1 /***************************************************************************/ 2 /* */ 3 /* fttrigon.c */ 4 /* */ 5 /* FreeType trigonometric functions (body). */ 6 /* */ 7 /* Copyright 2001-2018 by */ 8 /* David Turner, Robert Wilhelm, and Werner Lemberg. */ 9 /* */ 10 /* This file is part of the FreeType project, and may only be used, */ 11 /* modified, and distributed under the terms of the FreeType project */ 12 /* license, LICENSE.TXT. By continuing to use, modify, or distribute */ 13 /* this file you indicate that you have read the license and */ 14 /* understand and accept it fully. */ 15 /* */ 16 /***************************************************************************/ 17 18 /*************************************************************************/ 19 /* */ 20 /* This is a fixed-point CORDIC implementation of trigonometric */ 21 /* functions as well as transformations between Cartesian and polar */ 22 /* coordinates. The angles are represented as 16.16 fixed-point values */ 23 /* in degrees, i.e., the angular resolution is 2^-16 degrees. Note that */ 24 /* only vectors longer than 2^16*180/pi (or at least 22 bits) on a */ 25 /* discrete Cartesian grid can have the same or better angular */ 26 /* resolution. Therefore, to maintain this precision, some functions */ 27 /* require an interim upscaling of the vectors, whereas others operate */ 28 /* with 24-bit long vectors directly. */ 29 /* */ 30 /*************************************************************************/ 31 32 #include <ft2build.h> 33 #include FT_INTERNAL_OBJECTS_H 34 #include FT_INTERNAL_CALC_H 35 #include FT_TRIGONOMETRY_H 36 37 38 /* the Cordic shrink factor 0.858785336480436 * 2^32 */ 39 #define FT_TRIG_SCALE 0xDBD95B16UL 40 41 /* the highest bit in overflow-safe vector components, */ 42 /* MSB of 0.858785336480436 * sqrt(0.5) * 2^30 */ 43 #define FT_TRIG_SAFE_MSB 29 44 45 /* this table was generated for FT_PI = 180L << 16, i.e. degrees */ 46 #define FT_TRIG_MAX_ITERS 23 47 48 static const FT_Angle 49 ft_trig_arctan_table[] = 50 { 51 1740967L, 919879L, 466945L, 234379L, 117304L, 58666L, 29335L, 52 14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L, 53 57L, 29L, 14L, 7L, 4L, 2L, 1L 54 }; 55 56 57 #ifdef FT_LONG64 58 59 /* multiply a given value by the CORDIC shrink factor */ 60 static FT_Fixed 61 ft_trig_downscale( FT_Fixed val ) 62 { 63 FT_Int s = 1; 64 65 66 if ( val < 0 ) 67 { 68 val = -val; 69 s = -1; 70 } 71 72 /* 0x40000000 comes from regression analysis between true */ 73 /* and CORDIC hypotenuse, so it minimizes the error */ 74 val = (FT_Fixed)( 75 ( (FT_UInt64)val * FT_TRIG_SCALE + 0x40000000UL ) >> 32 ); 76 77 return s < 0 ? -val : val; 78 } 79 80 #else /* !FT_LONG64 */ 81 82 /* multiply a given value by the CORDIC shrink factor */ 83 static FT_Fixed 84 ft_trig_downscale( FT_Fixed val ) 85 { 86 FT_Int s = 1; 87 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2; 88 89 90 if ( val < 0 ) 91 { 92 val = -val; 93 s = -1; 94 } 95 96 lo1 = (FT_UInt32)val & 0x0000FFFFU; 97 hi1 = (FT_UInt32)val >> 16; 98 lo2 = FT_TRIG_SCALE & 0x0000FFFFU; 99 hi2 = FT_TRIG_SCALE >> 16; 100 101 lo = lo1 * lo2; 102 i1 = lo1 * hi2; 103 i2 = lo2 * hi1; 104 hi = hi1 * hi2; 105 106 /* Check carry overflow of i1 + i2 */ 107 i1 += i2; 108 hi += (FT_UInt32)( i1 < i2 ) << 16; 109 110 hi += i1 >> 16; 111 i1 = i1 << 16; 112 113 /* Check carry overflow of i1 + lo */ 114 lo += i1; 115 hi += ( lo < i1 ); 116 117 /* 0x40000000 comes from regression analysis between true */ 118 /* and CORDIC hypotenuse, so it minimizes the error */ 119 120 /* Check carry overflow of lo + 0x40000000 */ 121 lo += 0x40000000UL; 122 hi += ( lo < 0x40000000UL ); 123 124 val = (FT_Fixed)hi; 125 126 return s < 0 ? -val : val; 127 } 128 129 #endif /* !FT_LONG64 */ 130 131 132 /* undefined and never called for zero vector */ 133 static FT_Int 134 ft_trig_prenorm( FT_Vector* vec ) 135 { 136 FT_Pos x, y; 137 FT_Int shift; 138 139 140 x = vec->x; 141 y = vec->y; 142 143 shift = FT_MSB( (FT_UInt32)( FT_ABS( x ) | FT_ABS( y ) ) ); 144 145 if ( shift <= FT_TRIG_SAFE_MSB ) 146 { 147 shift = FT_TRIG_SAFE_MSB - shift; 148 vec->x = (FT_Pos)( (FT_ULong)x << shift ); 149 vec->y = (FT_Pos)( (FT_ULong)y << shift ); 150 } 151 else 152 { 153 shift -= FT_TRIG_SAFE_MSB; 154 vec->x = x >> shift; 155 vec->y = y >> shift; 156 shift = -shift; 157 } 158 159 return shift; 160 } 161 162 163 static void 164 ft_trig_pseudo_rotate( FT_Vector* vec, 165 FT_Angle theta ) 166 { 167 FT_Int i; 168 FT_Fixed x, y, xtemp, b; 169 const FT_Angle *arctanptr; 170 171 172 x = vec->x; 173 y = vec->y; 174 175 /* Rotate inside [-PI/4,PI/4] sector */ 176 while ( theta < -FT_ANGLE_PI4 ) 177 { 178 xtemp = y; 179 y = -x; 180 x = xtemp; 181 theta += FT_ANGLE_PI2; 182 } 183 184 while ( theta > FT_ANGLE_PI4 ) 185 { 186 xtemp = -y; 187 y = x; 188 x = xtemp; 189 theta -= FT_ANGLE_PI2; 190 } 191 192 arctanptr = ft_trig_arctan_table; 193 194 /* Pseudorotations, with right shifts */ 195 for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ ) 196 { 197 if ( theta < 0 ) 198 { 199 xtemp = x + ( ( y + b ) >> i ); 200 y = y - ( ( x + b ) >> i ); 201 x = xtemp; 202 theta += *arctanptr++; 203 } 204 else 205 { 206 xtemp = x - ( ( y + b ) >> i ); 207 y = y + ( ( x + b ) >> i ); 208 x = xtemp; 209 theta -= *arctanptr++; 210 } 211 } 212 213 vec->x = x; 214 vec->y = y; 215 } 216 217 218 static void 219 ft_trig_pseudo_polarize( FT_Vector* vec ) 220 { 221 FT_Angle theta; 222 FT_Int i; 223 FT_Fixed x, y, xtemp, b; 224 const FT_Angle *arctanptr; 225 226 227 x = vec->x; 228 y = vec->y; 229 230 /* Get the vector into [-PI/4,PI/4] sector */ 231 if ( y > x ) 232 { 233 if ( y > -x ) 234 { 235 theta = FT_ANGLE_PI2; 236 xtemp = y; 237 y = -x; 238 x = xtemp; 239 } 240 else 241 { 242 theta = y > 0 ? FT_ANGLE_PI : -FT_ANGLE_PI; 243 x = -x; 244 y = -y; 245 } 246 } 247 else 248 { 249 if ( y < -x ) 250 { 251 theta = -FT_ANGLE_PI2; 252 xtemp = -y; 253 y = x; 254 x = xtemp; 255 } 256 else 257 { 258 theta = 0; 259 } 260 } 261 262 arctanptr = ft_trig_arctan_table; 263 264 /* Pseudorotations, with right shifts */ 265 for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ ) 266 { 267 if ( y > 0 ) 268 { 269 xtemp = x + ( ( y + b ) >> i ); 270 y = y - ( ( x + b ) >> i ); 271 x = xtemp; 272 theta += *arctanptr++; 273 } 274 else 275 { 276 xtemp = x - ( ( y + b ) >> i ); 277 y = y + ( ( x + b ) >> i ); 278 x = xtemp; 279 theta -= *arctanptr++; 280 } 281 } 282 283 /* round theta to acknowledge its error that mostly comes */ 284 /* from accumulated rounding errors in the arctan table */ 285 if ( theta >= 0 ) 286 theta = FT_PAD_ROUND( theta, 16 ); 287 else 288 theta = -FT_PAD_ROUND( -theta, 16 ); 289 290 vec->x = x; 291 vec->y = theta; 292 } 293 294 295 /* documentation is in fttrigon.h */ 296 297 FT_EXPORT_DEF( FT_Fixed ) 298 FT_Cos( FT_Angle angle ) 299 { 300 FT_Vector v; 301 302 303 FT_Vector_Unit( &v, angle ); 304 305 return v.x; 306 } 307 308 309 /* documentation is in fttrigon.h */ 310 311 FT_EXPORT_DEF( FT_Fixed ) 312 FT_Sin( FT_Angle angle ) 313 { 314 FT_Vector v; 315 316 317 FT_Vector_Unit( &v, angle ); 318 319 return v.y; 320 } 321 322 323 /* documentation is in fttrigon.h */ 324 325 FT_EXPORT_DEF( FT_Fixed ) 326 FT_Tan( FT_Angle angle ) 327 { 328 FT_Vector v; 329 330 331 FT_Vector_Unit( &v, angle ); 332 333 return FT_DivFix( v.y, v.x ); 334 } 335 336 337 /* documentation is in fttrigon.h */ 338 339 FT_EXPORT_DEF( FT_Angle ) 340 FT_Atan2( FT_Fixed dx, 341 FT_Fixed dy ) 342 { 343 FT_Vector v; 344 345 346 if ( dx == 0 && dy == 0 ) 347 return 0; 348 349 v.x = dx; 350 v.y = dy; 351 ft_trig_prenorm( &v ); 352 ft_trig_pseudo_polarize( &v ); 353 354 return v.y; 355 } 356 357 358 /* documentation is in fttrigon.h */ 359 360 FT_EXPORT_DEF( void ) 361 FT_Vector_Unit( FT_Vector* vec, 362 FT_Angle angle ) 363 { 364 if ( !vec ) 365 return; 366 367 vec->x = FT_TRIG_SCALE >> 8; 368 vec->y = 0; 369 ft_trig_pseudo_rotate( vec, angle ); 370 vec->x = ( vec->x + 0x80L ) >> 8; 371 vec->y = ( vec->y + 0x80L ) >> 8; 372 } 373 374 375 /* these macros return 0 for positive numbers, 376 and -1 for negative ones */ 377 #define FT_SIGN_LONG( x ) ( (x) >> ( FT_SIZEOF_LONG * 8 - 1 ) ) 378 #define FT_SIGN_INT( x ) ( (x) >> ( FT_SIZEOF_INT * 8 - 1 ) ) 379 #define FT_SIGN_INT32( x ) ( (x) >> 31 ) 380 #define FT_SIGN_INT16( x ) ( (x) >> 15 ) 381 382 383 /* documentation is in fttrigon.h */ 384 385 FT_EXPORT_DEF( void ) 386 FT_Vector_Rotate( FT_Vector* vec, 387 FT_Angle angle ) 388 { 389 FT_Int shift; 390 FT_Vector v; 391 392 393 if ( !vec || !angle ) 394 return; 395 396 v = *vec; 397 398 if ( v.x == 0 && v.y == 0 ) 399 return; 400 401 shift = ft_trig_prenorm( &v ); 402 ft_trig_pseudo_rotate( &v, angle ); 403 v.x = ft_trig_downscale( v.x ); 404 v.y = ft_trig_downscale( v.y ); 405 406 if ( shift > 0 ) 407 { 408 FT_Int32 half = (FT_Int32)1L << ( shift - 1 ); 409 410 411 vec->x = ( v.x + half + FT_SIGN_LONG( v.x ) ) >> shift; 412 vec->y = ( v.y + half + FT_SIGN_LONG( v.y ) ) >> shift; 413 } 414 else 415 { 416 shift = -shift; 417 vec->x = (FT_Pos)( (FT_ULong)v.x << shift ); 418 vec->y = (FT_Pos)( (FT_ULong)v.y << shift ); 419 } 420 } 421 422 423 /* documentation is in fttrigon.h */ 424 425 FT_EXPORT_DEF( FT_Fixed ) 426 FT_Vector_Length( FT_Vector* vec ) 427 { 428 FT_Int shift; 429 FT_Vector v; 430 431 432 if ( !vec ) 433 return 0; 434 435 v = *vec; 436 437 /* handle trivial cases */ 438 if ( v.x == 0 ) 439 { 440 return FT_ABS( v.y ); 441 } 442 else if ( v.y == 0 ) 443 { 444 return FT_ABS( v.x ); 445 } 446 447 /* general case */ 448 shift = ft_trig_prenorm( &v ); 449 ft_trig_pseudo_polarize( &v ); 450 451 v.x = ft_trig_downscale( v.x ); 452 453 if ( shift > 0 ) 454 return ( v.x + ( 1L << ( shift - 1 ) ) ) >> shift; 455 456 return (FT_Fixed)( (FT_UInt32)v.x << -shift ); 457 } 458 459 460 /* documentation is in fttrigon.h */ 461 462 FT_EXPORT_DEF( void ) 463 FT_Vector_Polarize( FT_Vector* vec, 464 FT_Fixed *length, 465 FT_Angle *angle ) 466 { 467 FT_Int shift; 468 FT_Vector v; 469 470 471 if ( !vec || !length || !angle ) 472 return; 473 474 v = *vec; 475 476 if ( v.x == 0 && v.y == 0 ) 477 return; 478 479 shift = ft_trig_prenorm( &v ); 480 ft_trig_pseudo_polarize( &v ); 481 482 v.x = ft_trig_downscale( v.x ); 483 484 *length = shift >= 0 ? ( v.x >> shift ) 485 : (FT_Fixed)( (FT_UInt32)v.x << -shift ); 486 *angle = v.y; 487 } 488 489 490 /* documentation is in fttrigon.h */ 491 492 FT_EXPORT_DEF( void ) 493 FT_Vector_From_Polar( FT_Vector* vec, 494 FT_Fixed length, 495 FT_Angle angle ) 496 { 497 if ( !vec ) 498 return; 499 500 vec->x = length; 501 vec->y = 0; 502 503 FT_Vector_Rotate( vec, angle ); 504 } 505 506 507 /* documentation is in fttrigon.h */ 508 509 FT_EXPORT_DEF( FT_Angle ) 510 FT_Angle_Diff( FT_Angle angle1, 511 FT_Angle angle2 ) 512 { 513 FT_Angle delta = angle2 - angle1; 514 515 516 while ( delta <= -FT_ANGLE_PI ) 517 delta += FT_ANGLE_2PI; 518 519 while ( delta > FT_ANGLE_PI ) 520 delta -= FT_ANGLE_2PI; 521 522 return delta; 523 } 524 525 526 /* END */ 527
