1 /**************************************************************************** 2 * 3 * ftcalc.c 4 * 5 * Arithmetic computations (body). 6 * 7 * Copyright (C) 1996-2020 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 * Support for 1-complement arithmetic has been totally dropped in this 21 * release. You can still write your own code if you need it. 22 * 23 */ 24 25 /************************************************************************** 26 * 27 * Implementing basic computation routines. 28 * 29 * FT_MulDiv(), FT_MulFix(), FT_DivFix(), FT_RoundFix(), FT_CeilFix(), 30 * and FT_FloorFix() are declared in freetype.h. 31 * 32 */ 33 34 35 #include <freetype/ftglyph.h> 36 #include <freetype/fttrigon.h> 37 #include <freetype/internal/ftcalc.h> 38 #include <freetype/internal/ftdebug.h> 39 #include <freetype/internal/ftobjs.h> 40 41 42 #ifdef FT_MULFIX_ASSEMBLER 43 #undef FT_MulFix 44 #endif 45 46 /* we need to emulate a 64-bit data type if a real one isn't available */ 47 48 #ifndef FT_LONG64 49 50 typedef struct FT_Int64_ 51 { 52 FT_UInt32 lo; 53 FT_UInt32 hi; 54 55 } FT_Int64; 56 57 #endif /* !FT_LONG64 */ 58 59 60 /************************************************************************** 61 * 62 * The macro FT_COMPONENT is used in trace mode. It is an implicit 63 * parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log 64 * messages during execution. 65 */ 66 #undef FT_COMPONENT 67 #define FT_COMPONENT calc 68 69 70 /* transfer sign, leaving a positive number; */ 71 /* we need an unsigned value to safely negate INT_MIN (or LONG_MIN) */ 72 #define FT_MOVE_SIGN( x, x_unsigned, s ) \ 73 FT_BEGIN_STMNT \ 74 if ( x < 0 ) \ 75 { \ 76 x_unsigned = 0U - (x_unsigned); \ 77 s = -s; \ 78 } \ 79 FT_END_STMNT 80 81 /* The following three functions are available regardless of whether */ 82 /* FT_LONG64 is defined. */ 83 84 /* documentation is in freetype.h */ 85 86 FT_EXPORT_DEF( FT_Fixed ) FT_RoundFix(FT_Fixed a)87 FT_RoundFix( FT_Fixed a ) 88 { 89 return ( ADD_LONG( a, 0x8000L - ( a < 0 ) ) ) & ~0xFFFFL; 90 } 91 92 93 /* documentation is in freetype.h */ 94 95 FT_EXPORT_DEF( FT_Fixed ) FT_CeilFix(FT_Fixed a)96 FT_CeilFix( FT_Fixed a ) 97 { 98 return ( ADD_LONG( a, 0xFFFFL ) ) & ~0xFFFFL; 99 } 100 101 102 /* documentation is in freetype.h */ 103 104 FT_EXPORT_DEF( FT_Fixed ) FT_FloorFix(FT_Fixed a)105 FT_FloorFix( FT_Fixed a ) 106 { 107 return a & ~0xFFFFL; 108 } 109 110 #ifndef FT_MSB 111 112 FT_BASE_DEF ( FT_Int ) FT_MSB(FT_UInt32 z)113 FT_MSB( FT_UInt32 z ) 114 { 115 FT_Int shift = 0; 116 117 118 /* determine msb bit index in `shift' */ 119 if ( z & 0xFFFF0000UL ) 120 { 121 z >>= 16; 122 shift += 16; 123 } 124 if ( z & 0x0000FF00UL ) 125 { 126 z >>= 8; 127 shift += 8; 128 } 129 if ( z & 0x000000F0UL ) 130 { 131 z >>= 4; 132 shift += 4; 133 } 134 if ( z & 0x0000000CUL ) 135 { 136 z >>= 2; 137 shift += 2; 138 } 139 if ( z & 0x00000002UL ) 140 { 141 /* z >>= 1; */ 142 shift += 1; 143 } 144 145 return shift; 146 } 147 148 #endif /* !FT_MSB */ 149 150 151 /* documentation is in ftcalc.h */ 152 153 FT_BASE_DEF( FT_Fixed ) FT_Hypot(FT_Fixed x,FT_Fixed y)154 FT_Hypot( FT_Fixed x, 155 FT_Fixed y ) 156 { 157 FT_Vector v; 158 159 160 v.x = x; 161 v.y = y; 162 163 return FT_Vector_Length( &v ); 164 } 165 166 167 #ifdef FT_LONG64 168 169 170 /* documentation is in freetype.h */ 171 172 FT_EXPORT_DEF( FT_Long ) FT_MulDiv(FT_Long a_,FT_Long b_,FT_Long c_)173 FT_MulDiv( FT_Long a_, 174 FT_Long b_, 175 FT_Long c_ ) 176 { 177 FT_Int s = 1; 178 FT_UInt64 a, b, c, d; 179 FT_Long d_; 180 181 182 a = (FT_UInt64)a_; 183 b = (FT_UInt64)b_; 184 c = (FT_UInt64)c_; 185 186 FT_MOVE_SIGN( a_, a, s ); 187 FT_MOVE_SIGN( b_, b, s ); 188 FT_MOVE_SIGN( c_, c, s ); 189 190 d = c > 0 ? ( a * b + ( c >> 1 ) ) / c 191 : 0x7FFFFFFFUL; 192 193 d_ = (FT_Long)d; 194 195 return s < 0 ? NEG_LONG( d_ ) : d_; 196 } 197 198 199 /* documentation is in ftcalc.h */ 200 201 FT_BASE_DEF( FT_Long ) FT_MulDiv_No_Round(FT_Long a_,FT_Long b_,FT_Long c_)202 FT_MulDiv_No_Round( FT_Long a_, 203 FT_Long b_, 204 FT_Long c_ ) 205 { 206 FT_Int s = 1; 207 FT_UInt64 a, b, c, d; 208 FT_Long d_; 209 210 211 a = (FT_UInt64)a_; 212 b = (FT_UInt64)b_; 213 c = (FT_UInt64)c_; 214 215 FT_MOVE_SIGN( a_, a, s ); 216 FT_MOVE_SIGN( b_, b, s ); 217 FT_MOVE_SIGN( c_, c, s ); 218 219 d = c > 0 ? a * b / c 220 : 0x7FFFFFFFUL; 221 222 d_ = (FT_Long)d; 223 224 return s < 0 ? NEG_LONG( d_ ) : d_; 225 } 226 227 228 /* documentation is in freetype.h */ 229 230 FT_EXPORT_DEF( FT_Long ) FT_MulFix(FT_Long a_,FT_Long b_)231 FT_MulFix( FT_Long a_, 232 FT_Long b_ ) 233 { 234 #ifdef FT_MULFIX_ASSEMBLER 235 236 return FT_MULFIX_ASSEMBLER( (FT_Int32)a_, (FT_Int32)b_ ); 237 238 #else 239 240 FT_Int64 ab = (FT_Int64)a_ * (FT_Int64)b_; 241 242 /* this requires arithmetic right shift of signed numbers */ 243 return (FT_Long)( ( ab + 0x8000L - ( ab < 0 ) ) >> 16 ); 244 245 #endif /* FT_MULFIX_ASSEMBLER */ 246 } 247 248 249 /* documentation is in freetype.h */ 250 251 FT_EXPORT_DEF( FT_Long ) FT_DivFix(FT_Long a_,FT_Long b_)252 FT_DivFix( FT_Long a_, 253 FT_Long b_ ) 254 { 255 FT_Int s = 1; 256 FT_UInt64 a, b, q; 257 FT_Long q_; 258 259 260 a = (FT_UInt64)a_; 261 b = (FT_UInt64)b_; 262 263 FT_MOVE_SIGN( a_, a, s ); 264 FT_MOVE_SIGN( b_, b, s ); 265 266 q = b > 0 ? ( ( a << 16 ) + ( b >> 1 ) ) / b 267 : 0x7FFFFFFFUL; 268 269 q_ = (FT_Long)q; 270 271 return s < 0 ? NEG_LONG( q_ ) : q_; 272 } 273 274 275 #else /* !FT_LONG64 */ 276 277 278 static void ft_multo64(FT_UInt32 x,FT_UInt32 y,FT_Int64 * z)279 ft_multo64( FT_UInt32 x, 280 FT_UInt32 y, 281 FT_Int64 *z ) 282 { 283 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2; 284 285 286 lo1 = x & 0x0000FFFFU; hi1 = x >> 16; 287 lo2 = y & 0x0000FFFFU; hi2 = y >> 16; 288 289 lo = lo1 * lo2; 290 i1 = lo1 * hi2; 291 i2 = lo2 * hi1; 292 hi = hi1 * hi2; 293 294 /* Check carry overflow of i1 + i2 */ 295 i1 += i2; 296 hi += (FT_UInt32)( i1 < i2 ) << 16; 297 298 hi += i1 >> 16; 299 i1 = i1 << 16; 300 301 /* Check carry overflow of i1 + lo */ 302 lo += i1; 303 hi += ( lo < i1 ); 304 305 z->lo = lo; 306 z->hi = hi; 307 } 308 309 310 static FT_UInt32 ft_div64by32(FT_UInt32 hi,FT_UInt32 lo,FT_UInt32 y)311 ft_div64by32( FT_UInt32 hi, 312 FT_UInt32 lo, 313 FT_UInt32 y ) 314 { 315 FT_UInt32 r, q; 316 FT_Int i; 317 318 319 if ( hi >= y ) 320 return (FT_UInt32)0x7FFFFFFFL; 321 322 /* We shift as many bits as we can into the high register, perform */ 323 /* 32-bit division with modulo there, then work through the remaining */ 324 /* bits with long division. This optimization is especially noticeable */ 325 /* for smaller dividends that barely use the high register. */ 326 327 i = 31 - FT_MSB( hi ); 328 r = ( hi << i ) | ( lo >> ( 32 - i ) ); lo <<= i; /* left 64-bit shift */ 329 q = r / y; 330 r -= q * y; /* remainder */ 331 332 i = 32 - i; /* bits remaining in low register */ 333 do 334 { 335 q <<= 1; 336 r = ( r << 1 ) | ( lo >> 31 ); lo <<= 1; 337 338 if ( r >= y ) 339 { 340 r -= y; 341 q |= 1; 342 } 343 } while ( --i ); 344 345 return q; 346 } 347 348 349 static void FT_Add64(FT_Int64 * x,FT_Int64 * y,FT_Int64 * z)350 FT_Add64( FT_Int64* x, 351 FT_Int64* y, 352 FT_Int64 *z ) 353 { 354 FT_UInt32 lo, hi; 355 356 357 lo = x->lo + y->lo; 358 hi = x->hi + y->hi + ( lo < x->lo ); 359 360 z->lo = lo; 361 z->hi = hi; 362 } 363 364 365 /* The FT_MulDiv function has been optimized thanks to ideas from */ 366 /* Graham Asher and Alexei Podtelezhnikov. The trick is to optimize */ 367 /* a rather common case when everything fits within 32-bits. */ 368 /* */ 369 /* We compute 'a*b+c/2', then divide it by 'c' (all positive values). */ 370 /* */ 371 /* The product of two positive numbers never exceeds the square of */ 372 /* its mean values. Therefore, we always avoid the overflow by */ 373 /* imposing */ 374 /* */ 375 /* (a + b) / 2 <= sqrt(X - c/2) , */ 376 /* */ 377 /* where X = 2^32 - 1, the maximum unsigned 32-bit value, and using */ 378 /* unsigned arithmetic. Now we replace `sqrt' with a linear function */ 379 /* that is smaller or equal for all values of c in the interval */ 380 /* [0;X/2]; it should be equal to sqrt(X) and sqrt(3X/4) at the */ 381 /* endpoints. Substituting the linear solution and explicit numbers */ 382 /* we get */ 383 /* */ 384 /* a + b <= 131071.99 - c / 122291.84 . */ 385 /* */ 386 /* In practice, we should use a faster and even stronger inequality */ 387 /* */ 388 /* a + b <= 131071 - (c >> 16) */ 389 /* */ 390 /* or, alternatively, */ 391 /* */ 392 /* a + b <= 129894 - (c >> 17) . */ 393 /* */ 394 /* FT_MulFix, on the other hand, is optimized for a small value of */ 395 /* the first argument, when the second argument can be much larger. */ 396 /* This can be achieved by scaling the second argument and the limit */ 397 /* in the above inequalities. For example, */ 398 /* */ 399 /* a + (b >> 8) <= (131071 >> 4) */ 400 /* */ 401 /* covers the practical range of use. The actual test below is a bit */ 402 /* tighter to avoid the border case overflows. */ 403 /* */ 404 /* In the case of FT_DivFix, the exact overflow check */ 405 /* */ 406 /* a << 16 <= X - c/2 */ 407 /* */ 408 /* is scaled down by 2^16 and we use */ 409 /* */ 410 /* a <= 65535 - (c >> 17) . */ 411 412 /* documentation is in freetype.h */ 413 414 FT_EXPORT_DEF( FT_Long ) FT_MulDiv(FT_Long a_,FT_Long b_,FT_Long c_)415 FT_MulDiv( FT_Long a_, 416 FT_Long b_, 417 FT_Long c_ ) 418 { 419 FT_Int s = 1; 420 FT_UInt32 a, b, c; 421 422 423 /* XXX: this function does not allow 64-bit arguments */ 424 425 a = (FT_UInt32)a_; 426 b = (FT_UInt32)b_; 427 c = (FT_UInt32)c_; 428 429 FT_MOVE_SIGN( a_, a, s ); 430 FT_MOVE_SIGN( b_, b, s ); 431 FT_MOVE_SIGN( c_, c, s ); 432 433 if ( c == 0 ) 434 a = 0x7FFFFFFFUL; 435 436 else if ( a + b <= 129894UL - ( c >> 17 ) ) 437 a = ( a * b + ( c >> 1 ) ) / c; 438 439 else 440 { 441 FT_Int64 temp, temp2; 442 443 444 ft_multo64( a, b, &temp ); 445 446 temp2.hi = 0; 447 temp2.lo = c >> 1; 448 449 FT_Add64( &temp, &temp2, &temp ); 450 451 /* last attempt to ditch long division */ 452 a = ( temp.hi == 0 ) ? temp.lo / c 453 : ft_div64by32( temp.hi, temp.lo, c ); 454 } 455 456 a_ = (FT_Long)a; 457 458 return s < 0 ? NEG_LONG( a_ ) : a_; 459 } 460 461 462 FT_BASE_DEF( FT_Long ) FT_MulDiv_No_Round(FT_Long a_,FT_Long b_,FT_Long c_)463 FT_MulDiv_No_Round( FT_Long a_, 464 FT_Long b_, 465 FT_Long c_ ) 466 { 467 FT_Int s = 1; 468 FT_UInt32 a, b, c; 469 470 471 /* XXX: this function does not allow 64-bit arguments */ 472 473 a = (FT_UInt32)a_; 474 b = (FT_UInt32)b_; 475 c = (FT_UInt32)c_; 476 477 FT_MOVE_SIGN( a_, a, s ); 478 FT_MOVE_SIGN( b_, b, s ); 479 FT_MOVE_SIGN( c_, c, s ); 480 481 if ( c == 0 ) 482 a = 0x7FFFFFFFUL; 483 484 else if ( a + b <= 131071UL ) 485 a = a * b / c; 486 487 else 488 { 489 FT_Int64 temp; 490 491 492 ft_multo64( a, b, &temp ); 493 494 /* last attempt to ditch long division */ 495 a = ( temp.hi == 0 ) ? temp.lo / c 496 : ft_div64by32( temp.hi, temp.lo, c ); 497 } 498 499 a_ = (FT_Long)a; 500 501 return s < 0 ? NEG_LONG( a_ ) : a_; 502 } 503 504 505 /* documentation is in freetype.h */ 506 507 FT_EXPORT_DEF( FT_Long ) FT_MulFix(FT_Long a_,FT_Long b_)508 FT_MulFix( FT_Long a_, 509 FT_Long b_ ) 510 { 511 #ifdef FT_MULFIX_ASSEMBLER 512 513 return FT_MULFIX_ASSEMBLER( a_, b_ ); 514 515 #elif 0 516 517 /* 518 * This code is nonportable. See comment below. 519 * 520 * However, on a platform where right-shift of a signed quantity fills 521 * the leftmost bits by copying the sign bit, it might be faster. 522 */ 523 524 FT_Long sa, sb; 525 FT_UInt32 a, b; 526 527 528 /* 529 * This is a clever way of converting a signed number `a' into its 530 * absolute value (stored back into `a') and its sign. The sign is 531 * stored in `sa'; 0 means `a' was positive or zero, and -1 means `a' 532 * was negative. (Similarly for `b' and `sb'). 533 * 534 * Unfortunately, it doesn't work (at least not portably). 535 * 536 * It makes the assumption that right-shift on a negative signed value 537 * fills the leftmost bits by copying the sign bit. This is wrong. 538 * According to K&R 2nd ed, section `A7.8 Shift Operators' on page 206, 539 * the result of right-shift of a negative signed value is 540 * implementation-defined. At least one implementation fills the 541 * leftmost bits with 0s (i.e., it is exactly the same as an unsigned 542 * right shift). This means that when `a' is negative, `sa' ends up 543 * with the value 1 rather than -1. After that, everything else goes 544 * wrong. 545 */ 546 sa = ( a_ >> ( sizeof ( a_ ) * 8 - 1 ) ); 547 a = ( a_ ^ sa ) - sa; 548 sb = ( b_ >> ( sizeof ( b_ ) * 8 - 1 ) ); 549 b = ( b_ ^ sb ) - sb; 550 551 a = (FT_UInt32)a_; 552 b = (FT_UInt32)b_; 553 554 if ( a + ( b >> 8 ) <= 8190UL ) 555 a = ( a * b + 0x8000U ) >> 16; 556 else 557 { 558 FT_UInt32 al = a & 0xFFFFUL; 559 560 561 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 562 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 563 } 564 565 sa ^= sb; 566 a = ( a ^ sa ) - sa; 567 568 return (FT_Long)a; 569 570 #else /* 0 */ 571 572 FT_Int s = 1; 573 FT_UInt32 a, b; 574 575 576 /* XXX: this function does not allow 64-bit arguments */ 577 578 a = (FT_UInt32)a_; 579 b = (FT_UInt32)b_; 580 581 FT_MOVE_SIGN( a_, a, s ); 582 FT_MOVE_SIGN( b_, b, s ); 583 584 if ( a + ( b >> 8 ) <= 8190UL ) 585 a = ( a * b + 0x8000UL ) >> 16; 586 else 587 { 588 FT_UInt32 al = a & 0xFFFFUL; 589 590 591 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 592 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 593 } 594 595 a_ = (FT_Long)a; 596 597 return s < 0 ? NEG_LONG( a_ ) : a_; 598 599 #endif /* 0 */ 600 601 } 602 603 604 /* documentation is in freetype.h */ 605 606 FT_EXPORT_DEF( FT_Long ) FT_DivFix(FT_Long a_,FT_Long b_)607 FT_DivFix( FT_Long a_, 608 FT_Long b_ ) 609 { 610 FT_Int s = 1; 611 FT_UInt32 a, b, q; 612 FT_Long q_; 613 614 615 /* XXX: this function does not allow 64-bit arguments */ 616 617 a = (FT_UInt32)a_; 618 b = (FT_UInt32)b_; 619 620 FT_MOVE_SIGN( a_, a, s ); 621 FT_MOVE_SIGN( b_, b, s ); 622 623 if ( b == 0 ) 624 { 625 /* check for division by 0 */ 626 q = 0x7FFFFFFFUL; 627 } 628 else if ( a <= 65535UL - ( b >> 17 ) ) 629 { 630 /* compute result directly */ 631 q = ( ( a << 16 ) + ( b >> 1 ) ) / b; 632 } 633 else 634 { 635 /* we need more bits; we have to do it by hand */ 636 FT_Int64 temp, temp2; 637 638 639 temp.hi = a >> 16; 640 temp.lo = a << 16; 641 temp2.hi = 0; 642 temp2.lo = b >> 1; 643 644 FT_Add64( &temp, &temp2, &temp ); 645 q = ft_div64by32( temp.hi, temp.lo, b ); 646 } 647 648 q_ = (FT_Long)q; 649 650 return s < 0 ? NEG_LONG( q_ ) : q_; 651 } 652 653 654 #endif /* !FT_LONG64 */ 655 656 657 /* documentation is in ftglyph.h */ 658 659 FT_EXPORT_DEF( void ) FT_Matrix_Multiply(const FT_Matrix * a,FT_Matrix * b)660 FT_Matrix_Multiply( const FT_Matrix* a, 661 FT_Matrix *b ) 662 { 663 FT_Fixed xx, xy, yx, yy; 664 665 666 if ( !a || !b ) 667 return; 668 669 xx = ADD_LONG( FT_MulFix( a->xx, b->xx ), 670 FT_MulFix( a->xy, b->yx ) ); 671 xy = ADD_LONG( FT_MulFix( a->xx, b->xy ), 672 FT_MulFix( a->xy, b->yy ) ); 673 yx = ADD_LONG( FT_MulFix( a->yx, b->xx ), 674 FT_MulFix( a->yy, b->yx ) ); 675 yy = ADD_LONG( FT_MulFix( a->yx, b->xy ), 676 FT_MulFix( a->yy, b->yy ) ); 677 678 b->xx = xx; 679 b->xy = xy; 680 b->yx = yx; 681 b->yy = yy; 682 } 683 684 685 /* documentation is in ftglyph.h */ 686 687 FT_EXPORT_DEF( FT_Error ) FT_Matrix_Invert(FT_Matrix * matrix)688 FT_Matrix_Invert( FT_Matrix* matrix ) 689 { 690 FT_Pos delta, xx, yy; 691 692 693 if ( !matrix ) 694 return FT_THROW( Invalid_Argument ); 695 696 /* compute discriminant */ 697 delta = FT_MulFix( matrix->xx, matrix->yy ) - 698 FT_MulFix( matrix->xy, matrix->yx ); 699 700 if ( !delta ) 701 return FT_THROW( Invalid_Argument ); /* matrix can't be inverted */ 702 703 matrix->xy = -FT_DivFix( matrix->xy, delta ); 704 matrix->yx = -FT_DivFix( matrix->yx, delta ); 705 706 xx = matrix->xx; 707 yy = matrix->yy; 708 709 matrix->xx = FT_DivFix( yy, delta ); 710 matrix->yy = FT_DivFix( xx, delta ); 711 712 return FT_Err_Ok; 713 } 714 715 716 /* documentation is in ftcalc.h */ 717 718 FT_BASE_DEF( void ) FT_Matrix_Multiply_Scaled(const FT_Matrix * a,FT_Matrix * b,FT_Long scaling)719 FT_Matrix_Multiply_Scaled( const FT_Matrix* a, 720 FT_Matrix *b, 721 FT_Long scaling ) 722 { 723 FT_Fixed xx, xy, yx, yy; 724 725 FT_Long val = 0x10000L * scaling; 726 727 728 if ( !a || !b ) 729 return; 730 731 xx = ADD_LONG( FT_MulDiv( a->xx, b->xx, val ), 732 FT_MulDiv( a->xy, b->yx, val ) ); 733 xy = ADD_LONG( FT_MulDiv( a->xx, b->xy, val ), 734 FT_MulDiv( a->xy, b->yy, val ) ); 735 yx = ADD_LONG( FT_MulDiv( a->yx, b->xx, val ), 736 FT_MulDiv( a->yy, b->yx, val ) ); 737 yy = ADD_LONG( FT_MulDiv( a->yx, b->xy, val ), 738 FT_MulDiv( a->yy, b->yy, val ) ); 739 740 b->xx = xx; 741 b->xy = xy; 742 b->yx = yx; 743 b->yy = yy; 744 } 745 746 747 /* documentation is in ftcalc.h */ 748 749 FT_BASE_DEF( FT_Bool ) FT_Matrix_Check(const FT_Matrix * matrix)750 FT_Matrix_Check( const FT_Matrix* matrix ) 751 { 752 FT_Matrix m; 753 FT_Fixed val[4]; 754 FT_Fixed nonzero_minval, maxval; 755 FT_Fixed temp1, temp2; 756 FT_UInt i; 757 758 759 if ( !matrix ) 760 return 0; 761 762 val[0] = FT_ABS( matrix->xx ); 763 val[1] = FT_ABS( matrix->xy ); 764 val[2] = FT_ABS( matrix->yx ); 765 val[3] = FT_ABS( matrix->yy ); 766 767 /* 768 * To avoid overflow, we ensure that each value is not larger than 769 * 770 * int(sqrt(2^31 / 4)) = 23170 ; 771 * 772 * we also check that no value becomes zero if we have to scale. 773 */ 774 775 maxval = 0; 776 nonzero_minval = FT_LONG_MAX; 777 778 for ( i = 0; i < 4; i++ ) 779 { 780 if ( val[i] > maxval ) 781 maxval = val[i]; 782 if ( val[i] && val[i] < nonzero_minval ) 783 nonzero_minval = val[i]; 784 } 785 786 /* we only handle 32bit values */ 787 if ( maxval > 0x7FFFFFFFL ) 788 return 0; 789 790 if ( maxval > 23170 ) 791 { 792 FT_Fixed scale = FT_DivFix( maxval, 23170 ); 793 794 795 if ( !FT_DivFix( nonzero_minval, scale ) ) 796 return 0; /* value range too large */ 797 798 m.xx = FT_DivFix( matrix->xx, scale ); 799 m.xy = FT_DivFix( matrix->xy, scale ); 800 m.yx = FT_DivFix( matrix->yx, scale ); 801 m.yy = FT_DivFix( matrix->yy, scale ); 802 } 803 else 804 m = *matrix; 805 806 temp1 = FT_ABS( m.xx * m.yy - m.xy * m.yx ); 807 temp2 = m.xx * m.xx + m.xy * m.xy + m.yx * m.yx + m.yy * m.yy; 808 809 if ( temp1 == 0 || 810 temp2 / temp1 > 50 ) 811 return 0; 812 813 return 1; 814 } 815 816 817 /* documentation is in ftcalc.h */ 818 819 FT_BASE_DEF( void ) FT_Vector_Transform_Scaled(FT_Vector * vector,const FT_Matrix * matrix,FT_Long scaling)820 FT_Vector_Transform_Scaled( FT_Vector* vector, 821 const FT_Matrix* matrix, 822 FT_Long scaling ) 823 { 824 FT_Pos xz, yz; 825 826 FT_Long val = 0x10000L * scaling; 827 828 829 if ( !vector || !matrix ) 830 return; 831 832 xz = ADD_LONG( FT_MulDiv( vector->x, matrix->xx, val ), 833 FT_MulDiv( vector->y, matrix->xy, val ) ); 834 yz = ADD_LONG( FT_MulDiv( vector->x, matrix->yx, val ), 835 FT_MulDiv( vector->y, matrix->yy, val ) ); 836 837 vector->x = xz; 838 vector->y = yz; 839 } 840 841 842 /* documentation is in ftcalc.h */ 843 844 FT_BASE_DEF( FT_UInt32 ) FT_Vector_NormLen(FT_Vector * vector)845 FT_Vector_NormLen( FT_Vector* vector ) 846 { 847 FT_Int32 x_ = vector->x; 848 FT_Int32 y_ = vector->y; 849 FT_Int32 b, z; 850 FT_UInt32 x, y, u, v, l; 851 FT_Int sx = 1, sy = 1, shift; 852 853 854 x = (FT_UInt32)x_; 855 y = (FT_UInt32)y_; 856 857 FT_MOVE_SIGN( x_, x, sx ); 858 FT_MOVE_SIGN( y_, y, sy ); 859 860 /* trivial cases */ 861 if ( x == 0 ) 862 { 863 if ( y > 0 ) 864 vector->y = sy * 0x10000; 865 return y; 866 } 867 else if ( y == 0 ) 868 { 869 if ( x > 0 ) 870 vector->x = sx * 0x10000; 871 return x; 872 } 873 874 /* Estimate length and prenormalize by shifting so that */ 875 /* the new approximate length is between 2/3 and 4/3. */ 876 /* The magic constant 0xAAAAAAAAUL (2/3 of 2^32) helps */ 877 /* achieve this in 16.16 fixed-point representation. */ 878 l = x > y ? x + ( y >> 1 ) 879 : y + ( x >> 1 ); 880 881 shift = 31 - FT_MSB( l ); 882 shift -= 15 + ( l >= ( 0xAAAAAAAAUL >> shift ) ); 883 884 if ( shift > 0 ) 885 { 886 x <<= shift; 887 y <<= shift; 888 889 /* re-estimate length for tiny vectors */ 890 l = x > y ? x + ( y >> 1 ) 891 : y + ( x >> 1 ); 892 } 893 else 894 { 895 x >>= -shift; 896 y >>= -shift; 897 l >>= -shift; 898 } 899 900 /* lower linear approximation for reciprocal length minus one */ 901 b = 0x10000 - (FT_Int32)l; 902 903 x_ = (FT_Int32)x; 904 y_ = (FT_Int32)y; 905 906 /* Newton's iterations */ 907 do 908 { 909 u = (FT_UInt32)( x_ + ( x_ * b >> 16 ) ); 910 v = (FT_UInt32)( y_ + ( y_ * b >> 16 ) ); 911 912 /* Normalized squared length in the parentheses approaches 2^32. */ 913 /* On two's complement systems, converting to signed gives the */ 914 /* difference with 2^32 even if the expression wraps around. */ 915 z = -(FT_Int32)( u * u + v * v ) / 0x200; 916 z = z * ( ( 0x10000 + b ) >> 8 ) / 0x10000; 917 918 b += z; 919 920 } while ( z > 0 ); 921 922 vector->x = sx < 0 ? -(FT_Pos)u : (FT_Pos)u; 923 vector->y = sy < 0 ? -(FT_Pos)v : (FT_Pos)v; 924 925 /* Conversion to signed helps to recover from likely wrap around */ 926 /* in calculating the prenormalized length, because it gives the */ 927 /* correct difference with 2^32 on two's complement systems. */ 928 l = (FT_UInt32)( 0x10000 + (FT_Int32)( u * x + v * y ) / 0x10000 ); 929 if ( shift > 0 ) 930 l = ( l + ( 1 << ( shift - 1 ) ) ) >> shift; 931 else 932 l <<= -shift; 933 934 return l; 935 } 936 937 938 #if 0 939 940 /* documentation is in ftcalc.h */ 941 942 FT_BASE_DEF( FT_Int32 ) 943 FT_SqrtFixed( FT_Int32 x ) 944 { 945 FT_UInt32 root, rem_hi, rem_lo, test_div; 946 FT_Int count; 947 948 949 root = 0; 950 951 if ( x > 0 ) 952 { 953 rem_hi = 0; 954 rem_lo = (FT_UInt32)x; 955 count = 24; 956 do 957 { 958 rem_hi = ( rem_hi << 2 ) | ( rem_lo >> 30 ); 959 rem_lo <<= 2; 960 root <<= 1; 961 test_div = ( root << 1 ) + 1; 962 963 if ( rem_hi >= test_div ) 964 { 965 rem_hi -= test_div; 966 root += 1; 967 } 968 } while ( --count ); 969 } 970 971 return (FT_Int32)root; 972 } 973 974 #endif /* 0 */ 975 976 977 /* documentation is in ftcalc.h */ 978 979 FT_BASE_DEF( FT_Int ) ft_corner_orientation(FT_Pos in_x,FT_Pos in_y,FT_Pos out_x,FT_Pos out_y)980 ft_corner_orientation( FT_Pos in_x, 981 FT_Pos in_y, 982 FT_Pos out_x, 983 FT_Pos out_y ) 984 { 985 /* we silently ignore overflow errors since such large values */ 986 /* lead to even more (harmless) rendering errors later on */ 987 988 #ifdef FT_LONG64 989 990 FT_Int64 delta = SUB_INT64( MUL_INT64( in_x, out_y ), 991 MUL_INT64( in_y, out_x ) ); 992 993 994 return ( delta > 0 ) - ( delta < 0 ); 995 996 #else 997 998 FT_Int result; 999 1000 1001 if ( ADD_LONG( FT_ABS( in_x ), FT_ABS( out_y ) ) <= 131071L && 1002 ADD_LONG( FT_ABS( in_y ), FT_ABS( out_x ) ) <= 131071L ) 1003 { 1004 FT_Long z1 = MUL_LONG( in_x, out_y ); 1005 FT_Long z2 = MUL_LONG( in_y, out_x ); 1006 1007 1008 if ( z1 > z2 ) 1009 result = +1; 1010 else if ( z1 < z2 ) 1011 result = -1; 1012 else 1013 result = 0; 1014 } 1015 else /* products might overflow 32 bits */ 1016 { 1017 FT_Int64 z1, z2; 1018 1019 1020 /* XXX: this function does not allow 64-bit arguments */ 1021 ft_multo64( (FT_UInt32)in_x, (FT_UInt32)out_y, &z1 ); 1022 ft_multo64( (FT_UInt32)in_y, (FT_UInt32)out_x, &z2 ); 1023 1024 if ( z1.hi > z2.hi ) 1025 result = +1; 1026 else if ( z1.hi < z2.hi ) 1027 result = -1; 1028 else if ( z1.lo > z2.lo ) 1029 result = +1; 1030 else if ( z1.lo < z2.lo ) 1031 result = -1; 1032 else 1033 result = 0; 1034 } 1035 1036 /* XXX: only the sign of return value, +1/0/-1 must be used */ 1037 return result; 1038 1039 #endif 1040 } 1041 1042 1043 /* documentation is in ftcalc.h */ 1044 1045 FT_BASE_DEF( FT_Int ) ft_corner_is_flat(FT_Pos in_x,FT_Pos in_y,FT_Pos out_x,FT_Pos out_y)1046 ft_corner_is_flat( FT_Pos in_x, 1047 FT_Pos in_y, 1048 FT_Pos out_x, 1049 FT_Pos out_y ) 1050 { 1051 FT_Pos ax = in_x + out_x; 1052 FT_Pos ay = in_y + out_y; 1053 1054 FT_Pos d_in, d_out, d_hypot; 1055 1056 1057 /* The idea of this function is to compare the length of the */ 1058 /* hypotenuse with the `in' and `out' length. The `corner' */ 1059 /* represented by `in' and `out' is flat if the hypotenuse's */ 1060 /* length isn't too large. */ 1061 /* */ 1062 /* This approach has the advantage that the angle between */ 1063 /* `in' and `out' is not checked. In case one of the two */ 1064 /* vectors is `dominant', this is, much larger than the */ 1065 /* other vector, we thus always have a flat corner. */ 1066 /* */ 1067 /* hypotenuse */ 1068 /* x---------------------------x */ 1069 /* \ / */ 1070 /* \ / */ 1071 /* in \ / out */ 1072 /* \ / */ 1073 /* o */ 1074 /* Point */ 1075 1076 d_in = FT_HYPOT( in_x, in_y ); 1077 d_out = FT_HYPOT( out_x, out_y ); 1078 d_hypot = FT_HYPOT( ax, ay ); 1079 1080 /* now do a simple length comparison: */ 1081 /* */ 1082 /* d_in + d_out < 17/16 d_hypot */ 1083 1084 return ( d_in + d_out - d_hypot ) < ( d_hypot >> 4 ); 1085 } 1086 1087 1088 /* END */ 1089