1/***************************************************************************/
2/*                                                                         */
3/*  fttrigon.c                                                             */
4/*                                                                         */
5/*    FreeType trigonometric functions (body).                             */
6/*                                                                         */
7/*  Copyright 2001-2005, 2012-2013 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_Fixed
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_Fixed  s;
64    FT_Int64  v;
65
66
67    s   = val;
68    val = FT_ABS( val );
69
70    v   = ( val * (FT_Int64)FT_TRIG_SCALE ) + 0x100000000UL;
71    val = (FT_Fixed)( v >> 32 );
72
73    return ( s >= 0 ) ? val : -val;
74  }
75
76#else /* !FT_LONG64 */
77
78  /* multiply a given value by the CORDIC shrink factor */
79  static FT_Fixed
80  ft_trig_downscale( FT_Fixed  val )
81  {
82    FT_Fixed   s;
83    FT_UInt32  v1, v2, k1, k2, hi, lo1, lo2, lo3;
84
85
86    s   = val;
87    val = FT_ABS( val );
88
89    v1 = (FT_UInt32)val >> 16;
90    v2 = (FT_UInt32)( val & 0xFFFFL );
91
92    k1 = (FT_UInt32)FT_TRIG_SCALE >> 16;           /* constant */
93    k2 = (FT_UInt32)( FT_TRIG_SCALE & 0xFFFFL );   /* constant */
94
95    hi   = k1 * v1;
96    lo1  = k1 * v2 + k2 * v1;       /* can't overflow */
97
98    lo2  = ( k2 * v2 ) >> 16;
99    lo3  = FT_MAX( lo1, lo2 );
100    lo1 += lo2;
101
102    hi  += lo1 >> 16;
103    if ( lo1 < lo3 )
104      hi += (FT_UInt32)0x10000UL;
105
106    val  = (FT_Fixed)hi;
107
108    return ( s >= 0 ) ? val : -val;
109  }
110
111#endif /* !FT_LONG64 */
112
113
114  static FT_Int
115  ft_trig_prenorm( FT_Vector*  vec )
116  {
117    FT_Pos  x, y;
118    FT_Int  shift;
119
120
121    x = vec->x;
122    y = vec->y;
123
124    shift = FT_MSB( FT_ABS( x ) | FT_ABS( y ) );
125
126    if ( shift <= FT_TRIG_SAFE_MSB )
127    {
128      shift  = FT_TRIG_SAFE_MSB - shift;
129      vec->x = (FT_Pos)( (FT_ULong)x << shift );
130      vec->y = (FT_Pos)( (FT_ULong)y << shift );
131    }
132    else
133    {
134      shift -= FT_TRIG_SAFE_MSB;
135      vec->x = x >> shift;
136      vec->y = y >> shift;
137      shift  = -shift;
138    }
139
140    return shift;
141  }
142
143
144  static void
145  ft_trig_pseudo_rotate( FT_Vector*  vec,
146                         FT_Angle    theta )
147  {
148    FT_Int           i;
149    FT_Fixed         x, y, xtemp, b;
150    const FT_Fixed  *arctanptr;
151
152
153    x = vec->x;
154    y = vec->y;
155
156    /* Rotate inside [-PI/4,PI/4] sector */
157    while ( theta < -FT_ANGLE_PI4 )
158    {
159      xtemp  =  y;
160      y      = -x;
161      x      =  xtemp;
162      theta +=  FT_ANGLE_PI2;
163    }
164
165    while ( theta > FT_ANGLE_PI4 )
166    {
167      xtemp  = -y;
168      y      =  x;
169      x      =  xtemp;
170      theta -=  FT_ANGLE_PI2;
171    }
172
173    arctanptr = ft_trig_arctan_table;
174
175    /* Pseudorotations, with right shifts */
176    for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
177    {
178      if ( theta < 0 )
179      {
180        xtemp  = x + ( ( y + b ) >> i );
181        y      = y - ( ( x + b ) >> i );
182        x      = xtemp;
183        theta += *arctanptr++;
184      }
185      else
186      {
187        xtemp  = x - ( ( y + b ) >> i );
188        y      = y + ( ( x + b ) >> i );
189        x      = xtemp;
190        theta -= *arctanptr++;
191      }
192    }
193
194    vec->x = x;
195    vec->y = y;
196  }
197
198
199  static void
200  ft_trig_pseudo_polarize( FT_Vector*  vec )
201  {
202    FT_Angle         theta;
203    FT_Int           i;
204    FT_Fixed         x, y, xtemp, b;
205    const FT_Fixed  *arctanptr;
206
207
208    x = vec->x;
209    y = vec->y;
210
211    /* Get the vector into [-PI/4,PI/4] sector */
212    if ( y > x )
213    {
214      if ( y > -x )
215      {
216        theta =  FT_ANGLE_PI2;
217        xtemp =  y;
218        y     = -x;
219        x     =  xtemp;
220      }
221      else
222      {
223        theta =  y > 0 ? FT_ANGLE_PI : -FT_ANGLE_PI;
224        x     = -x;
225        y     = -y;
226      }
227    }
228    else
229    {
230      if ( y < -x )
231      {
232        theta = -FT_ANGLE_PI2;
233        xtemp = -y;
234        y     =  x;
235        x     =  xtemp;
236      }
237      else
238      {
239        theta = 0;
240      }
241    }
242
243    arctanptr = ft_trig_arctan_table;
244
245    /* Pseudorotations, with right shifts */
246    for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
247    {
248      if ( y > 0 )
249      {
250        xtemp  = x + ( ( y + b ) >> i );
251        y      = y - ( ( x + b ) >> i );
252        x      = xtemp;
253        theta += *arctanptr++;
254      }
255      else
256      {
257        xtemp  = x - ( ( y + b ) >> i );
258        y      = y + ( ( x + b ) >> i );
259        x      = xtemp;
260        theta -= *arctanptr++;
261      }
262    }
263
264    /* round theta */
265    if ( theta >= 0 )
266      theta = FT_PAD_ROUND( theta, 32 );
267    else
268      theta = -FT_PAD_ROUND( -theta, 32 );
269
270    vec->x = x;
271    vec->y = theta;
272  }
273
274
275  /* documentation is in fttrigon.h */
276
277  FT_EXPORT_DEF( FT_Fixed )
278  FT_Cos( FT_Angle  angle )
279  {
280    FT_Vector  v;
281
282
283    v.x = FT_TRIG_SCALE >> 8;
284    v.y = 0;
285    ft_trig_pseudo_rotate( &v, angle );
286
287    return ( v.x + 0x80L ) >> 8;
288  }
289
290
291  /* documentation is in fttrigon.h */
292
293  FT_EXPORT_DEF( FT_Fixed )
294  FT_Sin( FT_Angle  angle )
295  {
296    return FT_Cos( FT_ANGLE_PI2 - angle );
297  }
298
299
300  /* documentation is in fttrigon.h */
301
302  FT_EXPORT_DEF( FT_Fixed )
303  FT_Tan( FT_Angle  angle )
304  {
305    FT_Vector  v;
306
307
308    v.x = FT_TRIG_SCALE >> 8;
309    v.y = 0;
310    ft_trig_pseudo_rotate( &v, angle );
311
312    return FT_DivFix( v.y, v.x );
313  }
314
315
316  /* documentation is in fttrigon.h */
317
318  FT_EXPORT_DEF( FT_Angle )
319  FT_Atan2( FT_Fixed  dx,
320            FT_Fixed  dy )
321  {
322    FT_Vector  v;
323
324
325    if ( dx == 0 && dy == 0 )
326      return 0;
327
328    v.x = dx;
329    v.y = dy;
330    ft_trig_prenorm( &v );
331    ft_trig_pseudo_polarize( &v );
332
333    return v.y;
334  }
335
336
337  /* documentation is in fttrigon.h */
338
339  FT_EXPORT_DEF( void )
340  FT_Vector_Unit( FT_Vector*  vec,
341                  FT_Angle    angle )
342  {
343    vec->x = FT_TRIG_SCALE >> 8;
344    vec->y = 0;
345    ft_trig_pseudo_rotate( vec, angle );
346    vec->x = ( vec->x + 0x80L ) >> 8;
347    vec->y = ( vec->y + 0x80L ) >> 8;
348  }
349
350
351  /* these macros return 0 for positive numbers,
352     and -1 for negative ones */
353#define FT_SIGN_LONG( x )   ( (x) >> ( FT_SIZEOF_LONG * 8 - 1 ) )
354#define FT_SIGN_INT( x )    ( (x) >> ( FT_SIZEOF_INT * 8 - 1 ) )
355#define FT_SIGN_INT32( x )  ( (x) >> 31 )
356#define FT_SIGN_INT16( x )  ( (x) >> 15 )
357
358
359  /* documentation is in fttrigon.h */
360
361  FT_EXPORT_DEF( void )
362  FT_Vector_Rotate( FT_Vector*  vec,
363                    FT_Angle    angle )
364  {
365    FT_Int     shift;
366    FT_Vector  v;
367
368
369    v.x   = vec->x;
370    v.y   = vec->y;
371
372    if ( angle && ( v.x != 0 || v.y != 0 ) )
373    {
374      shift = ft_trig_prenorm( &v );
375      ft_trig_pseudo_rotate( &v, angle );
376      v.x = ft_trig_downscale( v.x );
377      v.y = ft_trig_downscale( v.y );
378
379      if ( shift > 0 )
380      {
381        FT_Int32  half = (FT_Int32)1L << ( shift - 1 );
382
383
384        vec->x = ( v.x + half + FT_SIGN_LONG( v.x ) ) >> shift;
385        vec->y = ( v.y + half + FT_SIGN_LONG( v.y ) ) >> shift;
386      }
387      else
388      {
389        shift  = -shift;
390        vec->x = (FT_Pos)( (FT_ULong)v.x << shift );
391        vec->y = (FT_Pos)( (FT_ULong)v.y << shift );
392      }
393    }
394  }
395
396
397  /* documentation is in fttrigon.h */
398
399  FT_EXPORT_DEF( FT_Fixed )
400  FT_Vector_Length( FT_Vector*  vec )
401  {
402    FT_Int     shift;
403    FT_Vector  v;
404
405
406    v = *vec;
407
408    /* handle trivial cases */
409    if ( v.x == 0 )
410    {
411      return FT_ABS( v.y );
412    }
413    else if ( v.y == 0 )
414    {
415      return FT_ABS( v.x );
416    }
417
418    /* general case */
419    shift = ft_trig_prenorm( &v );
420    ft_trig_pseudo_polarize( &v );
421
422    v.x = ft_trig_downscale( v.x );
423
424    if ( shift > 0 )
425      return ( v.x + ( 1 << ( shift - 1 ) ) ) >> shift;
426
427    return (FT_Fixed)( (FT_UInt32)v.x << -shift );
428  }
429
430
431  /* documentation is in fttrigon.h */
432
433  FT_EXPORT_DEF( void )
434  FT_Vector_Polarize( FT_Vector*  vec,
435                      FT_Fixed   *length,
436                      FT_Angle   *angle )
437  {
438    FT_Int     shift;
439    FT_Vector  v;
440
441
442    v = *vec;
443
444    if ( v.x == 0 && v.y == 0 )
445      return;
446
447    shift = ft_trig_prenorm( &v );
448    ft_trig_pseudo_polarize( &v );
449
450    v.x = ft_trig_downscale( v.x );
451
452    *length = ( shift >= 0 ) ?                      ( v.x >>  shift )
453                             : (FT_Fixed)( (FT_UInt32)v.x << -shift );
454    *angle  = v.y;
455  }
456
457
458  /* documentation is in fttrigon.h */
459
460  FT_EXPORT_DEF( void )
461  FT_Vector_From_Polar( FT_Vector*  vec,
462                        FT_Fixed    length,
463                        FT_Angle    angle )
464  {
465    vec->x = length;
466    vec->y = 0;
467
468    FT_Vector_Rotate( vec, angle );
469  }
470
471
472  /* documentation is in fttrigon.h */
473
474  FT_EXPORT_DEF( FT_Angle )
475  FT_Angle_Diff( FT_Angle  angle1,
476                 FT_Angle  angle2 )
477  {
478    FT_Angle  delta = angle2 - angle1;
479
480
481    delta %= FT_ANGLE_2PI;
482    if ( delta < 0 )
483      delta += FT_ANGLE_2PI;
484
485    if ( delta > FT_ANGLE_PI )
486      delta -= FT_ANGLE_2PI;
487
488    return delta;
489  }
490
491
492/* END */
493