m_matrix.c revision 27558a160a9fe91745728d7626995cd88f8fe339
1/* $Id: m_matrix.c,v 1.16 2003/03/01 01:50:24 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 42#include "m_matrix.h" 43 44 45static const char *types[] = { 46 "MATRIX_GENERAL", 47 "MATRIX_IDENTITY", 48 "MATRIX_3D_NO_ROT", 49 "MATRIX_PERSPECTIVE", 50 "MATRIX_2D", 51 "MATRIX_2D_NO_ROT", 52 "MATRIX_3D" 53}; 54 55 56static GLfloat Identity[16] = { 57 1.0, 0.0, 0.0, 0.0, 58 0.0, 1.0, 0.0, 0.0, 59 0.0, 0.0, 1.0, 0.0, 60 0.0, 0.0, 0.0, 1.0 61}; 62 63 64 65 66/* 67 * This matmul was contributed by Thomas Malik 68 * 69 * Perform a 4x4 matrix multiplication (product = a x b). 70 * Input: a, b - matrices to multiply 71 * Output: product - product of a and b 72 * WARNING: (product != b) assumed 73 * NOTE: (product == a) allowed 74 * 75 * KW: 4*16 = 64 muls 76 */ 77#define A(row,col) a[(col<<2)+row] 78#define B(row,col) b[(col<<2)+row] 79#define P(row,col) product[(col<<2)+row] 80 81static void matmul4( GLfloat *product, const GLfloat *a, const GLfloat *b ) 82{ 83 GLint i; 84 for (i = 0; i < 4; i++) { 85 const GLfloat ai0=A(i,0), ai1=A(i,1), ai2=A(i,2), ai3=A(i,3); 86 P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0) + ai3 * B(3,0); 87 P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1) + ai3 * B(3,1); 88 P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2) + ai3 * B(3,2); 89 P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3 * B(3,3); 90 } 91} 92 93 94/* Multiply two matrices known to occupy only the top three rows, such 95 * as typical model matrices, and ortho matrices. 96 */ 97static void matmul34( GLfloat *product, const GLfloat *a, const GLfloat *b ) 98{ 99 GLint i; 100 for (i = 0; i < 3; i++) { 101 const GLfloat ai0=A(i,0), ai1=A(i,1), ai2=A(i,2), ai3=A(i,3); 102 P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0); 103 P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1); 104 P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2); 105 P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3; 106 } 107 P(3,0) = 0; 108 P(3,1) = 0; 109 P(3,2) = 0; 110 P(3,3) = 1; 111} 112 113 114#undef A 115#undef B 116#undef P 117 118 119/* 120 * Multiply a matrix by an array of floats with known properties. 121 */ 122static void matrix_multf( GLmatrix *mat, const GLfloat *m, GLuint flags ) 123{ 124 mat->flags |= (flags | MAT_DIRTY_TYPE | MAT_DIRTY_INVERSE); 125 126 if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D)) 127 matmul34( mat->m, mat->m, m ); 128 else 129 matmul4( mat->m, mat->m, m ); 130} 131 132 133static void print_matrix_floats( const GLfloat m[16] ) 134{ 135 int i; 136 for (i=0;i<4;i++) { 137 _mesa_debug(NULL,"\t%f %f %f %f\n", m[i], m[4+i], m[8+i], m[12+i] ); 138 } 139} 140 141void 142_math_matrix_print( const GLmatrix *m ) 143{ 144 _mesa_debug(NULL, "Matrix type: %s, flags: %x\n", types[m->type], m->flags); 145 print_matrix_floats(m->m); 146 _mesa_debug(NULL, "Inverse: \n"); 147 if (m->inv) { 148 GLfloat prod[16]; 149 print_matrix_floats(m->inv); 150 matmul4(prod, m->m, m->inv); 151 _mesa_debug(NULL, "Mat * Inverse:\n"); 152 print_matrix_floats(prod); 153 } 154 else { 155 _mesa_debug(NULL, " - not available\n"); 156 } 157} 158 159 160 161 162#define SWAP_ROWS(a, b) { GLfloat *_tmp = a; (a)=(b); (b)=_tmp; } 163#define MAT(m,r,c) (m)[(c)*4+(r)] 164 165/* 166 * Compute inverse of 4x4 transformation matrix. 167 * Code contributed by Jacques Leroy jle@star.be 168 * Return GL_TRUE for success, GL_FALSE for failure (singular matrix) 169 */ 170static GLboolean invert_matrix_general( GLmatrix *mat ) 171{ 172 const GLfloat *m = mat->m; 173 GLfloat *out = mat->inv; 174 GLfloat wtmp[4][8]; 175 GLfloat m0, m1, m2, m3, s; 176 GLfloat *r0, *r1, *r2, *r3; 177 178 r0 = wtmp[0], r1 = wtmp[1], r2 = wtmp[2], r3 = wtmp[3]; 179 180 r0[0] = MAT(m,0,0), r0[1] = MAT(m,0,1), 181 r0[2] = MAT(m,0,2), r0[3] = MAT(m,0,3), 182 r0[4] = 1.0, r0[5] = r0[6] = r0[7] = 0.0, 183 184 r1[0] = MAT(m,1,0), r1[1] = MAT(m,1,1), 185 r1[2] = MAT(m,1,2), r1[3] = MAT(m,1,3), 186 r1[5] = 1.0, r1[4] = r1[6] = r1[7] = 0.0, 187 188 r2[0] = MAT(m,2,0), r2[1] = MAT(m,2,1), 189 r2[2] = MAT(m,2,2), r2[3] = MAT(m,2,3), 190 r2[6] = 1.0, r2[4] = r2[5] = r2[7] = 0.0, 191 192 r3[0] = MAT(m,3,0), r3[1] = MAT(m,3,1), 193 r3[2] = MAT(m,3,2), r3[3] = MAT(m,3,3), 194 r3[7] = 1.0, r3[4] = r3[5] = r3[6] = 0.0; 195 196 /* choose pivot - or die */ 197 if (fabs(r3[0])>fabs(r2[0])) SWAP_ROWS(r3, r2); 198 if (fabs(r2[0])>fabs(r1[0])) SWAP_ROWS(r2, r1); 199 if (fabs(r1[0])>fabs(r0[0])) SWAP_ROWS(r1, r0); 200 if (0.0 == r0[0]) return GL_FALSE; 201 202 /* eliminate first variable */ 203 m1 = r1[0]/r0[0]; m2 = r2[0]/r0[0]; m3 = r3[0]/r0[0]; 204 s = r0[1]; r1[1] -= m1 * s; r2[1] -= m2 * s; r3[1] -= m3 * s; 205 s = r0[2]; r1[2] -= m1 * s; r2[2] -= m2 * s; r3[2] -= m3 * s; 206 s = r0[3]; r1[3] -= m1 * s; r2[3] -= m2 * s; r3[3] -= m3 * s; 207 s = r0[4]; 208 if (s != 0.0) { r1[4] -= m1 * s; r2[4] -= m2 * s; r3[4] -= m3 * s; } 209 s = r0[5]; 210 if (s != 0.0) { r1[5] -= m1 * s; r2[5] -= m2 * s; r3[5] -= m3 * s; } 211 s = r0[6]; 212 if (s != 0.0) { r1[6] -= m1 * s; r2[6] -= m2 * s; r3[6] -= m3 * s; } 213 s = r0[7]; 214 if (s != 0.0) { r1[7] -= m1 * s; r2[7] -= m2 * s; r3[7] -= m3 * s; } 215 216 /* choose pivot - or die */ 217 if (fabs(r3[1])>fabs(r2[1])) SWAP_ROWS(r3, r2); 218 if (fabs(r2[1])>fabs(r1[1])) SWAP_ROWS(r2, r1); 219 if (0.0 == r1[1]) return GL_FALSE; 220 221 /* eliminate second variable */ 222 m2 = r2[1]/r1[1]; m3 = r3[1]/r1[1]; 223 r2[2] -= m2 * r1[2]; r3[2] -= m3 * r1[2]; 224 r2[3] -= m2 * r1[3]; r3[3] -= m3 * r1[3]; 225 s = r1[4]; if (0.0 != s) { r2[4] -= m2 * s; r3[4] -= m3 * s; } 226 s = r1[5]; if (0.0 != s) { r2[5] -= m2 * s; r3[5] -= m3 * s; } 227 s = r1[6]; if (0.0 != s) { r2[6] -= m2 * s; r3[6] -= m3 * s; } 228 s = r1[7]; if (0.0 != s) { r2[7] -= m2 * s; r3[7] -= m3 * s; } 229 230 /* choose pivot - or die */ 231 if (fabs(r3[2])>fabs(r2[2])) SWAP_ROWS(r3, r2); 232 if (0.0 == r2[2]) return GL_FALSE; 233 234 /* eliminate third variable */ 235 m3 = r3[2]/r2[2]; 236 r3[3] -= m3 * r2[3], r3[4] -= m3 * r2[4], 237 r3[5] -= m3 * r2[5], r3[6] -= m3 * r2[6], 238 r3[7] -= m3 * r2[7]; 239 240 /* last check */ 241 if (0.0 == r3[3]) return GL_FALSE; 242 243 s = 1.0F/r3[3]; /* now back substitute row 3 */ 244 r3[4] *= s; r3[5] *= s; r3[6] *= s; r3[7] *= s; 245 246 m2 = r2[3]; /* now back substitute row 2 */ 247 s = 1.0F/r2[2]; 248 r2[4] = s * (r2[4] - r3[4] * m2), r2[5] = s * (r2[5] - r3[5] * m2), 249 r2[6] = s * (r2[6] - r3[6] * m2), r2[7] = s * (r2[7] - r3[7] * m2); 250 m1 = r1[3]; 251 r1[4] -= r3[4] * m1, r1[5] -= r3[5] * m1, 252 r1[6] -= r3[6] * m1, r1[7] -= r3[7] * m1; 253 m0 = r0[3]; 254 r0[4] -= r3[4] * m0, r0[5] -= r3[5] * m0, 255 r0[6] -= r3[6] * m0, r0[7] -= r3[7] * m0; 256 257 m1 = r1[2]; /* now back substitute row 1 */ 258 s = 1.0F/r1[1]; 259 r1[4] = s * (r1[4] - r2[4] * m1), r1[5] = s * (r1[5] - r2[5] * m1), 260 r1[6] = s * (r1[6] - r2[6] * m1), r1[7] = s * (r1[7] - r2[7] * m1); 261 m0 = r0[2]; 262 r0[4] -= r2[4] * m0, r0[5] -= r2[5] * m0, 263 r0[6] -= r2[6] * m0, r0[7] -= r2[7] * m0; 264 265 m0 = r0[1]; /* now back substitute row 0 */ 266 s = 1.0F/r0[0]; 267 r0[4] = s * (r0[4] - r1[4] * m0), r0[5] = s * (r0[5] - r1[5] * m0), 268 r0[6] = s * (r0[6] - r1[6] * m0), r0[7] = s * (r0[7] - r1[7] * m0); 269 270 MAT(out,0,0) = r0[4]; MAT(out,0,1) = r0[5], 271 MAT(out,0,2) = r0[6]; MAT(out,0,3) = r0[7], 272 MAT(out,1,0) = r1[4]; MAT(out,1,1) = r1[5], 273 MAT(out,1,2) = r1[6]; MAT(out,1,3) = r1[7], 274 MAT(out,2,0) = r2[4]; MAT(out,2,1) = r2[5], 275 MAT(out,2,2) = r2[6]; MAT(out,2,3) = r2[7], 276 MAT(out,3,0) = r3[4]; MAT(out,3,1) = r3[5], 277 MAT(out,3,2) = r3[6]; MAT(out,3,3) = r3[7]; 278 279 return GL_TRUE; 280} 281#undef SWAP_ROWS 282 283 284/* Adapted from graphics gems II. 285 */ 286static GLboolean invert_matrix_3d_general( GLmatrix *mat ) 287{ 288 const GLfloat *in = mat->m; 289 GLfloat *out = mat->inv; 290 GLfloat pos, neg, t; 291 GLfloat det; 292 293 /* Calculate the determinant of upper left 3x3 submatrix and 294 * determine if the matrix is singular. 295 */ 296 pos = neg = 0.0; 297 t = MAT(in,0,0) * MAT(in,1,1) * MAT(in,2,2); 298 if (t >= 0.0) pos += t; else neg += t; 299 300 t = MAT(in,1,0) * MAT(in,2,1) * MAT(in,0,2); 301 if (t >= 0.0) pos += t; else neg += t; 302 303 t = MAT(in,2,0) * MAT(in,0,1) * MAT(in,1,2); 304 if (t >= 0.0) pos += t; else neg += t; 305 306 t = -MAT(in,2,0) * MAT(in,1,1) * MAT(in,0,2); 307 if (t >= 0.0) pos += t; else neg += t; 308 309 t = -MAT(in,1,0) * MAT(in,0,1) * MAT(in,2,2); 310 if (t >= 0.0) pos += t; else neg += t; 311 312 t = -MAT(in,0,0) * MAT(in,2,1) * MAT(in,1,2); 313 if (t >= 0.0) pos += t; else neg += t; 314 315 det = pos + neg; 316 317 if (det*det < 1e-25) 318 return GL_FALSE; 319 320 det = 1.0F / det; 321 MAT(out,0,0) = ( (MAT(in,1,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,1,2) )*det); 322 MAT(out,0,1) = (- (MAT(in,0,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,0,2) )*det); 323 MAT(out,0,2) = ( (MAT(in,0,1)*MAT(in,1,2) - MAT(in,1,1)*MAT(in,0,2) )*det); 324 MAT(out,1,0) = (- (MAT(in,1,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,1,2) )*det); 325 MAT(out,1,1) = ( (MAT(in,0,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,0,2) )*det); 326 MAT(out,1,2) = (- (MAT(in,0,0)*MAT(in,1,2) - MAT(in,1,0)*MAT(in,0,2) )*det); 327 MAT(out,2,0) = ( (MAT(in,1,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,1,1) )*det); 328 MAT(out,2,1) = (- (MAT(in,0,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,0,1) )*det); 329 MAT(out,2,2) = ( (MAT(in,0,0)*MAT(in,1,1) - MAT(in,1,0)*MAT(in,0,1) )*det); 330 331 /* Do the translation part */ 332 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) + 333 MAT(in,1,3) * MAT(out,0,1) + 334 MAT(in,2,3) * MAT(out,0,2) ); 335 MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) + 336 MAT(in,1,3) * MAT(out,1,1) + 337 MAT(in,2,3) * MAT(out,1,2) ); 338 MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) + 339 MAT(in,1,3) * MAT(out,2,1) + 340 MAT(in,2,3) * MAT(out,2,2) ); 341 342 return GL_TRUE; 343} 344 345 346static GLboolean invert_matrix_3d( GLmatrix *mat ) 347{ 348 const GLfloat *in = mat->m; 349 GLfloat *out = mat->inv; 350 351 if (!TEST_MAT_FLAGS(mat, MAT_FLAGS_ANGLE_PRESERVING)) { 352 return invert_matrix_3d_general( mat ); 353 } 354 355 if (mat->flags & MAT_FLAG_UNIFORM_SCALE) { 356 GLfloat scale = (MAT(in,0,0) * MAT(in,0,0) + 357 MAT(in,0,1) * MAT(in,0,1) + 358 MAT(in,0,2) * MAT(in,0,2)); 359 360 if (scale == 0.0) 361 return GL_FALSE; 362 363 scale = 1.0F / scale; 364 365 /* Transpose and scale the 3 by 3 upper-left submatrix. */ 366 MAT(out,0,0) = scale * MAT(in,0,0); 367 MAT(out,1,0) = scale * MAT(in,0,1); 368 MAT(out,2,0) = scale * MAT(in,0,2); 369 MAT(out,0,1) = scale * MAT(in,1,0); 370 MAT(out,1,1) = scale * MAT(in,1,1); 371 MAT(out,2,1) = scale * MAT(in,1,2); 372 MAT(out,0,2) = scale * MAT(in,2,0); 373 MAT(out,1,2) = scale * MAT(in,2,1); 374 MAT(out,2,2) = scale * MAT(in,2,2); 375 } 376 else if (mat->flags & MAT_FLAG_ROTATION) { 377 /* Transpose the 3 by 3 upper-left submatrix. */ 378 MAT(out,0,0) = MAT(in,0,0); 379 MAT(out,1,0) = MAT(in,0,1); 380 MAT(out,2,0) = MAT(in,0,2); 381 MAT(out,0,1) = MAT(in,1,0); 382 MAT(out,1,1) = MAT(in,1,1); 383 MAT(out,2,1) = MAT(in,1,2); 384 MAT(out,0,2) = MAT(in,2,0); 385 MAT(out,1,2) = MAT(in,2,1); 386 MAT(out,2,2) = MAT(in,2,2); 387 } 388 else { 389 /* pure translation */ 390 MEMCPY( out, Identity, sizeof(Identity) ); 391 MAT(out,0,3) = - MAT(in,0,3); 392 MAT(out,1,3) = - MAT(in,1,3); 393 MAT(out,2,3) = - MAT(in,2,3); 394 return GL_TRUE; 395 } 396 397 if (mat->flags & MAT_FLAG_TRANSLATION) { 398 /* Do the translation part */ 399 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) + 400 MAT(in,1,3) * MAT(out,0,1) + 401 MAT(in,2,3) * MAT(out,0,2) ); 402 MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) + 403 MAT(in,1,3) * MAT(out,1,1) + 404 MAT(in,2,3) * MAT(out,1,2) ); 405 MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) + 406 MAT(in,1,3) * MAT(out,2,1) + 407 MAT(in,2,3) * MAT(out,2,2) ); 408 } 409 else { 410 MAT(out,0,3) = MAT(out,1,3) = MAT(out,2,3) = 0.0; 411 } 412 413 return GL_TRUE; 414} 415 416 417 418static GLboolean invert_matrix_identity( GLmatrix *mat ) 419{ 420 MEMCPY( mat->inv, Identity, sizeof(Identity) ); 421 return GL_TRUE; 422} 423 424 425static GLboolean invert_matrix_3d_no_rot( GLmatrix *mat ) 426{ 427 const GLfloat *in = mat->m; 428 GLfloat *out = mat->inv; 429 430 if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0 || MAT(in,2,2) == 0 ) 431 return GL_FALSE; 432 433 MEMCPY( out, Identity, 16 * sizeof(GLfloat) ); 434 MAT(out,0,0) = 1.0F / MAT(in,0,0); 435 MAT(out,1,1) = 1.0F / MAT(in,1,1); 436 MAT(out,2,2) = 1.0F / MAT(in,2,2); 437 438 if (mat->flags & MAT_FLAG_TRANSLATION) { 439 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0)); 440 MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1)); 441 MAT(out,2,3) = - (MAT(in,2,3) * MAT(out,2,2)); 442 } 443 444 return GL_TRUE; 445} 446 447 448static GLboolean invert_matrix_2d_no_rot( GLmatrix *mat ) 449{ 450 const GLfloat *in = mat->m; 451 GLfloat *out = mat->inv; 452 453 if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0) 454 return GL_FALSE; 455 456 MEMCPY( out, Identity, 16 * sizeof(GLfloat) ); 457 MAT(out,0,0) = 1.0F / MAT(in,0,0); 458 MAT(out,1,1) = 1.0F / MAT(in,1,1); 459 460 if (mat->flags & MAT_FLAG_TRANSLATION) { 461 MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0)); 462 MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1)); 463 } 464 465 return GL_TRUE; 466} 467 468 469#if 0 470/* broken */ 471static GLboolean invert_matrix_perspective( GLmatrix *mat ) 472{ 473 const GLfloat *in = mat->m; 474 GLfloat *out = mat->inv; 475 476 if (MAT(in,2,3) == 0) 477 return GL_FALSE; 478 479 MEMCPY( out, Identity, 16 * sizeof(GLfloat) ); 480 481 MAT(out,0,0) = 1.0F / MAT(in,0,0); 482 MAT(out,1,1) = 1.0F / MAT(in,1,1); 483 484 MAT(out,0,3) = MAT(in,0,2); 485 MAT(out,1,3) = MAT(in,1,2); 486 487 MAT(out,2,2) = 0; 488 MAT(out,2,3) = -1; 489 490 MAT(out,3,2) = 1.0F / MAT(in,2,3); 491 MAT(out,3,3) = MAT(in,2,2) * MAT(out,3,2); 492 493 return GL_TRUE; 494} 495#endif 496 497 498typedef GLboolean (*inv_mat_func)( GLmatrix *mat ); 499 500 501static inv_mat_func inv_mat_tab[7] = { 502 invert_matrix_general, 503 invert_matrix_identity, 504 invert_matrix_3d_no_rot, 505#if 0 506 /* Don't use this function for now - it fails when the projection matrix 507 * is premultiplied by a translation (ala Chromium's tilesort SPU). 508 */ 509 invert_matrix_perspective, 510#else 511 invert_matrix_general, 512#endif 513 invert_matrix_3d, /* lazy! */ 514 invert_matrix_2d_no_rot, 515 invert_matrix_3d 516}; 517 518 519static GLboolean matrix_invert( GLmatrix *mat ) 520{ 521 if (inv_mat_tab[mat->type](mat)) { 522 mat->flags &= ~MAT_FLAG_SINGULAR; 523 return GL_TRUE; 524 } else { 525 mat->flags |= MAT_FLAG_SINGULAR; 526 MEMCPY( mat->inv, Identity, sizeof(Identity) ); 527 return GL_FALSE; 528 } 529} 530 531 532 533 534 535 536/* 537 * Generate a 4x4 transformation matrix from glRotate parameters, and 538 * postmultiply the input matrix by it. 539 * This function contributed by Erich Boleyn (erich@uruk.org). 540 * Optimizatios contributed by Rudolf Opalla (rudi@khm.de). 541 */ 542void 543_math_matrix_rotate( GLmatrix *mat, 544 GLfloat angle, GLfloat x, GLfloat y, GLfloat z ) 545{ 546 GLfloat xx, yy, zz, xy, yz, zx, xs, ys, zs, one_c, s, c; 547 GLfloat m[16]; 548 GLboolean optimized; 549 550 s = (GLfloat) sin( angle * DEG2RAD ); 551 c = (GLfloat) cos( angle * DEG2RAD ); 552 553 MEMCPY(m, Identity, sizeof(GLfloat)*16); 554 optimized = GL_FALSE; 555 556#define M(row,col) m[col*4+row] 557 558 if (x == 0.0F) { 559 if (y == 0.0F) { 560 if (z != 0.0F) { 561 optimized = GL_TRUE; 562 /* rotate only around z-axis */ 563 M(0,0) = c; 564 M(1,1) = c; 565 if (z < 0.0F) { 566 M(0,1) = s; 567 M(1,0) = -s; 568 } 569 else { 570 M(0,1) = -s; 571 M(1,0) = s; 572 } 573 } 574 } 575 else if (z == 0.0F) { 576 optimized = GL_TRUE; 577 /* rotate only around y-axis */ 578 M(0,0) = c; 579 M(2,2) = c; 580 if (y < 0.0F) { 581 M(0,2) = -s; 582 M(2,0) = s; 583 } 584 else { 585 M(0,2) = s; 586 M(2,0) = -s; 587 } 588 } 589 } 590 else if (y == 0.0F) { 591 if (z == 0.0F) { 592 optimized = GL_TRUE; 593 /* rotate only around x-axis */ 594 M(1,1) = c; 595 M(2,2) = c; 596 if (x < 0.0F) { 597 M(1,2) = s; 598 M(2,1) = -s; 599 } 600 else { 601 M(1,2) = -s; 602 M(2,1) = s; 603 } 604 } 605 } 606 607 if (!optimized) { 608 const GLfloat mag = SQRTF(x * x + y * y + z * z); 609 610 if (mag <= 1.0e-4) { 611 /* no rotation, leave mat as-is */ 612 return; 613 } 614 615 x /= mag; 616 y /= mag; 617 z /= mag; 618 619 620 /* 621 * Arbitrary axis rotation matrix. 622 * 623 * This is composed of 5 matrices, Rz, Ry, T, Ry', Rz', multiplied 624 * like so: Rz * Ry * T * Ry' * Rz'. T is the final rotation 625 * (which is about the X-axis), and the two composite transforms 626 * Ry' * Rz' and Rz * Ry are (respectively) the rotations necessary 627 * from the arbitrary axis to the X-axis then back. They are 628 * all elementary rotations. 629 * 630 * Rz' is a rotation about the Z-axis, to bring the axis vector 631 * into the x-z plane. Then Ry' is applied, rotating about the 632 * Y-axis to bring the axis vector parallel with the X-axis. The 633 * rotation about the X-axis is then performed. Ry and Rz are 634 * simply the respective inverse transforms to bring the arbitrary 635 * axis back to it's original orientation. The first transforms 636 * Rz' and Ry' are considered inverses, since the data from the 637 * arbitrary axis gives you info on how to get to it, not how 638 * to get away from it, and an inverse must be applied. 639 * 640 * The basic calculation used is to recognize that the arbitrary 641 * axis vector (x, y, z), since it is of unit length, actually 642 * represents the sines and cosines of the angles to rotate the 643 * X-axis to the same orientation, with theta being the angle about 644 * Z and phi the angle about Y (in the order described above) 645 * as follows: 646 * 647 * cos ( theta ) = x / sqrt ( 1 - z^2 ) 648 * sin ( theta ) = y / sqrt ( 1 - z^2 ) 649 * 650 * cos ( phi ) = sqrt ( 1 - z^2 ) 651 * sin ( phi ) = z 652 * 653 * Note that cos ( phi ) can further be inserted to the above 654 * formulas: 655 * 656 * cos ( theta ) = x / cos ( phi ) 657 * sin ( theta ) = y / sin ( phi ) 658 * 659 * ...etc. Because of those relations and the standard trigonometric 660 * relations, it is pssible to reduce the transforms down to what 661 * is used below. It may be that any primary axis chosen will give the 662 * same results (modulo a sign convention) using thie method. 663 * 664 * Particularly nice is to notice that all divisions that might 665 * have caused trouble when parallel to certain planes or 666 * axis go away with care paid to reducing the expressions. 667 * After checking, it does perform correctly under all cases, since 668 * in all the cases of division where the denominator would have 669 * been zero, the numerator would have been zero as well, giving 670 * the expected result. 671 */ 672 673 xx = x * x; 674 yy = y * y; 675 zz = z * z; 676 xy = x * y; 677 yz = y * z; 678 zx = z * x; 679 xs = x * s; 680 ys = y * s; 681 zs = z * s; 682 one_c = 1.0F - c; 683 684 /* We already hold the identity-matrix so we can skip some statements */ 685 M(0,0) = (one_c * xx) + c; 686 M(0,1) = (one_c * xy) - zs; 687 M(0,2) = (one_c * zx) + ys; 688/* M(0,3) = 0.0F; */ 689 690 M(1,0) = (one_c * xy) + zs; 691 M(1,1) = (one_c * yy) + c; 692 M(1,2) = (one_c * yz) - xs; 693/* M(1,3) = 0.0F; */ 694 695 M(2,0) = (one_c * zx) - ys; 696 M(2,1) = (one_c * yz) + xs; 697 M(2,2) = (one_c * zz) + c; 698/* M(2,3) = 0.0F; */ 699 700/* 701 M(3,0) = 0.0F; 702 M(3,1) = 0.0F; 703 M(3,2) = 0.0F; 704 M(3,3) = 1.0F; 705*/ 706 } 707#undef M 708 709 matrix_multf( mat, m, MAT_FLAG_ROTATION ); 710} 711 712 713 714void 715_math_matrix_frustum( GLmatrix *mat, 716 GLfloat left, GLfloat right, 717 GLfloat bottom, GLfloat top, 718 GLfloat nearval, GLfloat farval ) 719{ 720 GLfloat x, y, a, b, c, d; 721 GLfloat m[16]; 722 723 x = (2.0F*nearval) / (right-left); 724 y = (2.0F*nearval) / (top-bottom); 725 a = (right+left) / (right-left); 726 b = (top+bottom) / (top-bottom); 727 c = -(farval+nearval) / ( farval-nearval); 728 d = -(2.0F*farval*nearval) / (farval-nearval); /* error? */ 729 730#define M(row,col) m[col*4+row] 731 M(0,0) = x; M(0,1) = 0.0F; M(0,2) = a; M(0,3) = 0.0F; 732 M(1,0) = 0.0F; M(1,1) = y; M(1,2) = b; M(1,3) = 0.0F; 733 M(2,0) = 0.0F; M(2,1) = 0.0F; M(2,2) = c; M(2,3) = d; 734 M(3,0) = 0.0F; M(3,1) = 0.0F; M(3,2) = -1.0F; M(3,3) = 0.0F; 735#undef M 736 737 matrix_multf( mat, m, MAT_FLAG_PERSPECTIVE ); 738} 739 740void 741_math_matrix_ortho( GLmatrix *mat, 742 GLfloat left, GLfloat right, 743 GLfloat bottom, GLfloat top, 744 GLfloat nearval, GLfloat farval ) 745{ 746 GLfloat x, y, z; 747 GLfloat tx, ty, tz; 748 GLfloat m[16]; 749 750 x = 2.0F / (right-left); 751 y = 2.0F / (top-bottom); 752 z = -2.0F / (farval-nearval); 753 tx = -(right+left) / (right-left); 754 ty = -(top+bottom) / (top-bottom); 755 tz = -(farval+nearval) / (farval-nearval); 756 757#define M(row,col) m[col*4+row] 758 M(0,0) = x; M(0,1) = 0.0F; M(0,2) = 0.0F; M(0,3) = tx; 759 M(1,0) = 0.0F; M(1,1) = y; M(1,2) = 0.0F; M(1,3) = ty; 760 M(2,0) = 0.0F; M(2,1) = 0.0F; M(2,2) = z; M(2,3) = tz; 761 M(3,0) = 0.0F; M(3,1) = 0.0F; M(3,2) = 0.0F; M(3,3) = 1.0F; 762#undef M 763 764 matrix_multf( mat, m, (MAT_FLAG_GENERAL_SCALE|MAT_FLAG_TRANSLATION)); 765} 766 767 768#define ZERO(x) (1<<x) 769#define ONE(x) (1<<(x+16)) 770 771#define MASK_NO_TRX (ZERO(12) | ZERO(13) | ZERO(14)) 772#define MASK_NO_2D_SCALE ( ONE(0) | ONE(5)) 773 774#define MASK_IDENTITY ( ONE(0) | ZERO(4) | ZERO(8) | ZERO(12) |\ 775 ZERO(1) | ONE(5) | ZERO(9) | ZERO(13) |\ 776 ZERO(2) | ZERO(6) | ONE(10) | ZERO(14) |\ 777 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 778 779#define MASK_2D_NO_ROT ( ZERO(4) | ZERO(8) | \ 780 ZERO(1) | ZERO(9) | \ 781 ZERO(2) | ZERO(6) | ONE(10) | ZERO(14) |\ 782 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 783 784#define MASK_2D ( ZERO(8) | \ 785 ZERO(9) | \ 786 ZERO(2) | ZERO(6) | ONE(10) | ZERO(14) |\ 787 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 788 789 790#define MASK_3D_NO_ROT ( ZERO(4) | ZERO(8) | \ 791 ZERO(1) | ZERO(9) | \ 792 ZERO(2) | ZERO(6) | \ 793 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 794 795#define MASK_3D ( \ 796 \ 797 \ 798 ZERO(3) | ZERO(7) | ZERO(11) | ONE(15) ) 799 800 801#define MASK_PERSPECTIVE ( ZERO(4) | ZERO(12) |\ 802 ZERO(1) | ZERO(13) |\ 803 ZERO(2) | ZERO(6) | \ 804 ZERO(3) | ZERO(7) | ZERO(15) ) 805 806#define SQ(x) ((x)*(x)) 807 808/* Determine type and flags from scratch. This is expensive enough to 809 * only want to do it once. 810 */ 811static void analyse_from_scratch( GLmatrix *mat ) 812{ 813 const GLfloat *m = mat->m; 814 GLuint mask = 0; 815 GLuint i; 816 817 for (i = 0 ; i < 16 ; i++) { 818 if (m[i] == 0.0) mask |= (1<<i); 819 } 820 821 if (m[0] == 1.0F) mask |= (1<<16); 822 if (m[5] == 1.0F) mask |= (1<<21); 823 if (m[10] == 1.0F) mask |= (1<<26); 824 if (m[15] == 1.0F) mask |= (1<<31); 825 826 mat->flags &= ~MAT_FLAGS_GEOMETRY; 827 828 /* Check for translation - no-one really cares 829 */ 830 if ((mask & MASK_NO_TRX) != MASK_NO_TRX) 831 mat->flags |= MAT_FLAG_TRANSLATION; 832 833 /* Do the real work 834 */ 835 if (mask == (GLuint) MASK_IDENTITY) { 836 mat->type = MATRIX_IDENTITY; 837 } 838 else if ((mask & MASK_2D_NO_ROT) == (GLuint) MASK_2D_NO_ROT) { 839 mat->type = MATRIX_2D_NO_ROT; 840 841 if ((mask & MASK_NO_2D_SCALE) != MASK_NO_2D_SCALE) 842 mat->flags = MAT_FLAG_GENERAL_SCALE; 843 } 844 else if ((mask & MASK_2D) == (GLuint) MASK_2D) { 845 GLfloat mm = DOT2(m, m); 846 GLfloat m4m4 = DOT2(m+4,m+4); 847 GLfloat mm4 = DOT2(m,m+4); 848 849 mat->type = MATRIX_2D; 850 851 /* Check for scale */ 852 if (SQ(mm-1) > SQ(1e-6) || 853 SQ(m4m4-1) > SQ(1e-6)) 854 mat->flags |= MAT_FLAG_GENERAL_SCALE; 855 856 /* Check for rotation */ 857 if (SQ(mm4) > SQ(1e-6)) 858 mat->flags |= MAT_FLAG_GENERAL_3D; 859 else 860 mat->flags |= MAT_FLAG_ROTATION; 861 862 } 863 else if ((mask & MASK_3D_NO_ROT) == (GLuint) MASK_3D_NO_ROT) { 864 mat->type = MATRIX_3D_NO_ROT; 865 866 /* Check for scale */ 867 if (SQ(m[0]-m[5]) < SQ(1e-6) && 868 SQ(m[0]-m[10]) < SQ(1e-6)) { 869 if (SQ(m[0]-1.0) > SQ(1e-6)) { 870 mat->flags |= MAT_FLAG_UNIFORM_SCALE; 871 } 872 } 873 else { 874 mat->flags |= MAT_FLAG_GENERAL_SCALE; 875 } 876 } 877 else if ((mask & MASK_3D) == (GLuint) MASK_3D) { 878 GLfloat c1 = DOT3(m,m); 879 GLfloat c2 = DOT3(m+4,m+4); 880 GLfloat c3 = DOT3(m+8,m+8); 881 GLfloat d1 = DOT3(m, m+4); 882 GLfloat cp[3]; 883 884 mat->type = MATRIX_3D; 885 886 /* Check for scale */ 887 if (SQ(c1-c2) < SQ(1e-6) && SQ(c1-c3) < SQ(1e-6)) { 888 if (SQ(c1-1.0) > SQ(1e-6)) 889 mat->flags |= MAT_FLAG_UNIFORM_SCALE; 890 /* else no scale at all */ 891 } 892 else { 893 mat->flags |= MAT_FLAG_GENERAL_SCALE; 894 } 895 896 /* Check for rotation */ 897 if (SQ(d1) < SQ(1e-6)) { 898 CROSS3( cp, m, m+4 ); 899 SUB_3V( cp, cp, (m+8) ); 900 if (LEN_SQUARED_3FV(cp) < SQ(1e-6)) 901 mat->flags |= MAT_FLAG_ROTATION; 902 else 903 mat->flags |= MAT_FLAG_GENERAL_3D; 904 } 905 else { 906 mat->flags |= MAT_FLAG_GENERAL_3D; /* shear, etc */ 907 } 908 } 909 else if ((mask & MASK_PERSPECTIVE) == MASK_PERSPECTIVE && m[11]==-1.0F) { 910 mat->type = MATRIX_PERSPECTIVE; 911 mat->flags |= MAT_FLAG_GENERAL; 912 } 913 else { 914 mat->type = MATRIX_GENERAL; 915 mat->flags |= MAT_FLAG_GENERAL; 916 } 917} 918 919 920/* Analyse a matrix given that its flags are accurate - this is the 921 * more common operation, hopefully. 922 */ 923static void analyse_from_flags( GLmatrix *mat ) 924{ 925 const GLfloat *m = mat->m; 926 927 if (TEST_MAT_FLAGS(mat, 0)) { 928 mat->type = MATRIX_IDENTITY; 929 } 930 else if (TEST_MAT_FLAGS(mat, (MAT_FLAG_TRANSLATION | 931 MAT_FLAG_UNIFORM_SCALE | 932 MAT_FLAG_GENERAL_SCALE))) { 933 if ( m[10]==1.0F && m[14]==0.0F ) { 934 mat->type = MATRIX_2D_NO_ROT; 935 } 936 else { 937 mat->type = MATRIX_3D_NO_ROT; 938 } 939 } 940 else if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D)) { 941 if ( m[ 8]==0.0F 942 && m[ 9]==0.0F 943 && m[2]==0.0F && m[6]==0.0F && m[10]==1.0F && m[14]==0.0F) { 944 mat->type = MATRIX_2D; 945 } 946 else { 947 mat->type = MATRIX_3D; 948 } 949 } 950 else if ( m[4]==0.0F && m[12]==0.0F 951 && m[1]==0.0F && m[13]==0.0F 952 && m[2]==0.0F && m[6]==0.0F 953 && m[3]==0.0F && m[7]==0.0F && m[11]==-1.0F && m[15]==0.0F) { 954 mat->type = MATRIX_PERSPECTIVE; 955 } 956 else { 957 mat->type = MATRIX_GENERAL; 958 } 959} 960 961 962void 963_math_matrix_analyse( GLmatrix *mat ) 964{ 965 if (mat->flags & MAT_DIRTY_TYPE) { 966 if (mat->flags & MAT_DIRTY_FLAGS) 967 analyse_from_scratch( mat ); 968 else 969 analyse_from_flags( mat ); 970 } 971 972 if (mat->inv && (mat->flags & MAT_DIRTY_INVERSE)) { 973 matrix_invert( mat ); 974 } 975 976 mat->flags &= ~(MAT_DIRTY_FLAGS| 977 MAT_DIRTY_TYPE| 978 MAT_DIRTY_INVERSE); 979} 980 981 982void 983_math_matrix_copy( GLmatrix *to, const GLmatrix *from ) 984{ 985 MEMCPY( to->m, from->m, sizeof(Identity) ); 986 to->flags = from->flags; 987 to->type = from->type; 988 989 if (to->inv != 0) { 990 if (from->inv == 0) { 991 matrix_invert( to ); 992 } 993 else { 994 MEMCPY(to->inv, from->inv, sizeof(GLfloat)*16); 995 } 996 } 997} 998 999 1000void 1001_math_matrix_scale( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z ) 1002{ 1003 GLfloat *m = mat->m; 1004 m[0] *= x; m[4] *= y; m[8] *= z; 1005 m[1] *= x; m[5] *= y; m[9] *= z; 1006 m[2] *= x; m[6] *= y; m[10] *= z; 1007 m[3] *= x; m[7] *= y; m[11] *= z; 1008 1009 if (fabs(x - y) < 1e-8 && fabs(x - z) < 1e-8) 1010 mat->flags |= MAT_FLAG_UNIFORM_SCALE; 1011 else 1012 mat->flags |= MAT_FLAG_GENERAL_SCALE; 1013 1014 mat->flags |= (MAT_DIRTY_TYPE | 1015 MAT_DIRTY_INVERSE); 1016} 1017 1018 1019void 1020_math_matrix_translate( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z ) 1021{ 1022 GLfloat *m = mat->m; 1023 m[12] = m[0] * x + m[4] * y + m[8] * z + m[12]; 1024 m[13] = m[1] * x + m[5] * y + m[9] * z + m[13]; 1025 m[14] = m[2] * x + m[6] * y + m[10] * z + m[14]; 1026 m[15] = m[3] * x + m[7] * y + m[11] * z + m[15]; 1027 1028 mat->flags |= (MAT_FLAG_TRANSLATION | 1029 MAT_DIRTY_TYPE | 1030 MAT_DIRTY_INVERSE); 1031} 1032 1033 1034void 1035_math_matrix_loadf( GLmatrix *mat, const GLfloat *m ) 1036{ 1037 MEMCPY( mat->m, m, 16*sizeof(GLfloat) ); 1038 mat->flags = (MAT_FLAG_GENERAL | MAT_DIRTY); 1039} 1040 1041void 1042_math_matrix_ctr( GLmatrix *m ) 1043{ 1044 m->m = (GLfloat *) ALIGN_MALLOC( 16 * sizeof(GLfloat), 16 ); 1045 if (m->m) 1046 MEMCPY( m->m, Identity, sizeof(Identity) ); 1047 m->inv = NULL; 1048 m->type = MATRIX_IDENTITY; 1049 m->flags = 0; 1050} 1051 1052void 1053_math_matrix_dtr( GLmatrix *m ) 1054{ 1055 if (m->m) { 1056 ALIGN_FREE( m->m ); 1057 m->m = NULL; 1058 } 1059 if (m->inv) { 1060 ALIGN_FREE( m->inv ); 1061 m->inv = NULL; 1062 } 1063} 1064 1065 1066void 1067_math_matrix_alloc_inv( GLmatrix *m ) 1068{ 1069 if (!m->inv) { 1070 m->inv = (GLfloat *) ALIGN_MALLOC( 16 * sizeof(GLfloat), 16 ); 1071 if (m->inv) 1072 MEMCPY( m->inv, Identity, 16 * sizeof(GLfloat) ); 1073 } 1074} 1075 1076 1077void 1078_math_matrix_mul_matrix( GLmatrix *dest, const GLmatrix *a, const GLmatrix *b ) 1079{ 1080 dest->flags = (a->flags | 1081 b->flags | 1082 MAT_DIRTY_TYPE | 1083 MAT_DIRTY_INVERSE); 1084 1085 if (TEST_MAT_FLAGS(dest, MAT_FLAGS_3D)) 1086 matmul34( dest->m, a->m, b->m ); 1087 else 1088 matmul4( dest->m, a->m, b->m ); 1089} 1090 1091 1092void 1093_math_matrix_mul_floats( GLmatrix *dest, const GLfloat *m ) 1094{ 1095 dest->flags |= (MAT_FLAG_GENERAL | 1096 MAT_DIRTY_TYPE | 1097 MAT_DIRTY_INVERSE); 1098 1099 matmul4( dest->m, dest->m, m ); 1100} 1101 1102void 1103_math_matrix_set_identity( GLmatrix *mat ) 1104{ 1105 MEMCPY( mat->m, Identity, 16*sizeof(GLfloat) ); 1106 1107 if (mat->inv) 1108 MEMCPY( mat->inv, Identity, 16*sizeof(GLfloat) ); 1109 1110 mat->type = MATRIX_IDENTITY; 1111 mat->flags &= ~(MAT_DIRTY_FLAGS| 1112 MAT_DIRTY_TYPE| 1113 MAT_DIRTY_INVERSE); 1114} 1115 1116 1117 1118void 1119_math_transposef( GLfloat to[16], const GLfloat from[16] ) 1120{ 1121 to[0] = from[0]; 1122 to[1] = from[4]; 1123 to[2] = from[8]; 1124 to[3] = from[12]; 1125 to[4] = from[1]; 1126 to[5] = from[5]; 1127 to[6] = from[9]; 1128 to[7] = from[13]; 1129 to[8] = from[2]; 1130 to[9] = from[6]; 1131 to[10] = from[10]; 1132 to[11] = from[14]; 1133 to[12] = from[3]; 1134 to[13] = from[7]; 1135 to[14] = from[11]; 1136 to[15] = from[15]; 1137} 1138 1139 1140void 1141_math_transposed( GLdouble to[16], const GLdouble from[16] ) 1142{ 1143 to[0] = from[0]; 1144 to[1] = from[4]; 1145 to[2] = from[8]; 1146 to[3] = from[12]; 1147 to[4] = from[1]; 1148 to[5] = from[5]; 1149 to[6] = from[9]; 1150 to[7] = from[13]; 1151 to[8] = from[2]; 1152 to[9] = from[6]; 1153 to[10] = from[10]; 1154 to[11] = from[14]; 1155 to[12] = from[3]; 1156 to[13] = from[7]; 1157 to[14] = from[11]; 1158 to[15] = from[15]; 1159} 1160 1161void 1162_math_transposefd( GLfloat to[16], const GLdouble from[16] ) 1163{ 1164 to[0] = (GLfloat) from[0]; 1165 to[1] = (GLfloat) from[4]; 1166 to[2] = (GLfloat) from[8]; 1167 to[3] = (GLfloat) from[12]; 1168 to[4] = (GLfloat) from[1]; 1169 to[5] = (GLfloat) from[5]; 1170 to[6] = (GLfloat) from[9]; 1171 to[7] = (GLfloat) from[13]; 1172 to[8] = (GLfloat) from[2]; 1173 to[9] = (GLfloat) from[6]; 1174 to[10] = (GLfloat) from[10]; 1175 to[11] = (GLfloat) from[14]; 1176 to[12] = (GLfloat) from[3]; 1177 to[13] = (GLfloat) from[7]; 1178 to[14] = (GLfloat) from[11]; 1179 to[15] = (GLfloat) from[15]; 1180} 1181