m_matrix.c revision 1e091f48f0434e8fb9713fbebc9d74ad68a75e34
1/* $Id: m_matrix.c,v 1.15 2003/01/08 16:42:47 brianp Exp $ */ 2 3/* 4 * Mesa 3-D graphics library 5 * Version: 5.1 6 * 7 * Copyright (C) 1999-2003 Brian Paul All Rights Reserved. 8 * 9 * Permission is hereby granted, free of charge, to any person obtaining a 10 * copy of this software and associated documentation files (the "Software"), 11 * to deal in the Software without restriction, including without limitation 12 * the rights to use, copy, modify, merge, publish, distribute, sublicense, 13 * and/or sell copies of the Software, and to permit persons to whom the 14 * Software is furnished to do so, subject to the following conditions: 15 * 16 * The above copyright notice and this permission notice shall be included 17 * in all copies or substantial portions of the Software. 18 * 19 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 20 * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 21 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 22 * BRIAN PAUL BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN 23 * AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN 24 * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. 25 */ 26 27 28/* 29 * Matrix operations 30 * 31 * NOTES: 32 * 1. 4x4 transformation matrices are stored in memory in column major order. 33 * 2. Points/vertices are to be thought of as column vectors. 34 * 3. Transformation of a point p by a matrix M is: p' = M * p 35 */ 36 37#include "glheader.h" 38#include "imports.h" 39#include "macros.h" 40#include "imports.h" 41#include "mmath.h" 42 43#include "m_matrix.h" 44 45 46static const char *types[] = { 47 "MATRIX_GENERAL", 48 "MATRIX_IDENTITY", 49 "MATRIX_3D_NO_ROT", 50 "MATRIX_PERSPECTIVE", 51 "MATRIX_2D", 52 "MATRIX_2D_NO_ROT", 53 "MATRIX_3D" 54}; 55 56 57static GLfloat Identity[16] = { 58 1.0, 0.0, 0.0, 0.0, 59 0.0, 1.0, 0.0, 0.0, 60 0.0, 0.0, 1.0, 0.0, 61 0.0, 0.0, 0.0, 1.0 62}; 63 64 65 66 67/* 68 * This matmul was contributed by Thomas Malik 69 * 70 * Perform a 4x4 matrix multiplication (product = a x b). 71 * Input: a, b - matrices to multiply 72 * Output: product - product of a and b 73 * WARNING: (product != b) assumed 74 * NOTE: (product == a) allowed 75 * 76 * KW: 4*16 = 64 muls 77 */ 78#define A(row,col) a[(col<<2)+row] 79#define B(row,col) b[(col<<2)+row] 80#define P(row,col) product[(col<<2)+row] 81 82static void matmul4( GLfloat *product, const GLfloat *a, const GLfloat *b ) 83{ 84 GLint i; 85 for (i = 0; i < 4; i++) { 86 const GLfloat ai0=A(i,0), ai1=A(i,1), ai2=A(i,2), ai3=A(i,3); 87 P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0) + ai3 * B(3,0); 88 P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1) + ai3 * B(3,1); 89 P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2) + ai3 * B(3,2); 90 P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3 * B(3,3); 91 } 92} 93 94 95/* Multiply two matrices known to occupy only the top three rows, such 96 * as typical model matrices, and ortho matrices. 97 */ 98static void matmul34( GLfloat *product, const GLfloat *a, const GLfloat *b ) 99{ 100 GLint i; 101 for (i = 0; i < 3; i++) { 102 const GLfloat ai0=A(i,0), ai1=A(i,1), ai2=A(i,2), ai3=A(i,3); 103 P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0); 104 P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1); 105 P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2); 106 P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3; 107 } 108 P(3,0) = 0; 109 P(3,1) = 0; 110 P(3,2) = 0; 111 P(3,3) = 1; 112} 113 114 115#undef A 116#undef B 117#undef P 118 119 120/* 121 * Multiply a matrix by an array of floats with known properties. 122 */ 123static void matrix_multf( GLmatrix *mat, const GLfloat *m, GLuint flags ) 124{ 125 mat->flags |= (flags | MAT_DIRTY_TYPE | MAT_DIRTY_INVERSE); 126 127 if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D)) 128 matmul34( mat->m, mat->m, m ); 129 else 130 matmul4( mat->m, mat->m, m ); 131} 132 133 134static void print_matrix_floats( const GLfloat m[16] ) 135{ 136 int i; 137 for (i=0;i<4;i++) { 138 _mesa_debug(NULL,"\t%f %f %f %f\n", m[i], m[4+i], m[8+i], m[12+i] ); 139 } 140} 141 142void 143_math_matrix_print( const GLmatrix *m ) 144{ 145 _mesa_debug(NULL, "Matrix type: %s, flags: %x\n", types[m->type], m->flags); 146 print_matrix_floats(m->m); 147 _mesa_debug(NULL, "Inverse: \n"); 148 if (m->inv) { 149 GLfloat prod[16]; 150 print_matrix_floats(m->inv); 151 matmul4(prod, m->m, m->inv); 152 _mesa_debug(NULL, "Mat * Inverse:\n"); 153 print_matrix_floats(prod); 154 } 155 else { 156 _mesa_debug(NULL, " - not available\n"); 157 } 158} 159 160 161 162 163#define SWAP_ROWS(a, b) { GLfloat *_tmp = a; (a)=(b); (b)=_tmp; } 164#define MAT(m,r,c) (m)[(c)*4+(r)] 165 166/* 167 * Compute inverse of 4x4 transformation matrix. 168 * Code contributed by Jacques Leroy jle@star.be 169 * Return GL_TRUE for success, GL_FALSE for failure (singular matrix) 170 */ 171static GLboolean invert_matrix_general( GLmatrix *mat ) 172{ 173 const GLfloat *m = mat->m; 174 GLfloat *out = mat->inv; 175 GLfloat wtmp[4][8]; 176 GLfloat m0, m1, m2, m3, s; 177 GLfloat *r0, *r1, *r2, *r3; 178 179 r0 = wtmp[0], r1 = wtmp[1], r2 = wtmp[2], r3 = wtmp[3]; 180 181 r0[0] = MAT(m,0,0), r0[1] = MAT(m,0,1), 182 r0[2] = MAT(m,0,2), r0[3] = MAT(m,0,3), 183 r0[4] = 1.0, r0[5] = r0[6] = r0[7] = 0.0, 184 185 r1[0] = MAT(m,1,0), r1[1] = MAT(m,1,1), 186 r1[2] = MAT(m,1,2), r1[3] = MAT(m,1,3), 187 r1[5] = 1.0, r1[4] = r1[6] = r1[7] = 0.0, 188 189 r2[0] = MAT(m,2,0), r2[1] = MAT(m,2,1), 190 r2[2] = MAT(m,2,2), r2[3] = MAT(m,2,3), 191 r2[6] = 1.0, r2[4] = r2[5] = r2[7] = 0.0, 192 193 r3[0] = MAT(m,3,0), r3[1] = MAT(m,3,1), 194 r3[2] = MAT(m,3,2), r3[3] = MAT(m,3,3), 195 r3[7] = 1.0, r3[4] = r3[5] = r3[6] = 0.0; 196 197 /* choose pivot - or die */ 198 if (fabs(r3[0])>fabs(r2[0])) SWAP_ROWS(r3, r2); 199 if (fabs(r2[0])>fabs(r1[0])) SWAP_ROWS(r2, r1); 200 if (fabs(r1[0])>fabs(r0[0])) SWAP_ROWS(r1, r0); 201 if (0.0 == r0[0]) return GL_FALSE; 202 203 /* eliminate first variable */ 204 m1 = r1[0]/r0[0]; m2 = r2[0]/r0[0]; m3 = r3[0]/r0[0]; 205 s = r0[1]; r1[1] -= m1 * s; r2[1] -= m2 * s; r3[1] -= m3 * s; 206 s = r0[2]; r1[2] -= m1 * s; r2[2] -= m2 * s; r3[2] -= m3 * s; 207 s = r0[3]; r1[3] -= m1 * s; r2[3] -= m2 * s; r3[3] -= m3 * s; 208 s = r0[4]; 209 if (s != 0.0) { r1[4] -= m1 * s; r2[4] -= m2 * s; r3[4] -= m3 * s; } 210 s = r0[5]; 211 if (s != 0.0) { r1[5] -= m1 * s; r2[5] -= m2 * s; r3[5] -= m3 * s; } 212 s = r0[6]; 213 if (s != 0.0) { r1[6] -= m1 * s; r2[6] -= m2 * s; r3[6] -= m3 * s; } 214 s = r0[7]; 215 if (s != 0.0) { r1[7] -= m1 * s; r2[7] -= m2 * s; r3[7] -= m3 * s; } 216 217 /* choose pivot - or die */ 218 if (fabs(r3[1])>fabs(r2[1])) SWAP_ROWS(r3, r2); 219 if (fabs(r2[1])>fabs(r1[1])) SWAP_ROWS(r2, r1); 220 if (0.0 == r1[1]) return GL_FALSE; 221 222 /* eliminate second variable */ 223 m2 = r2[1]/r1[1]; m3 = r3[1]/r1[1]; 224 r2[2] -= m2 * r1[2]; r3[2] -= m3 * r1[2]; 225 r2[3] -= m2 * r1[3]; r3[3] -= m3 * r1[3]; 226 s = r1[4]; if (0.0 != s) { r2[4] -= m2 * s; r3[4] -= m3 * s; } 227 s = r1[5]; if (0.0 != s) { r2[5] -= m2 * s; r3[5] -= m3 * s; } 228 s = r1[6]; if (0.0 != s) { r2[6] -= m2 * s; r3[6] -= m3 * s; } 229 s = r1[7]; if (0.0 != s) { r2[7] -= m2 * s; r3[7] -= m3 * s; } 230 231 /* choose pivot - or die */ 232 if (fabs(r3[2])>fabs(r2[2])) SWAP_ROWS(r3, r2); 233 if (0.0 == r2[2]) return GL_FALSE; 234 235 /* eliminate third variable */ 236 m3 = r3[2]/r2[2]; 237 r3[3] -= m3 * r2[3], r3[4] -= m3 * r2[4], 238 r3[5] -= m3 * r2[5], r3[6] -= m3 * r2[6], 239 r3[7] -= m3 * r2[7]; 240 241 /* last check */ 242 if (0.0 == r3[3]) return GL_FALSE; 243 244 s = 1.0F/r3[3]; /* now back substitute row 3 */ 245 r3[4] *= s; r3[5] *= s; r3[6] *= s; r3[7] *= s; 246 247 m2 = r2[3]; /* now back substitute row 2 */ 248 s = 1.0F/r2[2]; 249 r2[4] = s * (r2[4] - r3[4] * m2), r2[5] = s * (r2[5] - r3[5] * m2), 250 r2[6] = s * (r2[6] - r3[6] * m2), r2[7] = s * (r2[7] - r3[7] * m2); 251 m1 = r1[3]; 252 r1[4] -= r3[4] * m1, r1[5] -= r3[5] * m1, 253 r1[6] -= r3[6] * m1, r1[7] -= r3[7] * m1; 254 m0 = r0[3]; 255 r0[4] -= r3[4] * m0, r0[5] -= r3[5] * m0, 256 r0[6] -= r3[6] * m0, r0[7] -= r3[7] * m0; 257 258 m1 = r1[2]; /* now back substitute row 1 */ 259 s = 1.0F/r1[1]; 260 r1[4] = s * (r1[4] - r2[4] * m1), r1[5] = s * (r1[5] - r2[5] * m1), 261 r1[6] = s * (r1[6] - r2[6] * m1), r1[7] = s * (r1[7] - r2[7] * m1); 262 m0 = r0[2]; 263 r0[4] -= r2[4] * m0, r0[5] -= r2[5] * m0, 264 r0[6] -= r2[6] * m0, r0[7] -= r2[7] * m0; 265 266 m0 = r0[1]; /* now back substitute row 0 */ 267 s = 1.0F/r0[0]; 268 r0[4] = s * (r0[4] - r1[4] * m0), r0[5] = s * (r0[5] - r1[5] * m0), 269 r0[6] = s * (r0[6] - r1[6] * m0), r0[7] = s * (r0[7] - r1[7] * m0); 270 271 MAT(out,0,0) = r0[4]; MAT(out,0,1) = r0[5], 272 MAT(out,0,2) = r0[6]; MAT(out,0,3) = r0[7], 273 MAT(out,1,0) = r1[4]; MAT(out,1,1) = r1[5], 274 MAT(out,1,2) = r1[6]; MAT(out,1,3) = r1[7], 275 MAT(out,2,0) = r2[4]; MAT(out,2,1) = r2[5], 276 MAT(out,2,2) = r2[6]; MAT(out,2,3) = r2[7], 277 MAT(out,3,0) = r3[4]; MAT(out,3,1) = r3[5], 278 MAT(out,3,2) = r3[6]; MAT(out,3,3) = r3[7]; 279 280 return GL_TRUE; 281} 282#undef SWAP_ROWS 283 284 285/* Adapted from graphics gems II. 286 */ 287static GLboolean invert_matrix_3d_general( GLmatrix *mat ) 288{ 289 const GLfloat *in = mat->m; 290 GLfloat *out = mat->inv; 291 GLfloat pos, neg, t; 292 GLfloat det; 293 294 /* Calculate the determinant of upper left 3x3 submatrix and 295 * determine if the matrix is singular. 296 */ 297 pos = neg = 0.0; 298 t = MAT(in,0,0) * MAT(in,1,1) * MAT(in,2,2); 299 if (t >= 0.0) pos += t; else neg += t; 300 301 t = MAT(in,1,0) * MAT(in,2,1) * MAT(in,0,2); 302 if (t >= 0.0) pos += t; else neg += t; 303 304 t = MAT(in,2,0) * MAT(in,0,1) * MAT(in,1,2); 305 if (t >= 0.0) pos += t; else neg += t; 306 307 t = -MAT(in,2,0) * MAT(in,1,1) * MAT(in,0,2); 308 if (t >= 0.0) pos += t; else neg += t; 309 310 t = -MAT(in,1,0) * MAT(in,0,1) * MAT(in,2,2); 311 if (t >= 0.0) pos += t; else neg += t; 312 313 t = -MAT(in,0,0) * MAT(in,2,1) * MAT(in,1,2); 314 if (t >= 0.0) pos += t; else neg += t; 315 316 det = pos + neg; 317 318 if (det*det < 1e-25) 319 return GL_FALSE; 320 321 det = 1.0F / det; 322 MAT(out,0,0) = ( (MAT(in,1,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,1,2) )*det); 323 MAT(out,0,1) = (- (MAT(in,0,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,0,2) )*det); 324 MAT(out,0,2) = ( (MAT(in,0,1)*MAT(in,1,2) - MAT(in,1,1)*MAT(in,0,2) )*det); 325 MAT(out,1,0) = (- (MAT(in,1,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,1,2) )*det); 326 MAT(out,1,1) = ( (MAT(in,0,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,0,2) )*det); 327 MAT(out,1,2) = (- (MAT(in,0,0)*MAT(in,1,2) - MAT(in,1,0)*MAT(in,0,2) )*det); 328 MAT(out,2,0) = ( (MAT(in,1,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,1,1) )*det); 329 MAT(out,2,1) = (- (MAT(in,0,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,0,1) )*det); 330 MAT(out,2,2) = ( (MAT(in,0,0)*MAT(in,1,1) - MAT(in,1,0)*MAT(in,0,1) )*det); 331 332 /* Do the translation part */ 333 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) + 334 MAT(in,1,3) * MAT(out,0,1) + 335 MAT(in,2,3) * MAT(out,0,2) ); 336 MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) + 337 MAT(in,1,3) * MAT(out,1,1) + 338 MAT(in,2,3) * MAT(out,1,2) ); 339 MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) + 340 MAT(in,1,3) * MAT(out,2,1) + 341 MAT(in,2,3) * MAT(out,2,2) ); 342 343 return GL_TRUE; 344} 345 346 347static GLboolean invert_matrix_3d( GLmatrix *mat ) 348{ 349 const GLfloat *in = mat->m; 350 GLfloat *out = mat->inv; 351 352 if (!TEST_MAT_FLAGS(mat, MAT_FLAGS_ANGLE_PRESERVING)) { 353 return invert_matrix_3d_general( mat ); 354 } 355 356 if (mat->flags & MAT_FLAG_UNIFORM_SCALE) { 357 GLfloat scale = (MAT(in,0,0) * MAT(in,0,0) + 358 MAT(in,0,1) * MAT(in,0,1) + 359 MAT(in,0,2) * MAT(in,0,2)); 360 361 if (scale == 0.0) 362 return GL_FALSE; 363 364 scale = 1.0F / scale; 365 366 /* Transpose and scale the 3 by 3 upper-left submatrix. */ 367 MAT(out,0,0) = scale * MAT(in,0,0); 368 MAT(out,1,0) = scale * MAT(in,0,1); 369 MAT(out,2,0) = scale * MAT(in,0,2); 370 MAT(out,0,1) = scale * MAT(in,1,0); 371 MAT(out,1,1) = scale * MAT(in,1,1); 372 MAT(out,2,1) = scale * MAT(in,1,2); 373 MAT(out,0,2) = scale * MAT(in,2,0); 374 MAT(out,1,2) = scale * MAT(in,2,1); 375 MAT(out,2,2) = scale * MAT(in,2,2); 376 } 377 else if (mat->flags & MAT_FLAG_ROTATION) { 378 /* Transpose the 3 by 3 upper-left submatrix. */ 379 MAT(out,0,0) = MAT(in,0,0); 380 MAT(out,1,0) = MAT(in,0,1); 381 MAT(out,2,0) = MAT(in,0,2); 382 MAT(out,0,1) = MAT(in,1,0); 383 MAT(out,1,1) = MAT(in,1,1); 384 MAT(out,2,1) = MAT(in,1,2); 385 MAT(out,0,2) = MAT(in,2,0); 386 MAT(out,1,2) = MAT(in,2,1); 387 MAT(out,2,2) = MAT(in,2,2); 388 } 389 else { 390 /* pure translation */ 391 MEMCPY( out, Identity, sizeof(Identity) ); 392 MAT(out,0,3) = - MAT(in,0,3); 393 MAT(out,1,3) = - MAT(in,1,3); 394 MAT(out,2,3) = - MAT(in,2,3); 395 return GL_TRUE; 396 } 397 398 if (mat->flags & MAT_FLAG_TRANSLATION) { 399 /* Do the translation part */ 400 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) + 401 MAT(in,1,3) * MAT(out,0,1) + 402 MAT(in,2,3) * MAT(out,0,2) ); 403 MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) + 404 MAT(in,1,3) * MAT(out,1,1) + 405 MAT(in,2,3) * MAT(out,1,2) ); 406 MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) + 407 MAT(in,1,3) * MAT(out,2,1) + 408 MAT(in,2,3) * MAT(out,2,2) ); 409 } 410 else { 411 MAT(out,0,3) = MAT(out,1,3) = MAT(out,2,3) = 0.0; 412 } 413 414 return GL_TRUE; 415} 416 417 418 419static GLboolean invert_matrix_identity( GLmatrix *mat ) 420{ 421 MEMCPY( mat->inv, Identity, sizeof(Identity) ); 422 return GL_TRUE; 423} 424 425 426static GLboolean invert_matrix_3d_no_rot( GLmatrix *mat ) 427{ 428 const GLfloat *in = mat->m; 429 GLfloat *out = mat->inv; 430 431 if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0 || MAT(in,2,2) == 0 ) 432 return GL_FALSE; 433 434 MEMCPY( out, Identity, 16 * sizeof(GLfloat) ); 435 MAT(out,0,0) = 1.0F / MAT(in,0,0); 436 MAT(out,1,1) = 1.0F / MAT(in,1,1); 437 MAT(out,2,2) = 1.0F / MAT(in,2,2); 438 439 if (mat->flags & MAT_FLAG_TRANSLATION) { 440 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0)); 441 MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1)); 442 MAT(out,2,3) = - (MAT(in,2,3) * MAT(out,2,2)); 443 } 444 445 return GL_TRUE; 446} 447 448 449static GLboolean invert_matrix_2d_no_rot( GLmatrix *mat ) 450{ 451 const GLfloat *in = mat->m; 452 GLfloat *out = mat->inv; 453 454 if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0) 455 return GL_FALSE; 456 457 MEMCPY( out, Identity, 16 * sizeof(GLfloat) ); 458 MAT(out,0,0) = 1.0F / MAT(in,0,0); 459 MAT(out,1,1) = 1.0F / MAT(in,1,1); 460 461 if (mat->flags & MAT_FLAG_TRANSLATION) { 462 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0)); 463 MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1)); 464 } 465 466 return GL_TRUE; 467} 468 469 470#if 0 471/* broken */ 472static GLboolean invert_matrix_perspective( GLmatrix *mat ) 473{ 474 const GLfloat *in = mat->m; 475 GLfloat *out = mat->inv; 476 477 if (MAT(in,2,3) == 0) 478 return GL_FALSE; 479 480 MEMCPY( out, Identity, 16 * sizeof(GLfloat) ); 481 482 MAT(out,0,0) = 1.0F / MAT(in,0,0); 483 MAT(out,1,1) = 1.0F / MAT(in,1,1); 484 485 MAT(out,0,3) = MAT(in,0,2); 486 MAT(out,1,3) = MAT(in,1,2); 487 488 MAT(out,2,2) = 0; 489 MAT(out,2,3) = -1; 490 491 MAT(out,3,2) = 1.0F / MAT(in,2,3); 492 MAT(out,3,3) = MAT(in,2,2) * MAT(out,3,2); 493 494 return GL_TRUE; 495} 496#endif 497 498 499typedef GLboolean (*inv_mat_func)( GLmatrix *mat ); 500 501 502static inv_mat_func inv_mat_tab[7] = { 503 invert_matrix_general, 504 invert_matrix_identity, 505 invert_matrix_3d_no_rot, 506#if 0 507 /* Don't use this function for now - it fails when the projection matrix 508 * is premultiplied by a translation (ala Chromium's tilesort SPU). 509 */ 510 invert_matrix_perspective, 511#else 512 invert_matrix_general, 513#endif 514 invert_matrix_3d, /* lazy! */ 515 invert_matrix_2d_no_rot, 516 invert_matrix_3d 517}; 518 519 520static GLboolean matrix_invert( GLmatrix *mat ) 521{ 522 if (inv_mat_tab[mat->type](mat)) { 523 mat->flags &= ~MAT_FLAG_SINGULAR; 524 return GL_TRUE; 525 } else { 526 mat->flags |= MAT_FLAG_SINGULAR; 527 MEMCPY( mat->inv, Identity, sizeof(Identity) ); 528 return GL_FALSE; 529 } 530} 531 532 533 534 535 536 537/* 538 * Generate a 4x4 transformation matrix from glRotate parameters, and 539 * postmultiply the input matrix by it. 540 * This function contributed by Erich Boleyn (erich@uruk.org). 541 * Optimizatios contributed by Rudolf Opalla (rudi@khm.de). 542 */ 543void 544_math_matrix_rotate( GLmatrix *mat, 545 GLfloat angle, GLfloat x, GLfloat y, GLfloat z ) 546{ 547 GLfloat xx, yy, zz, xy, yz, zx, xs, ys, zs, one_c, s, c; 548 GLfloat m[16]; 549 GLboolean optimized; 550 551 s = (GLfloat) sin( angle * DEG2RAD ); 552 c = (GLfloat) cos( angle * DEG2RAD ); 553 554 MEMCPY(m, Identity, sizeof(GLfloat)*16); 555 optimized = GL_FALSE; 556 557#define M(row,col) m[col*4+row] 558 559 if (x == 0.0F) { 560 if (y == 0.0F) { 561 if (z != 0.0F) { 562 optimized = GL_TRUE; 563 /* rotate only around z-axis */ 564 M(0,0) = c; 565 M(1,1) = c; 566 if (z < 0.0F) { 567 M(0,1) = s; 568 M(1,0) = -s; 569 } 570 else { 571 M(0,1) = -s; 572 M(1,0) = s; 573 } 574 } 575 } 576 else if (z == 0.0F) { 577 optimized = GL_TRUE; 578 /* rotate only around y-axis */ 579 M(0,0) = c; 580 M(2,2) = c; 581 if (y < 0.0F) { 582 M(0,2) = -s; 583 M(2,0) = s; 584 } 585 else { 586 M(0,2) = s; 587 M(2,0) = -s; 588 } 589 } 590 } 591 else if (y == 0.0F) { 592 if (z == 0.0F) { 593 optimized = GL_TRUE; 594 /* rotate only around x-axis */ 595 M(1,1) = c; 596 M(2,2) = c; 597 if (x < 0.0F) { 598 M(1,2) = s; 599 M(2,1) = -s; 600 } 601 else { 602 M(1,2) = -s; 603 M(2,1) = s; 604 } 605 } 606 } 607 608 if (!optimized) { 609 const GLfloat mag = (GLfloat) GL_SQRT(x * x + y * y + z * z); 610 611 if (mag <= 1.0e-4) { 612 /* no rotation, leave mat as-is */ 613 return; 614 } 615 616 x /= mag; 617 y /= mag; 618 z /= mag; 619 620 621 /* 622 * Arbitrary axis rotation matrix. 623 * 624 * This is composed of 5 matrices, Rz, Ry, T, Ry', Rz', multiplied 625 * like so: Rz * Ry * T * Ry' * Rz'. T is the final rotation 626 * (which is about the X-axis), and the two composite transforms 627 * Ry' * Rz' and Rz * Ry are (respectively) the rotations necessary 628 * from the arbitrary axis to the X-axis then back. They are 629 * all elementary rotations. 630 * 631 * Rz' is a rotation about the Z-axis, to bring the axis vector 632 * into the x-z plane. Then Ry' is applied, rotating about the 633 * Y-axis to bring the axis vector parallel with the X-axis. The 634 * rotation about the X-axis is then performed. Ry and Rz are 635 * simply the respective inverse transforms to bring the arbitrary 636 * axis back to it's original orientation. The first transforms 637 * Rz' and Ry' are considered inverses, since the data from the 638 * arbitrary axis gives you info on how to get to it, not how 639 * to get away from it, and an inverse must be applied. 640 * 641 * The basic calculation used is to recognize that the arbitrary 642 * axis vector (x, y, z), since it is of unit length, actually 643 * represents the sines and cosines of the angles to rotate the 644 * X-axis to the same orientation, with theta being the angle about 645 * Z and phi the angle about Y (in the order described above) 646 * as follows: 647 * 648 * cos ( theta ) = x / sqrt ( 1 - z^2 ) 649 * sin ( theta ) = y / sqrt ( 1 - z^2 ) 650 * 651 * cos ( phi ) = sqrt ( 1 - z^2 ) 652 * sin ( phi ) = z 653 * 654 * Note that cos ( phi ) can further be inserted to the above 655 * formulas: 656 * 657 * cos ( theta ) = x / cos ( phi ) 658 * sin ( theta ) = y / sin ( phi ) 659 * 660 * ...etc. Because of those relations and the standard trigonometric 661 * relations, it is pssible to reduce the transforms down to what 662 * is used below. It may be that any primary axis chosen will give the 663 * same results (modulo a sign convention) using thie method. 664 * 665 * Particularly nice is to notice that all divisions that might 666 * have caused trouble when parallel to certain planes or 667 * axis go away with care paid to reducing the expressions. 668 * After checking, it does perform correctly under all cases, since 669 * in all the cases of division where the denominator would have 670 * been zero, the numerator would have been zero as well, giving 671 * the expected result. 672 */ 673 674 xx = x * x; 675 yy = y * y; 676 zz = z * z; 677 xy = x * y; 678 yz = y * z; 679 zx = z * x; 680 xs = x * s; 681 ys = y * s; 682 zs = z * s; 683 one_c = 1.0F - c; 684 685 /* We already hold the identity-matrix so we can skip some statements */ 686 M(0,0) = (one_c * xx) + c; 687 M(0,1) = (one_c * xy) - zs; 688 M(0,2) = (one_c * zx) + ys; 689/* M(0,3) = 0.0F; */ 690 691 M(1,0) = (one_c * xy) + zs; 692 M(1,1) = (one_c * yy) + c; 693 M(1,2) = (one_c * yz) - xs; 694/* M(1,3) = 0.0F; */ 695 696 M(2,0) = (one_c * zx) - ys; 697 M(2,1) = (one_c * yz) + xs; 698 M(2,2) = (one_c * zz) + c; 699/* M(2,3) = 0.0F; */ 700 701/* 702 M(3,0) = 0.0F; 703 M(3,1) = 0.0F; 704 M(3,2) = 0.0F; 705 M(3,3) = 1.0F; 706*/ 707 } 708#undef M 709 710 matrix_multf( mat, m, MAT_FLAG_ROTATION ); 711} 712 713 714 715void 716_math_matrix_frustum( GLmatrix *mat, 717 GLfloat left, GLfloat right, 718 GLfloat bottom, GLfloat top, 719 GLfloat nearval, GLfloat farval ) 720{ 721 GLfloat x, y, a, b, c, d; 722 GLfloat m[16]; 723 724 x = (2.0F*nearval) / (right-left); 725 y = (2.0F*nearval) / (top-bottom); 726 a = (right+left) / (right-left); 727 b = (top+bottom) / (top-bottom); 728 c = -(farval+nearval) / ( farval-nearval); 729 d = -(2.0F*farval*nearval) / (farval-nearval); /* error? */ 730 731#define M(row,col) m[col*4+row] 732 M(0,0) = x; M(0,1) = 0.0F; M(0,2) = a; M(0,3) = 0.0F; 733 M(1,0) = 0.0F; M(1,1) = y; M(1,2) = b; M(1,3) = 0.0F; 734 M(2,0) = 0.0F; M(2,1) = 0.0F; M(2,2) = c; M(2,3) = d; 735 M(3,0) = 0.0F; M(3,1) = 0.0F; M(3,2) = -1.0F; M(3,3) = 0.0F; 736#undef M 737 738 matrix_multf( mat, m, MAT_FLAG_PERSPECTIVE ); 739} 740 741void 742_math_matrix_ortho( GLmatrix *mat, 743 GLfloat left, GLfloat right, 744 GLfloat bottom, GLfloat top, 745 GLfloat nearval, GLfloat farval ) 746{ 747 GLfloat x, y, z; 748 GLfloat tx, ty, tz; 749 GLfloat m[16]; 750 751 x = 2.0F / (right-left); 752 y = 2.0F / (top-bottom); 753 z = -2.0F / (farval-nearval); 754 tx = -(right+left) / (right-left); 755 ty = -(top+bottom) / (top-bottom); 756 tz = -(farval+nearval) / (farval-nearval); 757 758#define M(row,col) m[col*4+row] 759 M(0,0) = x; M(0,1) = 0.0F; M(0,2) = 0.0F; M(0,3) = tx; 760 M(1,0) = 0.0F; M(1,1) = y; M(1,2) = 0.0F; M(1,3) = ty; 761 M(2,0) = 0.0F; M(2,1) = 0.0F; M(2,2) = z; M(2,3) = tz; 762 M(3,0) = 0.0F; M(3,1) = 0.0F; M(3,2) = 0.0F; M(3,3) = 1.0F; 763#undef M 764 765 matrix_multf( mat, m, (MAT_FLAG_GENERAL_SCALE|MAT_FLAG_TRANSLATION)); 766} 767 768 769#define ZERO(x) (1<<x) 770#define ONE(x) (1<<(x+16)) 771 772#define MASK_NO_TRX (ZERO(12) | ZERO(13) | ZERO(14)) 773#define MASK_NO_2D_SCALE ( ONE(0) | ONE(5)) 774 775#define MASK_IDENTITY ( ONE(0) | ZERO(4) | ZERO(8) | ZERO(12) |\ 776 ZERO(1) | ONE(5) | ZERO(9) | ZERO(13) |\ 777 ZERO(2) | ZERO(6) | ONE(10) | ZERO(14) |\ 778 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 779 780#define MASK_2D_NO_ROT ( ZERO(4) | ZERO(8) | \ 781 ZERO(1) | ZERO(9) | \ 782 ZERO(2) | ZERO(6) | ONE(10) | ZERO(14) |\ 783 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 784 785#define MASK_2D ( ZERO(8) | \ 786 ZERO(9) | \ 787 ZERO(2) | ZERO(6) | ONE(10) | ZERO(14) |\ 788 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 789 790 791#define MASK_3D_NO_ROT ( ZERO(4) | ZERO(8) | \ 792 ZERO(1) | ZERO(9) | \ 793 ZERO(2) | ZERO(6) | \ 794 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 795 796#define MASK_3D ( \ 797 \ 798 \ 799 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 800 801 802#define MASK_PERSPECTIVE ( ZERO(4) | ZERO(12) |\ 803 ZERO(1) | ZERO(13) |\ 804 ZERO(2) | ZERO(6) | \ 805 ZERO(3) | ZERO(7) | ZERO(15) ) 806 807#define SQ(x) ((x)*(x)) 808 809/* Determine type and flags from scratch. This is expensive enough to 810 * only want to do it once. 811 */ 812static void analyse_from_scratch( GLmatrix *mat ) 813{ 814 const GLfloat *m = mat->m; 815 GLuint mask = 0; 816 GLuint i; 817 818 for (i = 0 ; i < 16 ; i++) { 819 if (m[i] == 0.0) mask |= (1<<i); 820 } 821 822 if (m[0] == 1.0F) mask |= (1<<16); 823 if (m[5] == 1.0F) mask |= (1<<21); 824 if (m[10] == 1.0F) mask |= (1<<26); 825 if (m[15] == 1.0F) mask |= (1<<31); 826 827 mat->flags &= ~MAT_FLAGS_GEOMETRY; 828 829 /* Check for translation - no-one really cares 830 */ 831 if ((mask & MASK_NO_TRX) != MASK_NO_TRX) 832 mat->flags |= MAT_FLAG_TRANSLATION; 833 834 /* Do the real work 835 */ 836 if (mask == (GLuint) MASK_IDENTITY) { 837 mat->type = MATRIX_IDENTITY; 838 } 839 else if ((mask & MASK_2D_NO_ROT) == (GLuint) MASK_2D_NO_ROT) { 840 mat->type = MATRIX_2D_NO_ROT; 841 842 if ((mask & MASK_NO_2D_SCALE) != MASK_NO_2D_SCALE) 843 mat->flags = MAT_FLAG_GENERAL_SCALE; 844 } 845 else if ((mask & MASK_2D) == (GLuint) MASK_2D) { 846 GLfloat mm = DOT2(m, m); 847 GLfloat m4m4 = DOT2(m+4,m+4); 848 GLfloat mm4 = DOT2(m,m+4); 849 850 mat->type = MATRIX_2D; 851 852 /* Check for scale */ 853 if (SQ(mm-1) > SQ(1e-6) || 854 SQ(m4m4-1) > SQ(1e-6)) 855 mat->flags |= MAT_FLAG_GENERAL_SCALE; 856 857 /* Check for rotation */ 858 if (SQ(mm4) > SQ(1e-6)) 859 mat->flags |= MAT_FLAG_GENERAL_3D; 860 else 861 mat->flags |= MAT_FLAG_ROTATION; 862 863 } 864 else if ((mask & MASK_3D_NO_ROT) == (GLuint) MASK_3D_NO_ROT) { 865 mat->type = MATRIX_3D_NO_ROT; 866 867 /* Check for scale */ 868 if (SQ(m[0]-m[5]) < SQ(1e-6) && 869 SQ(m[0]-m[10]) < SQ(1e-6)) { 870 if (SQ(m[0]-1.0) > SQ(1e-6)) { 871 mat->flags |= MAT_FLAG_UNIFORM_SCALE; 872 } 873 } 874 else { 875 mat->flags |= MAT_FLAG_GENERAL_SCALE; 876 } 877 } 878 else if ((mask & MASK_3D) == (GLuint) MASK_3D) { 879 GLfloat c1 = DOT3(m,m); 880 GLfloat c2 = DOT3(m+4,m+4); 881 GLfloat c3 = DOT3(m+8,m+8); 882 GLfloat d1 = DOT3(m, m+4); 883 GLfloat cp[3]; 884 885 mat->type = MATRIX_3D; 886 887 /* Check for scale */ 888 if (SQ(c1-c2) < SQ(1e-6) && SQ(c1-c3) < SQ(1e-6)) { 889 if (SQ(c1-1.0) > SQ(1e-6)) 890 mat->flags |= MAT_FLAG_UNIFORM_SCALE; 891 /* else no scale at all */ 892 } 893 else { 894 mat->flags |= MAT_FLAG_GENERAL_SCALE; 895 } 896 897 /* Check for rotation */ 898 if (SQ(d1) < SQ(1e-6)) { 899 CROSS3( cp, m, m+4 ); 900 SUB_3V( cp, cp, (m+8) ); 901 if (LEN_SQUARED_3FV(cp) < SQ(1e-6)) 902 mat->flags |= MAT_FLAG_ROTATION; 903 else 904 mat->flags |= MAT_FLAG_GENERAL_3D; 905 } 906 else { 907 mat->flags |= MAT_FLAG_GENERAL_3D; /* shear, etc */ 908 } 909 } 910 else if ((mask & MASK_PERSPECTIVE) == MASK_PERSPECTIVE && m[11]==-1.0F) { 911 mat->type = MATRIX_PERSPECTIVE; 912 mat->flags |= MAT_FLAG_GENERAL; 913 } 914 else { 915 mat->type = MATRIX_GENERAL; 916 mat->flags |= MAT_FLAG_GENERAL; 917 } 918} 919 920 921/* Analyse a matrix given that its flags are accurate - this is the 922 * more common operation, hopefully. 923 */ 924static void analyse_from_flags( GLmatrix *mat ) 925{ 926 const GLfloat *m = mat->m; 927 928 if (TEST_MAT_FLAGS(mat, 0)) { 929 mat->type = MATRIX_IDENTITY; 930 } 931 else if (TEST_MAT_FLAGS(mat, (MAT_FLAG_TRANSLATION | 932 MAT_FLAG_UNIFORM_SCALE | 933 MAT_FLAG_GENERAL_SCALE))) { 934 if ( m[10]==1.0F && m[14]==0.0F ) { 935 mat->type = MATRIX_2D_NO_ROT; 936 } 937 else { 938 mat->type = MATRIX_3D_NO_ROT; 939 } 940 } 941 else if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D)) { 942 if ( m[ 8]==0.0F 943 && m[ 9]==0.0F 944 && m[2]==0.0F && m[6]==0.0F && m[10]==1.0F && m[14]==0.0F) { 945 mat->type = MATRIX_2D; 946 } 947 else { 948 mat->type = MATRIX_3D; 949 } 950 } 951 else if ( m[4]==0.0F && m[12]==0.0F 952 && m[1]==0.0F && m[13]==0.0F 953 && m[2]==0.0F && m[6]==0.0F 954 && m[3]==0.0F && m[7]==0.0F && m[11]==-1.0F && m[15]==0.0F) { 955 mat->type = MATRIX_PERSPECTIVE; 956 } 957 else { 958 mat->type = MATRIX_GENERAL; 959 } 960} 961 962 963void 964_math_matrix_analyse( GLmatrix *mat ) 965{ 966 if (mat->flags & MAT_DIRTY_TYPE) { 967 if (mat->flags & MAT_DIRTY_FLAGS) 968 analyse_from_scratch( mat ); 969 else 970 analyse_from_flags( mat ); 971 } 972 973 if (mat->inv && (mat->flags & MAT_DIRTY_INVERSE)) { 974 matrix_invert( mat ); 975 } 976 977 mat->flags &= ~(MAT_DIRTY_FLAGS| 978 MAT_DIRTY_TYPE| 979 MAT_DIRTY_INVERSE); 980} 981 982 983void 984_math_matrix_copy( GLmatrix *to, const GLmatrix *from ) 985{ 986 MEMCPY( to->m, from->m, sizeof(Identity) ); 987 to->flags = from->flags; 988 to->type = from->type; 989 990 if (to->inv != 0) { 991 if (from->inv == 0) { 992 matrix_invert( to ); 993 } 994 else { 995 MEMCPY(to->inv, from->inv, sizeof(GLfloat)*16); 996 } 997 } 998} 999 1000 1001void 1002_math_matrix_scale( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z ) 1003{ 1004 GLfloat *m = mat->m; 1005 m[0] *= x; m[4] *= y; m[8] *= z; 1006 m[1] *= x; m[5] *= y; m[9] *= z; 1007 m[2] *= x; m[6] *= y; m[10] *= z; 1008 m[3] *= x; m[7] *= y; m[11] *= z; 1009 1010 if (fabs(x - y) < 1e-8 && fabs(x - z) < 1e-8) 1011 mat->flags |= MAT_FLAG_UNIFORM_SCALE; 1012 else 1013 mat->flags |= MAT_FLAG_GENERAL_SCALE; 1014 1015 mat->flags |= (MAT_DIRTY_TYPE | 1016 MAT_DIRTY_INVERSE); 1017} 1018 1019 1020void 1021_math_matrix_translate( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z ) 1022{ 1023 GLfloat *m = mat->m; 1024 m[12] = m[0] * x + m[4] * y + m[8] * z + m[12]; 1025 m[13] = m[1] * x + m[5] * y + m[9] * z + m[13]; 1026 m[14] = m[2] * x + m[6] * y + m[10] * z + m[14]; 1027 m[15] = m[3] * x + m[7] * y + m[11] * z + m[15]; 1028 1029 mat->flags |= (MAT_FLAG_TRANSLATION | 1030 MAT_DIRTY_TYPE | 1031 MAT_DIRTY_INVERSE); 1032} 1033 1034 1035void 1036_math_matrix_loadf( GLmatrix *mat, const GLfloat *m ) 1037{ 1038 MEMCPY( mat->m, m, 16*sizeof(GLfloat) ); 1039 mat->flags = (MAT_FLAG_GENERAL | MAT_DIRTY); 1040} 1041 1042void 1043_math_matrix_ctr( GLmatrix *m ) 1044{ 1045 m->m = (GLfloat *) ALIGN_MALLOC( 16 * sizeof(GLfloat), 16 ); 1046 if (m->m) 1047 MEMCPY( m->m, Identity, sizeof(Identity) ); 1048 m->inv = NULL; 1049 m->type = MATRIX_IDENTITY; 1050 m->flags = 0; 1051} 1052 1053void 1054_math_matrix_dtr( GLmatrix *m ) 1055{ 1056 if (m->m) { 1057 ALIGN_FREE( m->m ); 1058 m->m = NULL; 1059 } 1060 if (m->inv) { 1061 ALIGN_FREE( m->inv ); 1062 m->inv = NULL; 1063 } 1064} 1065 1066 1067void 1068_math_matrix_alloc_inv( GLmatrix *m ) 1069{ 1070 if (!m->inv) { 1071 m->inv = (GLfloat *) ALIGN_MALLOC( 16 * sizeof(GLfloat), 16 ); 1072 if (m->inv) 1073 MEMCPY( m->inv, Identity, 16 * sizeof(GLfloat) ); 1074 } 1075} 1076 1077 1078void 1079_math_matrix_mul_matrix( GLmatrix *dest, const GLmatrix *a, const GLmatrix *b ) 1080{ 1081 dest->flags = (a->flags | 1082 b->flags | 1083 MAT_DIRTY_TYPE | 1084 MAT_DIRTY_INVERSE); 1085 1086 if (TEST_MAT_FLAGS(dest, MAT_FLAGS_3D)) 1087 matmul34( dest->m, a->m, b->m ); 1088 else 1089 matmul4( dest->m, a->m, b->m ); 1090} 1091 1092 1093void 1094_math_matrix_mul_floats( GLmatrix *dest, const GLfloat *m ) 1095{ 1096 dest->flags |= (MAT_FLAG_GENERAL | 1097 MAT_DIRTY_TYPE | 1098 MAT_DIRTY_INVERSE); 1099 1100 matmul4( dest->m, dest->m, m ); 1101} 1102 1103void 1104_math_matrix_set_identity( GLmatrix *mat ) 1105{ 1106 MEMCPY( mat->m, Identity, 16*sizeof(GLfloat) ); 1107 1108 if (mat->inv) 1109 MEMCPY( mat->inv, Identity, 16*sizeof(GLfloat) ); 1110 1111 mat->type = MATRIX_IDENTITY; 1112 mat->flags &= ~(MAT_DIRTY_FLAGS| 1113 MAT_DIRTY_TYPE| 1114 MAT_DIRTY_INVERSE); 1115} 1116 1117 1118 1119void 1120_math_transposef( GLfloat to[16], const GLfloat from[16] ) 1121{ 1122 to[0] = from[0]; 1123 to[1] = from[4]; 1124 to[2] = from[8]; 1125 to[3] = from[12]; 1126 to[4] = from[1]; 1127 to[5] = from[5]; 1128 to[6] = from[9]; 1129 to[7] = from[13]; 1130 to[8] = from[2]; 1131 to[9] = from[6]; 1132 to[10] = from[10]; 1133 to[11] = from[14]; 1134 to[12] = from[3]; 1135 to[13] = from[7]; 1136 to[14] = from[11]; 1137 to[15] = from[15]; 1138} 1139 1140 1141void 1142_math_transposed( GLdouble to[16], const GLdouble from[16] ) 1143{ 1144 to[0] = from[0]; 1145 to[1] = from[4]; 1146 to[2] = from[8]; 1147 to[3] = from[12]; 1148 to[4] = from[1]; 1149 to[5] = from[5]; 1150 to[6] = from[9]; 1151 to[7] = from[13]; 1152 to[8] = from[2]; 1153 to[9] = from[6]; 1154 to[10] = from[10]; 1155 to[11] = from[14]; 1156 to[12] = from[3]; 1157 to[13] = from[7]; 1158 to[14] = from[11]; 1159 to[15] = from[15]; 1160} 1161 1162void 1163_math_transposefd( GLfloat to[16], const GLdouble from[16] ) 1164{ 1165 to[0] = (GLfloat) from[0]; 1166 to[1] = (GLfloat) from[4]; 1167 to[2] = (GLfloat) from[8]; 1168 to[3] = (GLfloat) from[12]; 1169 to[4] = (GLfloat) from[1]; 1170 to[5] = (GLfloat) from[5]; 1171 to[6] = (GLfloat) from[9]; 1172 to[7] = (GLfloat) from[13]; 1173 to[8] = (GLfloat) from[2]; 1174 to[9] = (GLfloat) from[6]; 1175 to[10] = (GLfloat) from[10]; 1176 to[11] = (GLfloat) from[14]; 1177 to[12] = (GLfloat) from[3]; 1178 to[13] = (GLfloat) from[7]; 1179 to[14] = (GLfloat) from[11]; 1180 to[15] = (GLfloat) from[15]; 1181} 1182