1 /**************************************************************************** 2 * 3 * ftcalc.c 4 * 5 * Arithmetic computations (body). 6 * 7 * Copyright 1996-2018 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 * 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 <ft2build.h> 36 #include FT_GLYPH_H 37 #include FT_TRIGONOMETRY_H 38 #include FT_INTERNAL_CALC_H 39 #include FT_INTERNAL_DEBUG_H 40 #include FT_INTERNAL_OBJECTS_H 41 42 43 #ifdef FT_MULFIX_ASSEMBLER 44 #undef FT_MulFix 45 #endif 46 47 /* we need to emulate a 64-bit data type if a real one isn't available */ 48 49 #ifndef FT_LONG64 50 51 typedef struct FT_Int64_ 52 { 53 FT_UInt32 lo; 54 FT_UInt32 hi; 55 56 } FT_Int64; 57 58 #endif /* !FT_LONG64 */ 59 60 61 /************************************************************************** 62 * 63 * The macro FT_COMPONENT is used in trace mode. It is an implicit 64 * parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log 65 * messages during execution. 66 */ 67 #undef FT_COMPONENT 68 #define FT_COMPONENT trace_calc 69 70 71 /* transfer sign, leaving a positive number; */ 72 /* we need an unsigned value to safely negate INT_MIN (or LONG_MIN) */ 73 #define FT_MOVE_SIGN( x, x_unsigned, s ) \ 74 FT_BEGIN_STMNT \ 75 if ( x < 0 ) \ 76 { \ 77 x_unsigned = 0U - (x_unsigned); \ 78 s = -s; \ 79 } \ 80 FT_END_STMNT 81 82 /* The following three functions are available regardless of whether */ 83 /* FT_LONG64 is defined. */ 84 85 /* documentation is in freetype.h */ 86 87 FT_EXPORT_DEF( FT_Fixed ) FT_RoundFix(FT_Fixed a)88 FT_RoundFix( FT_Fixed a ) 89 { 90 return ( ADD_LONG( a, 0x8000L - ( a < 0 ) ) ) & ~0xFFFFL; 91 } 92 93 94 /* documentation is in freetype.h */ 95 96 FT_EXPORT_DEF( FT_Fixed ) FT_CeilFix(FT_Fixed a)97 FT_CeilFix( FT_Fixed a ) 98 { 99 return ( ADD_LONG( a, 0xFFFFL ) ) & ~0xFFFFL; 100 } 101 102 103 /* documentation is in freetype.h */ 104 105 FT_EXPORT_DEF( FT_Fixed ) FT_FloorFix(FT_Fixed a)106 FT_FloorFix( FT_Fixed a ) 107 { 108 return a & ~0xFFFFL; 109 } 110 111 #ifndef FT_MSB 112 113 FT_BASE_DEF ( FT_Int ) FT_MSB(FT_UInt32 z)114 FT_MSB( FT_UInt32 z ) 115 { 116 FT_Int shift = 0; 117 118 119 /* determine msb bit index in `shift' */ 120 if ( z & 0xFFFF0000UL ) 121 { 122 z >>= 16; 123 shift += 16; 124 } 125 if ( z & 0x0000FF00UL ) 126 { 127 z >>= 8; 128 shift += 8; 129 } 130 if ( z & 0x000000F0UL ) 131 { 132 z >>= 4; 133 shift += 4; 134 } 135 if ( z & 0x0000000CUL ) 136 { 137 z >>= 2; 138 shift += 2; 139 } 140 if ( z & 0x00000002UL ) 141 { 142 /* z >>= 1; */ 143 shift += 1; 144 } 145 146 return shift; 147 } 148 149 #endif /* !FT_MSB */ 150 151 152 /* documentation is in ftcalc.h */ 153 154 FT_BASE_DEF( FT_Fixed ) FT_Hypot(FT_Fixed x,FT_Fixed y)155 FT_Hypot( FT_Fixed x, 156 FT_Fixed y ) 157 { 158 FT_Vector v; 159 160 161 v.x = x; 162 v.y = y; 163 164 return FT_Vector_Length( &v ); 165 } 166 167 168 #ifdef FT_LONG64 169 170 171 /* documentation is in freetype.h */ 172 173 FT_EXPORT_DEF( FT_Long ) FT_MulDiv(FT_Long a_,FT_Long b_,FT_Long c_)174 FT_MulDiv( FT_Long a_, 175 FT_Long b_, 176 FT_Long c_ ) 177 { 178 FT_Int s = 1; 179 FT_UInt64 a, b, c, d; 180 FT_Long d_; 181 182 183 a = (FT_UInt64)a_; 184 b = (FT_UInt64)b_; 185 c = (FT_UInt64)c_; 186 187 FT_MOVE_SIGN( a_, a, s ); 188 FT_MOVE_SIGN( b_, b, s ); 189 FT_MOVE_SIGN( c_, c, s ); 190 191 d = c > 0 ? ( a * b + ( c >> 1 ) ) / c 192 : 0x7FFFFFFFUL; 193 194 d_ = (FT_Long)d; 195 196 return s < 0 ? NEG_LONG( d_ ) : d_; 197 } 198 199 200 /* documentation is in ftcalc.h */ 201 202 FT_BASE_DEF( FT_Long ) FT_MulDiv_No_Round(FT_Long a_,FT_Long b_,FT_Long c_)203 FT_MulDiv_No_Round( FT_Long a_, 204 FT_Long b_, 205 FT_Long c_ ) 206 { 207 FT_Int s = 1; 208 FT_UInt64 a, b, c, d; 209 FT_Long d_; 210 211 212 a = (FT_UInt64)a_; 213 b = (FT_UInt64)b_; 214 c = (FT_UInt64)c_; 215 216 FT_MOVE_SIGN( a_, a, s ); 217 FT_MOVE_SIGN( b_, b, s ); 218 FT_MOVE_SIGN( c_, c, s ); 219 220 d = c > 0 ? a * b / c 221 : 0x7FFFFFFFUL; 222 223 d_ = (FT_Long)d; 224 225 return s < 0 ? NEG_LONG( d_ ) : d_; 226 } 227 228 229 /* documentation is in freetype.h */ 230 231 FT_EXPORT_DEF( FT_Long ) FT_MulFix(FT_Long a_,FT_Long b_)232 FT_MulFix( FT_Long a_, 233 FT_Long b_ ) 234 { 235 #ifdef FT_MULFIX_ASSEMBLER 236 237 return FT_MULFIX_ASSEMBLER( (FT_Int32)a_, (FT_Int32)b_ ); 238 239 #else 240 241 FT_Int64 ab = (FT_Int64)a_ * (FT_Int64)b_; 242 243 /* this requires arithmetic right shift of signed numbers */ 244 return (FT_Long)( ( ab + 0x8000L - ( ab < 0 ) ) >> 16 ); 245 246 #endif /* FT_MULFIX_ASSEMBLER */ 247 } 248 249 250 /* documentation is in freetype.h */ 251 252 FT_EXPORT_DEF( FT_Long ) FT_DivFix(FT_Long a_,FT_Long b_)253 FT_DivFix( FT_Long a_, 254 FT_Long b_ ) 255 { 256 FT_Int s = 1; 257 FT_UInt64 a, b, q; 258 FT_Long q_; 259 260 261 a = (FT_UInt64)a_; 262 b = (FT_UInt64)b_; 263 264 FT_MOVE_SIGN( a_, a, s ); 265 FT_MOVE_SIGN( b_, b, s ); 266 267 q = b > 0 ? ( ( a << 16 ) + ( b >> 1 ) ) / b 268 : 0x7FFFFFFFUL; 269 270 q_ = (FT_Long)q; 271 272 return s < 0 ? NEG_LONG( q_ ) : q_; 273 } 274 275 276 #else /* !FT_LONG64 */ 277 278 279 static void ft_multo64(FT_UInt32 x,FT_UInt32 y,FT_Int64 * z)280 ft_multo64( FT_UInt32 x, 281 FT_UInt32 y, 282 FT_Int64 *z ) 283 { 284 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2; 285 286 287 lo1 = x & 0x0000FFFFU; hi1 = x >> 16; 288 lo2 = y & 0x0000FFFFU; hi2 = y >> 16; 289 290 lo = lo1 * lo2; 291 i1 = lo1 * hi2; 292 i2 = lo2 * hi1; 293 hi = hi1 * hi2; 294 295 /* Check carry overflow of i1 + i2 */ 296 i1 += i2; 297 hi += (FT_UInt32)( i1 < i2 ) << 16; 298 299 hi += i1 >> 16; 300 i1 = i1 << 16; 301 302 /* Check carry overflow of i1 + lo */ 303 lo += i1; 304 hi += ( lo < i1 ); 305 306 z->lo = lo; 307 z->hi = hi; 308 } 309 310 311 static FT_UInt32 ft_div64by32(FT_UInt32 hi,FT_UInt32 lo,FT_UInt32 y)312 ft_div64by32( FT_UInt32 hi, 313 FT_UInt32 lo, 314 FT_UInt32 y ) 315 { 316 FT_UInt32 r, q; 317 FT_Int i; 318 319 320 if ( hi >= y ) 321 return (FT_UInt32)0x7FFFFFFFL; 322 323 /* We shift as many bits as we can into the high register, perform */ 324 /* 32-bit division with modulo there, then work through the remaining */ 325 /* bits with long division. This optimization is especially noticeable */ 326 /* for smaller dividends that barely use the high register. */ 327 328 i = 31 - FT_MSB( hi ); 329 r = ( hi << i ) | ( lo >> ( 32 - i ) ); lo <<= i; /* left 64-bit shift */ 330 q = r / y; 331 r -= q * y; /* remainder */ 332 333 i = 32 - i; /* bits remaining in low register */ 334 do 335 { 336 q <<= 1; 337 r = ( r << 1 ) | ( lo >> 31 ); lo <<= 1; 338 339 if ( r >= y ) 340 { 341 r -= y; 342 q |= 1; 343 } 344 } while ( --i ); 345 346 return q; 347 } 348 349 350 static void FT_Add64(FT_Int64 * x,FT_Int64 * y,FT_Int64 * z)351 FT_Add64( FT_Int64* x, 352 FT_Int64* y, 353 FT_Int64 *z ) 354 { 355 FT_UInt32 lo, hi; 356 357 358 lo = x->lo + y->lo; 359 hi = x->hi + y->hi + ( lo < x->lo ); 360 361 z->lo = lo; 362 z->hi = hi; 363 } 364 365 366 /* The FT_MulDiv function has been optimized thanks to ideas from */ 367 /* Graham Asher and Alexei Podtelezhnikov. The trick is to optimize */ 368 /* a rather common case when everything fits within 32-bits. */ 369 /* */ 370 /* We compute 'a*b+c/2', then divide it by 'c' (all positive values). */ 371 /* */ 372 /* The product of two positive numbers never exceeds the square of */ 373 /* its mean values. Therefore, we always avoid the overflow by */ 374 /* imposing */ 375 /* */ 376 /* (a + b) / 2 <= sqrt(X - c/2) , */ 377 /* */ 378 /* where X = 2^32 - 1, the maximum unsigned 32-bit value, and using */ 379 /* unsigned arithmetic. Now we replace `sqrt' with a linear function */ 380 /* that is smaller or equal for all values of c in the interval */ 381 /* [0;X/2]; it should be equal to sqrt(X) and sqrt(3X/4) at the */ 382 /* endpoints. Substituting the linear solution and explicit numbers */ 383 /* we get */ 384 /* */ 385 /* a + b <= 131071.99 - c / 122291.84 . */ 386 /* */ 387 /* In practice, we should use a faster and even stronger inequality */ 388 /* */ 389 /* a + b <= 131071 - (c >> 16) */ 390 /* */ 391 /* or, alternatively, */ 392 /* */ 393 /* a + b <= 129894 - (c >> 17) . */ 394 /* */ 395 /* FT_MulFix, on the other hand, is optimized for a small value of */ 396 /* the first argument, when the second argument can be much larger. */ 397 /* This can be achieved by scaling the second argument and the limit */ 398 /* in the above inequalities. For example, */ 399 /* */ 400 /* a + (b >> 8) <= (131071 >> 4) */ 401 /* */ 402 /* covers the practical range of use. The actual test below is a bit */ 403 /* tighter to avoid the border case overflows. */ 404 /* */ 405 /* In the case of FT_DivFix, the exact overflow check */ 406 /* */ 407 /* a << 16 <= X - c/2 */ 408 /* */ 409 /* is scaled down by 2^16 and we use */ 410 /* */ 411 /* a <= 65535 - (c >> 17) . */ 412 413 /* documentation is in freetype.h */ 414 415 FT_EXPORT_DEF( FT_Long ) FT_MulDiv(FT_Long a_,FT_Long b_,FT_Long c_)416 FT_MulDiv( FT_Long a_, 417 FT_Long b_, 418 FT_Long c_ ) 419 { 420 FT_Int s = 1; 421 FT_UInt32 a, b, c; 422 423 424 /* XXX: this function does not allow 64-bit arguments */ 425 426 a = (FT_UInt32)a_; 427 b = (FT_UInt32)b_; 428 c = (FT_UInt32)c_; 429 430 FT_MOVE_SIGN( a_, a, s ); 431 FT_MOVE_SIGN( b_, b, s ); 432 FT_MOVE_SIGN( c_, c, s ); 433 434 if ( c == 0 ) 435 a = 0x7FFFFFFFUL; 436 437 else if ( a + b <= 129894UL - ( c >> 17 ) ) 438 a = ( a * b + ( c >> 1 ) ) / c; 439 440 else 441 { 442 FT_Int64 temp, temp2; 443 444 445 ft_multo64( a, b, &temp ); 446 447 temp2.hi = 0; 448 temp2.lo = c >> 1; 449 450 FT_Add64( &temp, &temp2, &temp ); 451 452 /* last attempt to ditch long division */ 453 a = ( temp.hi == 0 ) ? temp.lo / c 454 : ft_div64by32( temp.hi, temp.lo, c ); 455 } 456 457 a_ = (FT_Long)a; 458 459 return s < 0 ? NEG_LONG( a_ ) : a_; 460 } 461 462 463 FT_BASE_DEF( FT_Long ) FT_MulDiv_No_Round(FT_Long a_,FT_Long b_,FT_Long c_)464 FT_MulDiv_No_Round( FT_Long a_, 465 FT_Long b_, 466 FT_Long c_ ) 467 { 468 FT_Int s = 1; 469 FT_UInt32 a, b, c; 470 471 472 /* XXX: this function does not allow 64-bit arguments */ 473 474 a = (FT_UInt32)a_; 475 b = (FT_UInt32)b_; 476 c = (FT_UInt32)c_; 477 478 FT_MOVE_SIGN( a_, a, s ); 479 FT_MOVE_SIGN( b_, b, s ); 480 FT_MOVE_SIGN( c_, c, s ); 481 482 if ( c == 0 ) 483 a = 0x7FFFFFFFUL; 484 485 else if ( a + b <= 131071UL ) 486 a = a * b / c; 487 488 else 489 { 490 FT_Int64 temp; 491 492 493 ft_multo64( a, b, &temp ); 494 495 /* last attempt to ditch long division */ 496 a = ( temp.hi == 0 ) ? temp.lo / c 497 : ft_div64by32( temp.hi, temp.lo, c ); 498 } 499 500 a_ = (FT_Long)a; 501 502 return s < 0 ? NEG_LONG( a_ ) : a_; 503 } 504 505 506 /* documentation is in freetype.h */ 507 508 FT_EXPORT_DEF( FT_Long ) FT_MulFix(FT_Long a_,FT_Long b_)509 FT_MulFix( FT_Long a_, 510 FT_Long b_ ) 511 { 512 #ifdef FT_MULFIX_ASSEMBLER 513 514 return FT_MULFIX_ASSEMBLER( a_, b_ ); 515 516 #elif 0 517 518 /* 519 * This code is nonportable. See comment below. 520 * 521 * However, on a platform where right-shift of a signed quantity fills 522 * the leftmost bits by copying the sign bit, it might be faster. 523 */ 524 525 FT_Long sa, sb; 526 FT_UInt32 a, b; 527 528 529 /* 530 * This is a clever way of converting a signed number `a' into its 531 * absolute value (stored back into `a') and its sign. The sign is 532 * stored in `sa'; 0 means `a' was positive or zero, and -1 means `a' 533 * was negative. (Similarly for `b' and `sb'). 534 * 535 * Unfortunately, it doesn't work (at least not portably). 536 * 537 * It makes the assumption that right-shift on a negative signed value 538 * fills the leftmost bits by copying the sign bit. This is wrong. 539 * According to K&R 2nd ed, section `A7.8 Shift Operators' on page 206, 540 * the result of right-shift of a negative signed value is 541 * implementation-defined. At least one implementation fills the 542 * leftmost bits with 0s (i.e., it is exactly the same as an unsigned 543 * right shift). This means that when `a' is negative, `sa' ends up 544 * with the value 1 rather than -1. After that, everything else goes 545 * wrong. 546 */ 547 sa = ( a_ >> ( sizeof ( a_ ) * 8 - 1 ) ); 548 a = ( a_ ^ sa ) - sa; 549 sb = ( b_ >> ( sizeof ( b_ ) * 8 - 1 ) ); 550 b = ( b_ ^ sb ) - sb; 551 552 a = (FT_UInt32)a_; 553 b = (FT_UInt32)b_; 554 555 if ( a + ( b >> 8 ) <= 8190UL ) 556 a = ( a * b + 0x8000U ) >> 16; 557 else 558 { 559 FT_UInt32 al = a & 0xFFFFUL; 560 561 562 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 563 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 564 } 565 566 sa ^= sb; 567 a = ( a ^ sa ) - sa; 568 569 return (FT_Long)a; 570 571 #else /* 0 */ 572 573 FT_Int s = 1; 574 FT_UInt32 a, b; 575 576 577 /* XXX: this function does not allow 64-bit arguments */ 578 579 a = (FT_UInt32)a_; 580 b = (FT_UInt32)b_; 581 582 FT_MOVE_SIGN( a_, a, s ); 583 FT_MOVE_SIGN( b_, b, s ); 584 585 if ( a + ( b >> 8 ) <= 8190UL ) 586 a = ( a * b + 0x8000UL ) >> 16; 587 else 588 { 589 FT_UInt32 al = a & 0xFFFFUL; 590 591 592 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 593 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 594 } 595 596 a_ = (FT_Long)a; 597 598 return s < 0 ? NEG_LONG( a_ ) : a_; 599 600 #endif /* 0 */ 601 602 } 603 604 605 /* documentation is in freetype.h */ 606 607 FT_EXPORT_DEF( FT_Long ) FT_DivFix(FT_Long a_,FT_Long b_)608 FT_DivFix( FT_Long a_, 609 FT_Long b_ ) 610 { 611 FT_Int s = 1; 612 FT_UInt32 a, b, q; 613 FT_Long q_; 614 615 616 /* XXX: this function does not allow 64-bit arguments */ 617 618 a = (FT_UInt32)a_; 619 b = (FT_UInt32)b_; 620 621 FT_MOVE_SIGN( a_, a, s ); 622 FT_MOVE_SIGN( b_, b, s ); 623 624 if ( b == 0 ) 625 { 626 /* check for division by 0 */ 627 q = 0x7FFFFFFFUL; 628 } 629 else if ( a <= 65535UL - ( b >> 17 ) ) 630 { 631 /* compute result directly */ 632 q = ( ( a << 16 ) + ( b >> 1 ) ) / b; 633 } 634 else 635 { 636 /* we need more bits; we have to do it by hand */ 637 FT_Int64 temp, temp2; 638 639 640 temp.hi = a >> 16; 641 temp.lo = a << 16; 642 temp2.hi = 0; 643 temp2.lo = b >> 1; 644 645 FT_Add64( &temp, &temp2, &temp ); 646 q = ft_div64by32( temp.hi, temp.lo, b ); 647 } 648 649 q_ = (FT_Long)q; 650 651 return s < 0 ? NEG_LONG( q_ ) : q_; 652 } 653 654 655 #endif /* !FT_LONG64 */ 656 657 658 /* documentation is in ftglyph.h */ 659 660 FT_EXPORT_DEF( void ) FT_Matrix_Multiply(const FT_Matrix * a,FT_Matrix * b)661 FT_Matrix_Multiply( const FT_Matrix* a, 662 FT_Matrix *b ) 663 { 664 FT_Fixed xx, xy, yx, yy; 665 666 667 if ( !a || !b ) 668 return; 669 670 xx = ADD_LONG( FT_MulFix( a->xx, b->xx ), 671 FT_MulFix( a->xy, b->yx ) ); 672 xy = ADD_LONG( FT_MulFix( a->xx, b->xy ), 673 FT_MulFix( a->xy, b->yy ) ); 674 yx = ADD_LONG( FT_MulFix( a->yx, b->xx ), 675 FT_MulFix( a->yy, b->yx ) ); 676 yy = ADD_LONG( FT_MulFix( a->yx, b->xy ), 677 FT_MulFix( a->yy, b->yy ) ); 678 679 b->xx = xx; 680 b->xy = xy; 681 b->yx = yx; 682 b->yy = yy; 683 } 684 685 686 /* documentation is in ftglyph.h */ 687 688 FT_EXPORT_DEF( FT_Error ) FT_Matrix_Invert(FT_Matrix * matrix)689 FT_Matrix_Invert( FT_Matrix* matrix ) 690 { 691 FT_Pos delta, xx, yy; 692 693 694 if ( !matrix ) 695 return FT_THROW( Invalid_Argument ); 696 697 /* compute discriminant */ 698 delta = FT_MulFix( matrix->xx, matrix->yy ) - 699 FT_MulFix( matrix->xy, matrix->yx ); 700 701 if ( !delta ) 702 return FT_THROW( Invalid_Argument ); /* matrix can't be inverted */ 703 704 matrix->xy = - FT_DivFix( matrix->xy, delta ); 705 matrix->yx = - FT_DivFix( matrix->yx, delta ); 706 707 xx = matrix->xx; 708 yy = matrix->yy; 709 710 matrix->xx = FT_DivFix( yy, delta ); 711 matrix->yy = FT_DivFix( xx, delta ); 712 713 return FT_Err_Ok; 714 } 715 716 717 /* documentation is in ftcalc.h */ 718 719 FT_BASE_DEF( void ) FT_Matrix_Multiply_Scaled(const FT_Matrix * a,FT_Matrix * b,FT_Long scaling)720 FT_Matrix_Multiply_Scaled( const FT_Matrix* a, 721 FT_Matrix *b, 722 FT_Long scaling ) 723 { 724 FT_Fixed xx, xy, yx, yy; 725 726 FT_Long val = 0x10000L * scaling; 727 728 729 if ( !a || !b ) 730 return; 731 732 xx = ADD_LONG( FT_MulDiv( a->xx, b->xx, val ), 733 FT_MulDiv( a->xy, b->yx, val ) ); 734 xy = ADD_LONG( FT_MulDiv( a->xx, b->xy, val ), 735 FT_MulDiv( a->xy, b->yy, val ) ); 736 yx = ADD_LONG( FT_MulDiv( a->yx, b->xx, val ), 737 FT_MulDiv( a->yy, b->yx, val ) ); 738 yy = ADD_LONG( FT_MulDiv( a->yx, b->xy, val ), 739 FT_MulDiv( a->yy, b->yy, val ) ); 740 741 b->xx = xx; 742 b->xy = xy; 743 b->yx = yx; 744 b->yy = yy; 745 } 746 747 748 /* documentation is in ftcalc.h */ 749 750 FT_BASE_DEF( FT_Bool ) FT_Matrix_Check(const FT_Matrix * matrix)751 FT_Matrix_Check( const FT_Matrix* matrix ) 752 { 753 FT_Matrix m; 754 FT_Fixed val[4]; 755 FT_Fixed nonzero_minval, maxval; 756 FT_Fixed temp1, temp2; 757 FT_UInt i; 758 759 760 if ( !matrix ) 761 return 0; 762 763 val[0] = FT_ABS( matrix->xx ); 764 val[1] = FT_ABS( matrix->xy ); 765 val[2] = FT_ABS( matrix->yx ); 766 val[3] = FT_ABS( matrix->yy ); 767 768 /* 769 * To avoid overflow, we ensure that each value is not larger than 770 * 771 * int(sqrt(2^31 / 4)) = 23170 ; 772 * 773 * we also check that no value becomes zero if we have to scale. 774 */ 775 776 maxval = 0; 777 nonzero_minval = FT_LONG_MAX; 778 779 for ( i = 0; i < 4; i++ ) 780 { 781 if ( val[i] > maxval ) 782 maxval = val[i]; 783 if ( val[i] && val[i] < nonzero_minval ) 784 nonzero_minval = val[i]; 785 } 786 787 if ( maxval > 23170 ) 788 { 789 FT_Fixed scale = FT_DivFix( maxval, 23170 ); 790 791 792 if ( !FT_DivFix( nonzero_minval, scale ) ) 793 return 0; /* value range too large */ 794 795 m.xx = FT_DivFix( matrix->xx, scale ); 796 m.xy = FT_DivFix( matrix->xy, scale ); 797 m.yx = FT_DivFix( matrix->yx, scale ); 798 m.yy = FT_DivFix( matrix->yy, scale ); 799 } 800 else 801 m = *matrix; 802 803 temp1 = FT_ABS( m.xx * m.yy - m.xy * m.yx ); 804 temp2 = m.xx * m.xx + m.xy * m.xy + m.yx * m.yx + m.yy * m.yy; 805 806 if ( temp1 == 0 || 807 temp2 / temp1 > 50 ) 808 return 0; 809 810 return 1; 811 } 812 813 814 /* documentation is in ftcalc.h */ 815 816 FT_BASE_DEF( void ) FT_Vector_Transform_Scaled(FT_Vector * vector,const FT_Matrix * matrix,FT_Long scaling)817 FT_Vector_Transform_Scaled( FT_Vector* vector, 818 const FT_Matrix* matrix, 819 FT_Long scaling ) 820 { 821 FT_Pos xz, yz; 822 823 FT_Long val = 0x10000L * scaling; 824 825 826 if ( !vector || !matrix ) 827 return; 828 829 xz = ADD_LONG( FT_MulDiv( vector->x, matrix->xx, val ), 830 FT_MulDiv( vector->y, matrix->xy, val ) ); 831 yz = ADD_LONG( FT_MulDiv( vector->x, matrix->yx, val ), 832 FT_MulDiv( vector->y, matrix->yy, val ) ); 833 834 vector->x = xz; 835 vector->y = yz; 836 } 837 838 839 /* documentation is in ftcalc.h */ 840 841 FT_BASE_DEF( FT_UInt32 ) FT_Vector_NormLen(FT_Vector * vector)842 FT_Vector_NormLen( FT_Vector* vector ) 843 { 844 FT_Int32 x_ = vector->x; 845 FT_Int32 y_ = vector->y; 846 FT_Int32 b, z; 847 FT_UInt32 x, y, u, v, l; 848 FT_Int sx = 1, sy = 1, shift; 849 850 851 x = (FT_UInt32)x_; 852 y = (FT_UInt32)y_; 853 854 FT_MOVE_SIGN( x_, x, sx ); 855 FT_MOVE_SIGN( y_, y, sy ); 856 857 /* trivial cases */ 858 if ( x == 0 ) 859 { 860 if ( y > 0 ) 861 vector->y = sy * 0x10000; 862 return y; 863 } 864 else if ( y == 0 ) 865 { 866 if ( x > 0 ) 867 vector->x = sx * 0x10000; 868 return x; 869 } 870 871 /* Estimate length and prenormalize by shifting so that */ 872 /* the new approximate length is between 2/3 and 4/3. */ 873 /* The magic constant 0xAAAAAAAAUL (2/3 of 2^32) helps */ 874 /* achieve this in 16.16 fixed-point representation. */ 875 l = x > y ? x + ( y >> 1 ) 876 : y + ( x >> 1 ); 877 878 shift = 31 - FT_MSB( l ); 879 shift -= 15 + ( l >= ( 0xAAAAAAAAUL >> shift ) ); 880 881 if ( shift > 0 ) 882 { 883 x <<= shift; 884 y <<= shift; 885 886 /* re-estimate length for tiny vectors */ 887 l = x > y ? x + ( y >> 1 ) 888 : y + ( x >> 1 ); 889 } 890 else 891 { 892 x >>= -shift; 893 y >>= -shift; 894 l >>= -shift; 895 } 896 897 /* lower linear approximation for reciprocal length minus one */ 898 b = 0x10000 - (FT_Int32)l; 899 900 x_ = (FT_Int32)x; 901 y_ = (FT_Int32)y; 902 903 /* Newton's iterations */ 904 do 905 { 906 u = (FT_UInt32)( x_ + ( x_ * b >> 16 ) ); 907 v = (FT_UInt32)( y_ + ( y_ * b >> 16 ) ); 908 909 /* Normalized squared length in the parentheses approaches 2^32. */ 910 /* On two's complement systems, converting to signed gives the */ 911 /* difference with 2^32 even if the expression wraps around. */ 912 z = -(FT_Int32)( u * u + v * v ) / 0x200; 913 z = z * ( ( 0x10000 + b ) >> 8 ) / 0x10000; 914 915 b += z; 916 917 } while ( z > 0 ); 918 919 vector->x = sx < 0 ? -(FT_Pos)u : (FT_Pos)u; 920 vector->y = sy < 0 ? -(FT_Pos)v : (FT_Pos)v; 921 922 /* Conversion to signed helps to recover from likely wrap around */ 923 /* in calculating the prenormalized length, because it gives the */ 924 /* correct difference with 2^32 on two's complement systems. */ 925 l = (FT_UInt32)( 0x10000 + (FT_Int32)( u * x + v * y ) / 0x10000 ); 926 if ( shift > 0 ) 927 l = ( l + ( 1 << ( shift - 1 ) ) ) >> shift; 928 else 929 l <<= -shift; 930 931 return l; 932 } 933 934 935 #if 0 936 937 /* documentation is in ftcalc.h */ 938 939 FT_BASE_DEF( FT_Int32 ) 940 FT_SqrtFixed( FT_Int32 x ) 941 { 942 FT_UInt32 root, rem_hi, rem_lo, test_div; 943 FT_Int count; 944 945 946 root = 0; 947 948 if ( x > 0 ) 949 { 950 rem_hi = 0; 951 rem_lo = (FT_UInt32)x; 952 count = 24; 953 do 954 { 955 rem_hi = ( rem_hi << 2 ) | ( rem_lo >> 30 ); 956 rem_lo <<= 2; 957 root <<= 1; 958 test_div = ( root << 1 ) + 1; 959 960 if ( rem_hi >= test_div ) 961 { 962 rem_hi -= test_div; 963 root += 1; 964 } 965 } while ( --count ); 966 } 967 968 return (FT_Int32)root; 969 } 970 971 #endif /* 0 */ 972 973 974 /* documentation is in ftcalc.h */ 975 976 FT_BASE_DEF( FT_Int ) ft_corner_orientation(FT_Pos in_x,FT_Pos in_y,FT_Pos out_x,FT_Pos out_y)977 ft_corner_orientation( FT_Pos in_x, 978 FT_Pos in_y, 979 FT_Pos out_x, 980 FT_Pos out_y ) 981 { 982 #ifdef FT_LONG64 983 984 FT_Int64 delta = (FT_Int64)in_x * out_y - (FT_Int64)in_y * out_x; 985 986 987 return ( delta > 0 ) - ( delta < 0 ); 988 989 #else 990 991 FT_Int result; 992 993 994 /* we silently ignore overflow errors, since such large values */ 995 /* lead to even more (harmless) rendering errors later on */ 996 if ( ADD_LONG( FT_ABS( in_x ), FT_ABS( out_y ) ) <= 131071L && 997 ADD_LONG( FT_ABS( in_y ), FT_ABS( out_x ) ) <= 131071L ) 998 { 999 FT_Long z1 = MUL_LONG( in_x, out_y ); 1000 FT_Long z2 = MUL_LONG( in_y, out_x ); 1001 1002 1003 if ( z1 > z2 ) 1004 result = +1; 1005 else if ( z1 < z2 ) 1006 result = -1; 1007 else 1008 result = 0; 1009 } 1010 else /* products might overflow 32 bits */ 1011 { 1012 FT_Int64 z1, z2; 1013 1014 1015 /* XXX: this function does not allow 64-bit arguments */ 1016 ft_multo64( (FT_UInt32)in_x, (FT_UInt32)out_y, &z1 ); 1017 ft_multo64( (FT_UInt32)in_y, (FT_UInt32)out_x, &z2 ); 1018 1019 if ( z1.hi > z2.hi ) 1020 result = +1; 1021 else if ( z1.hi < z2.hi ) 1022 result = -1; 1023 else if ( z1.lo > z2.lo ) 1024 result = +1; 1025 else if ( z1.lo < z2.lo ) 1026 result = -1; 1027 else 1028 result = 0; 1029 } 1030 1031 /* XXX: only the sign of return value, +1/0/-1 must be used */ 1032 return result; 1033 1034 #endif 1035 } 1036 1037 1038 /* documentation is in ftcalc.h */ 1039 1040 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)1041 ft_corner_is_flat( FT_Pos in_x, 1042 FT_Pos in_y, 1043 FT_Pos out_x, 1044 FT_Pos out_y ) 1045 { 1046 FT_Pos ax = in_x + out_x; 1047 FT_Pos ay = in_y + out_y; 1048 1049 FT_Pos d_in, d_out, d_hypot; 1050 1051 1052 /* The idea of this function is to compare the length of the */ 1053 /* hypotenuse with the `in' and `out' length. The `corner' */ 1054 /* represented by `in' and `out' is flat if the hypotenuse's */ 1055 /* length isn't too large. */ 1056 /* */ 1057 /* This approach has the advantage that the angle between */ 1058 /* `in' and `out' is not checked. In case one of the two */ 1059 /* vectors is `dominant', this is, much larger than the */ 1060 /* other vector, we thus always have a flat corner. */ 1061 /* */ 1062 /* hypotenuse */ 1063 /* x---------------------------x */ 1064 /* \ / */ 1065 /* \ / */ 1066 /* in \ / out */ 1067 /* \ / */ 1068 /* o */ 1069 /* Point */ 1070 1071 d_in = FT_HYPOT( in_x, in_y ); 1072 d_out = FT_HYPOT( out_x, out_y ); 1073 d_hypot = FT_HYPOT( ax, ay ); 1074 1075 /* now do a simple length comparison: */ 1076 /* */ 1077 /* d_in + d_out < 17/16 d_hypot */ 1078 1079 return ( d_in + d_out - d_hypot ) < ( d_hypot >> 4 ); 1080 } 1081 1082 1083 /* END */ 1084