1 /***************************************************************************/ 2 /* */ 3 /* fttrigon.c */ 4 /* */ 5 /* FreeType trigonometric functions (body). */ 6 /* */ 7 /* Copyright 2001-2015 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)( ( (FT_Int64)val * FT_TRIG_SCALE + 0x40000000UL ) >> 32 ); 75 76 return s < 0 ? -val : val; 77 } 78 79 #else /* !FT_LONG64 */ 80 81 /* multiply a given value by the CORDIC shrink factor */ 82 static FT_Fixed 83 ft_trig_downscale( FT_Fixed val ) 84 { 85 FT_Int s = 1; 86 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2; 87 88 89 if ( val < 0 ) 90 { 91 val = -val; 92 s = -1; 93 } 94 95 lo1 = (FT_UInt32)val & 0x0000FFFFU; 96 hi1 = (FT_UInt32)val >> 16; 97 lo2 = FT_TRIG_SCALE & 0x0000FFFFU; 98 hi2 = FT_TRIG_SCALE >> 16; 99 100 lo = lo1 * lo2; 101 i1 = lo1 * hi2; 102 i2 = lo2 * hi1; 103 hi = hi1 * hi2; 104 105 /* Check carry overflow of i1 + i2 */ 106 i1 += i2; 107 hi += (FT_UInt32)( i1 < i2 ) << 16; 108 109 hi += i1 >> 16; 110 i1 = i1 << 16; 111 112 /* Check carry overflow of i1 + lo */ 113 lo += i1; 114 hi += ( lo < i1 ); 115 116 /* 0x40000000 comes from regression analysis between true */ 117 /* and CORDIC hypotenuse, so it minimizes the error */ 118 119 /* Check carry overflow of lo + 0x40000000 */ 120 lo += 0x40000000UL; 121 hi += ( lo < 0x40000000UL ); 122 123 val = (FT_Fixed)hi; 124 125 return s < 0 ? -val : val; 126 } 127 128 #endif /* !FT_LONG64 */ 129 130 131 /* undefined and never called for zero vector */ 132 static FT_Int 133 ft_trig_prenorm( FT_Vector* vec ) 134 { 135 FT_Pos x, y; 136 FT_Int shift; 137 138 139 x = vec->x; 140 y = vec->y; 141 142 shift = FT_MSB( (FT_UInt32)( FT_ABS( x ) | FT_ABS( y ) ) ); 143 144 if ( shift <= FT_TRIG_SAFE_MSB ) 145 { 146 shift = FT_TRIG_SAFE_MSB - shift; 147 vec->x = (FT_Pos)( (FT_ULong)x << shift ); 148 vec->y = (FT_Pos)( (FT_ULong)y << shift ); 149 } 150 else 151 { 152 shift -= FT_TRIG_SAFE_MSB; 153 vec->x = x >> shift; 154 vec->y = y >> shift; 155 shift = -shift; 156 } 157 158 return shift; 159 } 160 161 162 static void 163 ft_trig_pseudo_rotate( FT_Vector* vec, 164 FT_Angle theta ) 165 { 166 FT_Int i; 167 FT_Fixed x, y, xtemp, b; 168 const FT_Angle *arctanptr; 169 170 171 x = vec->x; 172 y = vec->y; 173 174 /* Rotate inside [-PI/4,PI/4] sector */ 175 while ( theta < -FT_ANGLE_PI4 ) 176 { 177 xtemp = y; 178 y = -x; 179 x = xtemp; 180 theta += FT_ANGLE_PI2; 181 } 182 183 while ( theta > FT_ANGLE_PI4 ) 184 { 185 xtemp = -y; 186 y = x; 187 x = xtemp; 188 theta -= FT_ANGLE_PI2; 189 } 190 191 arctanptr = ft_trig_arctan_table; 192 193 /* Pseudorotations, with right shifts */ 194 for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ ) 195 { 196 if ( theta < 0 ) 197 { 198 xtemp = x + ( ( y + b ) >> i ); 199 y = y - ( ( x + b ) >> i ); 200 x = xtemp; 201 theta += *arctanptr++; 202 } 203 else 204 { 205 xtemp = x - ( ( y + b ) >> i ); 206 y = y + ( ( x + b ) >> i ); 207 x = xtemp; 208 theta -= *arctanptr++; 209 } 210 } 211 212 vec->x = x; 213 vec->y = y; 214 } 215 216 217 static void 218 ft_trig_pseudo_polarize( FT_Vector* vec ) 219 { 220 FT_Angle theta; 221 FT_Int i; 222 FT_Fixed x, y, xtemp, b; 223 const FT_Angle *arctanptr; 224 225 226 x = vec->x; 227 y = vec->y; 228 229 /* Get the vector into [-PI/4,PI/4] sector */ 230 if ( y > x ) 231 { 232 if ( y > -x ) 233 { 234 theta = FT_ANGLE_PI2; 235 xtemp = y; 236 y = -x; 237 x = xtemp; 238 } 239 else 240 { 241 theta = y > 0 ? FT_ANGLE_PI : -FT_ANGLE_PI; 242 x = -x; 243 y = -y; 244 } 245 } 246 else 247 { 248 if ( y < -x ) 249 { 250 theta = -FT_ANGLE_PI2; 251 xtemp = -y; 252 y = x; 253 x = xtemp; 254 } 255 else 256 { 257 theta = 0; 258 } 259 } 260 261 arctanptr = ft_trig_arctan_table; 262 263 /* Pseudorotations, with right shifts */ 264 for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ ) 265 { 266 if ( y > 0 ) 267 { 268 xtemp = x + ( ( y + b ) >> i ); 269 y = y - ( ( x + b ) >> i ); 270 x = xtemp; 271 theta += *arctanptr++; 272 } 273 else 274 { 275 xtemp = x - ( ( y + b ) >> i ); 276 y = y + ( ( x + b ) >> i ); 277 x = xtemp; 278 theta -= *arctanptr++; 279 } 280 } 281 282 /* round theta to acknowledge its error that mostly comes */ 283 /* from accumulated rounding errors in the arctan table */ 284 if ( theta >= 0 ) 285 theta = FT_PAD_ROUND( theta, 16 ); 286 else 287 theta = -FT_PAD_ROUND( -theta, 16 ); 288 289 vec->x = x; 290 vec->y = theta; 291 } 292 293 294 /* documentation is in fttrigon.h */ 295 296 FT_EXPORT_DEF( FT_Fixed ) 297 FT_Cos( FT_Angle angle ) 298 { 299 FT_Vector v; 300 301 302 FT_Vector_Unit( &v, angle ); 303 304 return v.x; 305 } 306 307 308 /* documentation is in fttrigon.h */ 309 310 FT_EXPORT_DEF( FT_Fixed ) 311 FT_Sin( FT_Angle angle ) 312 { 313 FT_Vector v; 314 315 316 FT_Vector_Unit( &v, angle ); 317 318 return v.y; 319 } 320 321 322 /* documentation is in fttrigon.h */ 323 324 FT_EXPORT_DEF( FT_Fixed ) 325 FT_Tan( FT_Angle angle ) 326 { 327 FT_Vector v; 328 329 330 FT_Vector_Unit( &v, angle ); 331 332 return FT_DivFix( v.y, v.x ); 333 } 334 335 336 /* documentation is in fttrigon.h */ 337 338 FT_EXPORT_DEF( FT_Angle ) 339 FT_Atan2( FT_Fixed dx, 340 FT_Fixed dy ) 341 { 342 FT_Vector v; 343 344 345 if ( dx == 0 && dy == 0 ) 346 return 0; 347 348 v.x = dx; 349 v.y = dy; 350 ft_trig_prenorm( &v ); 351 ft_trig_pseudo_polarize( &v ); 352 353 return v.y; 354 } 355 356 357 /* documentation is in fttrigon.h */ 358 359 FT_EXPORT_DEF( void ) 360 FT_Vector_Unit( FT_Vector* vec, 361 FT_Angle angle ) 362 { 363 if ( !vec ) 364 return; 365 366 vec->x = FT_TRIG_SCALE >> 8; 367 vec->y = 0; 368 ft_trig_pseudo_rotate( vec, angle ); 369 vec->x = ( vec->x + 0x80L ) >> 8; 370 vec->y = ( vec->y + 0x80L ) >> 8; 371 } 372 373 374 /* these macros return 0 for positive numbers, 375 and -1 for negative ones */ 376 #define FT_SIGN_LONG( x ) ( (x) >> ( FT_SIZEOF_LONG * 8 - 1 ) ) 377 #define FT_SIGN_INT( x ) ( (x) >> ( FT_SIZEOF_INT * 8 - 1 ) ) 378 #define FT_SIGN_INT32( x ) ( (x) >> 31 ) 379 #define FT_SIGN_INT16( x ) ( (x) >> 15 ) 380 381 382 /* documentation is in fttrigon.h */ 383 384 FT_EXPORT_DEF( void ) 385 FT_Vector_Rotate( FT_Vector* vec, 386 FT_Angle angle ) 387 { 388 FT_Int shift; 389 FT_Vector v; 390 391 392 if ( !vec || !angle ) 393 return; 394 395 v = *vec; 396 397 if ( v.x == 0 && v.y == 0 ) 398 return; 399 400 shift = ft_trig_prenorm( &v ); 401 ft_trig_pseudo_rotate( &v, angle ); 402 v.x = ft_trig_downscale( v.x ); 403 v.y = ft_trig_downscale( v.y ); 404 405 if ( shift > 0 ) 406 { 407 FT_Int32 half = (FT_Int32)1L << ( shift - 1 ); 408 409 410 vec->x = ( v.x + half + FT_SIGN_LONG( v.x ) ) >> shift; 411 vec->y = ( v.y + half + FT_SIGN_LONG( v.y ) ) >> shift; 412 } 413 else 414 { 415 shift = -shift; 416 vec->x = (FT_Pos)( (FT_ULong)v.x << shift ); 417 vec->y = (FT_Pos)( (FT_ULong)v.y << shift ); 418 } 419 } 420 421 422 /* documentation is in fttrigon.h */ 423 424 FT_EXPORT_DEF( FT_Fixed ) 425 FT_Vector_Length( FT_Vector* vec ) 426 { 427 FT_Int shift; 428 FT_Vector v; 429 430 431 if ( !vec ) 432 return 0; 433 434 v = *vec; 435 436 /* handle trivial cases */ 437 if ( v.x == 0 ) 438 { 439 return FT_ABS( v.y ); 440 } 441 else if ( v.y == 0 ) 442 { 443 return FT_ABS( v.x ); 444 } 445 446 /* general case */ 447 shift = ft_trig_prenorm( &v ); 448 ft_trig_pseudo_polarize( &v ); 449 450 v.x = ft_trig_downscale( v.x ); 451 452 if ( shift > 0 ) 453 return ( v.x + ( 1L << ( shift - 1 ) ) ) >> shift; 454 455 return (FT_Fixed)( (FT_UInt32)v.x << -shift ); 456 } 457 458 459 /* documentation is in fttrigon.h */ 460 461 FT_EXPORT_DEF( void ) 462 FT_Vector_Polarize( FT_Vector* vec, 463 FT_Fixed *length, 464 FT_Angle *angle ) 465 { 466 FT_Int shift; 467 FT_Vector v; 468 469 470 if ( !vec || !length || !angle ) 471 return; 472 473 v = *vec; 474 475 if ( v.x == 0 && v.y == 0 ) 476 return; 477 478 shift = ft_trig_prenorm( &v ); 479 ft_trig_pseudo_polarize( &v ); 480 481 v.x = ft_trig_downscale( v.x ); 482 483 *length = shift >= 0 ? ( v.x >> shift ) 484 : (FT_Fixed)( (FT_UInt32)v.x << -shift ); 485 *angle = v.y; 486 } 487 488 489 /* documentation is in fttrigon.h */ 490 491 FT_EXPORT_DEF( void ) 492 FT_Vector_From_Polar( FT_Vector* vec, 493 FT_Fixed length, 494 FT_Angle angle ) 495 { 496 if ( !vec ) 497 return; 498 499 vec->x = length; 500 vec->y = 0; 501 502 FT_Vector_Rotate( vec, angle ); 503 } 504 505 506 /* documentation is in fttrigon.h */ 507 508 FT_EXPORT_DEF( FT_Angle ) 509 FT_Angle_Diff( FT_Angle angle1, 510 FT_Angle angle2 ) 511 { 512 FT_Angle delta = angle2 - angle1; 513 514 515 while ( delta <= -FT_ANGLE_PI ) 516 delta += FT_ANGLE_2PI; 517 518 while ( delta > FT_ANGLE_PI ) 519 delta -= FT_ANGLE_2PI; 520 521 return delta; 522 } 523 524 525 /* END */ 526
