1], up[1], -forward[1], T(0)},
{right[2], up[2], -forward[2], T(0)},
{T(0), T(0), T(0), T(1)}
}};
return translate(m, negate(position));
}
template
constexpr matrix mul(const matrix& a, const matrix& b) noexcept
{
matrix c = matrix::zero();
for (std::size_t i = 0; i < P; ++i)
{
for (std::size_t j = 0; j < M; ++j)
{
for (std::size_t k = 0; k < N; ++k)
{
c[i][j] += a[k][j] * b[i][k];
}
}
}
return c;
}
/// @private
template
constexpr inline matrix mul(const matrix& a, T b, std::index_sequence) noexcept
{
return {(a[I] * b) ...};
}
template
constexpr matrix mul(const matrix& a, T b) noexcept
{
return mul(a, b, std::make_index_sequence{});
}
/// @private
template
constexpr inline typename matrix::column_vector_type mul(const matrix& a, const typename matrix::row_vector_type& b, std::index_sequence) noexcept
{
return ((a[I] * b[I]) + ...);
}
template
constexpr typename matrix::column_vector_type mul(const matrix& a, const typename matrix::row_vector_type& b) noexcept
{
return mul(a, b, std::make_index_sequence{});
}
/// @private
template
constexpr inline typename matrix::row_vector_type mul(const typename matrix::column_vector_type& a, const matrix& b, std::index_sequence) noexcept
{
return {dot(a, b[I]) ...};
}
template
constexpr typename matrix::row_vector_type mul(const typename matrix::column_vector_type& a, const matrix& b) noexcept
{
return mul(a, b, std::make_index_sequence{});
}
template
matrix rotate(T angle, const vector& axis)
{
const T c = std::cos(angle);
const T s = std::sin(angle);
const vector temp = mul(axis, T(1) - c);
matrix rotation;
rotation[0][0] = axis[0] * temp[0] + c;
rotation[0][1] = axis[1] * temp[0] + axis[2] * s;
rotation[0][2] = axis[2] * temp[0] - axis[1] * s;
rotation[1][0] = axis[0] * temp[1] - axis[2] * s;
rotation[1][1] = axis[1] * temp[1] + c;
rotation[1][2] = axis[2] * temp[1] + axis[0] * s;
rotation[2][0] = axis[0] * temp[2] + axis[1] * s;
rotation[2][1] = axis[1] * temp[2] - axis[0] * s;
rotation[2][2] = axis[2] * temp[2] + c;
return rotation;
}
template
matrix3 rotate_x(T angle)
{
const T c = std::cos(angle);
const T s = std::sin(angle);
return matrix3
{
T(1), T(0), T(0),
T(0), c, s,
T(0), -s, c
};
}
template