/************************************************************************* * Copyright (C) 2009-2014 Tavian Barnes * * * * This file is part of The Dimension Library. * * * * The Dimension Library is free software; you can redistribute it and/ * * or modify it under the terms of the GNU Lesser General Public License * * as published by the Free Software Foundation; either version 3 of the * * License, or (at your option) any later version. * * * * The Dimension Library is distributed in the hope that it will be * * useful, but WITHOUT ANY WARRANTY; without even the implied warranty * * of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU * * Lesser General Public License for more details. * * * * You should have received a copy of the GNU Lesser General Public * * License along with this program. If not, see * * . * *************************************************************************/ /** * @file * Affine transformation matrices. */ #ifndef DMNSN_MATH_H #error "Please include instead of this header directly." #endif /** A 4x4 affine transformation matrix, with implied [0 0 0 1] bottom row. */ typedef struct dmnsn_matrix { double n[3][4]; /**< The matrix elements in row-major order. */ } dmnsn_matrix; /** A standard format string for matricies. */ #define DMNSN_MATRIX_FORMAT \ "[%g\t%g\t%g\t%g]\n" \ "[%g\t%g\t%g\t%g]\n" \ "[%g\t%g\t%g\t%g]\n" \ "[%g\t%g\t%g\t%g]" /** The appropriate arguements to printf() a matrix. */ #define DMNSN_MATRIX_PRINTF(m) \ (m).n[0][0], (m).n[0][1], (m).n[0][2], (m).n[0][3], \ (m).n[1][0], (m).n[1][1], (m).n[1][2], (m).n[1][3], \ (m).n[2][0], (m).n[2][1], (m).n[2][2], (m).n[2][3], \ 0.0, 0.0, 0.0, 1.0 /** Construct a new transformation matrix. */ DMNSN_INLINE dmnsn_matrix dmnsn_new_matrix(double a0, double a1, double a2, double a3, double b0, double b1, double b2, double b3, double c0, double c1, double c2, double c3) { dmnsn_matrix m = { { { a0, a1, a2, a3 }, { b0, b1, b2, b3 }, { c0, c1, c2, c3 } } }; return m; } /** Construct a new transformation matrix from column vectors. */ DMNSN_INLINE dmnsn_matrix dmnsn_new_matrix4(dmnsn_vector a, dmnsn_vector b, dmnsn_vector c, dmnsn_vector d) { dmnsn_matrix m = { { { a.x, b.x, c.x, d.x }, { a.y, b.y, c.y, d.y }, { a.z, b.z, c.z, d.z } } }; return m; } /** Extract column vectors from a matrix. */ DMNSN_INLINE dmnsn_vector dmnsn_matrix_column(dmnsn_matrix M, unsigned int i) { return dmnsn_new_vector(M.n[0][i], M.n[1][i], M.n[2][i]); } /** Return the identity matrix. */ dmnsn_matrix dmnsn_identity_matrix(void); /** * A scale transformation. * @param[in] s A vector with components representing the scaling factor in * each axis. * @return The transformation matrix. */ dmnsn_matrix dmnsn_scale_matrix(dmnsn_vector s); /** * A translation. * @param[in] d The vector to translate by. * @return The transformation matrix. */ dmnsn_matrix dmnsn_translation_matrix(dmnsn_vector d); /** * A left-handed rotation. * @param[in] theta A vector representing an axis and angle. * @f$ axis = \vec{\theta}/|\vec{\theta}| @f$, * @f$ angle = |\vec{\theta}| @f$ * @return The transformation matrix. */ dmnsn_matrix dmnsn_rotation_matrix(dmnsn_vector theta); /** * An alignment matrix. * @param[in] from The initial vector. * @param[in] to The desired direction. * @param[in] axis1 The first axis about which to rotate. * @param[in] axis2 The second axis about which to rotate. * @return A transformation matrix that will rotate \p from to \p to. */ dmnsn_matrix dmnsn_alignment_matrix(dmnsn_vector from, dmnsn_vector to, dmnsn_vector axis1, dmnsn_vector axis2); /** Invert a matrix. */ dmnsn_matrix dmnsn_matrix_inverse(dmnsn_matrix A); /** Multiply two matricies. */ dmnsn_matrix dmnsn_matrix_mul(dmnsn_matrix lhs, dmnsn_matrix rhs); /** Transform a point by a matrix. */ DMNSN_INLINE dmnsn_vector dmnsn_transform_point(dmnsn_matrix T, dmnsn_vector v) { /* 9 multiplications, 9 additions */ dmnsn_vector r; r.x = T.n[0][0]*v.x + T.n[0][1]*v.y + T.n[0][2]*v.z + T.n[0][3]; r.y = T.n[1][0]*v.x + T.n[1][1]*v.y + T.n[1][2]*v.z + T.n[1][3]; r.z = T.n[2][0]*v.x + T.n[2][1]*v.y + T.n[2][2]*v.z + T.n[2][3]; return r; } /** Transform a direction by a matrix. */ DMNSN_INLINE dmnsn_vector dmnsn_transform_direction(dmnsn_matrix T, dmnsn_vector v) { /* 9 multiplications, 6 additions */ dmnsn_vector r; r.x = T.n[0][0]*v.x + T.n[0][1]*v.y + T.n[0][2]*v.z; r.y = T.n[1][0]*v.x + T.n[1][1]*v.y + T.n[1][2]*v.z; r.z = T.n[2][0]*v.x + T.n[2][1]*v.y + T.n[2][2]*v.z; return r; } /** * Transform a pseudovector by a matrix. * @param[in] Tinv The inverse of the transformation matrix. * @param[in] v The pseudovector to transform * @return The transformed pseudovector. */ DMNSN_INLINE dmnsn_vector dmnsn_transform_normal(dmnsn_matrix Tinv, dmnsn_vector v) { /* Multiply by the transpose of the inverse (9 multiplications, 6 additions) */ dmnsn_vector r; r.x = Tinv.n[0][0]*v.x + Tinv.n[1][0]*v.y + Tinv.n[2][0]*v.z; r.y = Tinv.n[0][1]*v.x + Tinv.n[1][1]*v.y + Tinv.n[2][1]*v.z; r.z = Tinv.n[0][2]*v.x + Tinv.n[1][2]*v.y + Tinv.n[2][2]*v.z; return r; } /** * Transform a ray by a matrix. * \f$ n' = T(l.\vec{x_0} + l.\vec{n}) - T(l.\vec{x_0}) \f$, * \f$ \vec{x_0}' = T(l.\vec{x_0}) \f$ */ DMNSN_INLINE dmnsn_ray dmnsn_transform_ray(dmnsn_matrix T, dmnsn_ray l) { /* 18 multiplications, 15 additions */ dmnsn_ray ret; ret.x0 = dmnsn_transform_point(T, l.x0); ret.n = dmnsn_transform_direction(T, l.n); return ret; } /** Transform a bounding box by a matrix. */ dmnsn_aabb dmnsn_transform_aabb(dmnsn_matrix T, dmnsn_aabb box); /** Return whether a matrix contains any NaN components. */ DMNSN_INLINE bool dmnsn_matrix_isnan(dmnsn_matrix m) { size_t i, j; for (i = 0; i < 3; ++i) { for (j = 0; j < 4; ++j) { if (dmnsn_isnan(m.n[i][j])) { return true; } } } return false; }