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-/***************************************************************************/
-/* */
-/* fttrigon.c */
-/* */
-/* FreeType trigonometric functions (body). */
-/* */
-/* Copyright 2001-2017 by */
-/* David Turner, Robert Wilhelm, and Werner Lemberg. */
-/* */
-/* This file is part of the FreeType project, and may only be used, */
-/* modified, and distributed under the terms of the FreeType project */
-/* license, LICENSE.TXT. By continuing to use, modify, or distribute */
-/* this file you indicate that you have read the license and */
-/* understand and accept it fully. */
-/* */
-/***************************************************************************/
-
- /*************************************************************************/
- /* */
- /* This is a fixed-point CORDIC implementation of trigonometric */
- /* functions as well as transformations between Cartesian and polar */
- /* coordinates. The angles are represented as 16.16 fixed-point values */
- /* in degrees, i.e., the angular resolution is 2^-16 degrees. Note that */
- /* only vectors longer than 2^16*180/pi (or at least 22 bits) on a */
- /* discrete Cartesian grid can have the same or better angular */
- /* resolution. Therefore, to maintain this precision, some functions */
- /* require an interim upscaling of the vectors, whereas others operate */
- /* with 24-bit long vectors directly. */
- /* */
- /*************************************************************************/
-
-#include <ft2build.h>
-#include FT_INTERNAL_OBJECTS_H
-#include FT_INTERNAL_CALC_H
-#include FT_TRIGONOMETRY_H
-
-
- /* the Cordic shrink factor 0.858785336480436 * 2^32 */
-#define FT_TRIG_SCALE 0xDBD95B16UL
-
- /* the highest bit in overflow-safe vector components, */
- /* MSB of 0.858785336480436 * sqrt(0.5) * 2^30 */
-#define FT_TRIG_SAFE_MSB 29
-
- /* this table was generated for FT_PI = 180L << 16, i.e. degrees */
-#define FT_TRIG_MAX_ITERS 23
-
- static const FT_Angle
- ft_trig_arctan_table[] =
- {
- 1740967L, 919879L, 466945L, 234379L, 117304L, 58666L, 29335L,
- 14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L,
- 57L, 29L, 14L, 7L, 4L, 2L, 1L
- };
-
-
-#ifdef FT_LONG64
-
- /* multiply a given value by the CORDIC shrink factor */
- static FT_Fixed
- ft_trig_downscale( FT_Fixed val )
- {
- FT_Int s = 1;
-
-
- if ( val < 0 )
- {
- val = -val;
- s = -1;
- }
-
- /* 0x40000000 comes from regression analysis between true */
- /* and CORDIC hypotenuse, so it minimizes the error */
- val = (FT_Fixed)(
- ( (FT_UInt64)val * FT_TRIG_SCALE + 0x40000000UL ) >> 32 );
-
- return s < 0 ? -val : val;
- }
-
-#else /* !FT_LONG64 */
-
- /* multiply a given value by the CORDIC shrink factor */
- static FT_Fixed
- ft_trig_downscale( FT_Fixed val )
- {
- FT_Int s = 1;
- FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2;
-
-
- if ( val < 0 )
- {
- val = -val;
- s = -1;
- }
-
- lo1 = (FT_UInt32)val & 0x0000FFFFU;
- hi1 = (FT_UInt32)val >> 16;
- lo2 = FT_TRIG_SCALE & 0x0000FFFFU;
- hi2 = FT_TRIG_SCALE >> 16;
-
- lo = lo1 * lo2;
- i1 = lo1 * hi2;
- i2 = lo2 * hi1;
- hi = hi1 * hi2;
-
- /* Check carry overflow of i1 + i2 */
- i1 += i2;
- hi += (FT_UInt32)( i1 < i2 ) << 16;
-
- hi += i1 >> 16;
- i1 = i1 << 16;
-
- /* Check carry overflow of i1 + lo */
- lo += i1;
- hi += ( lo < i1 );
-
- /* 0x40000000 comes from regression analysis between true */
- /* and CORDIC hypotenuse, so it minimizes the error */
-
- /* Check carry overflow of lo + 0x40000000 */
- lo += 0x40000000UL;
- hi += ( lo < 0x40000000UL );
-
- val = (FT_Fixed)hi;
-
- return s < 0 ? -val : val;
- }
-
-#endif /* !FT_LONG64 */
-
-
- /* undefined and never called for zero vector */
- static FT_Int
- ft_trig_prenorm( FT_Vector* vec )
- {
- FT_Pos x, y;
- FT_Int shift;
-
-
- x = vec->x;
- y = vec->y;
-
- shift = FT_MSB( (FT_UInt32)( FT_ABS( x ) | FT_ABS( y ) ) );
-
- if ( shift <= FT_TRIG_SAFE_MSB )
- {
- shift = FT_TRIG_SAFE_MSB - shift;
- vec->x = (FT_Pos)( (FT_ULong)x << shift );
- vec->y = (FT_Pos)( (FT_ULong)y << shift );
- }
- else
- {
- shift -= FT_TRIG_SAFE_MSB;
- vec->x = x >> shift;
- vec->y = y >> shift;
- shift = -shift;
- }
-
- return shift;
- }
-
-
- static void
- ft_trig_pseudo_rotate( FT_Vector* vec,
- FT_Angle theta )
- {
- FT_Int i;
- FT_Fixed x, y, xtemp, b;
- const FT_Angle *arctanptr;
-
-
- x = vec->x;
- y = vec->y;
-
- /* Rotate inside [-PI/4,PI/4] sector */
- while ( theta < -FT_ANGLE_PI4 )
- {
- xtemp = y;
- y = -x;
- x = xtemp;
- theta += FT_ANGLE_PI2;
- }
-
- while ( theta > FT_ANGLE_PI4 )
- {
- xtemp = -y;
- y = x;
- x = xtemp;
- theta -= FT_ANGLE_PI2;
- }
-
- arctanptr = ft_trig_arctan_table;
-
- /* Pseudorotations, with right shifts */
- for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
- {
- if ( theta < 0 )
- {
- xtemp = x + ( ( y + b ) >> i );
- y = y - ( ( x + b ) >> i );
- x = xtemp;
- theta += *arctanptr++;
- }
- else
- {
- xtemp = x - ( ( y + b ) >> i );
- y = y + ( ( x + b ) >> i );
- x = xtemp;
- theta -= *arctanptr++;
- }
- }
-
- vec->x = x;
- vec->y = y;
- }
-
-
- static void
- ft_trig_pseudo_polarize( FT_Vector* vec )
- {
- FT_Angle theta;
- FT_Int i;
- FT_Fixed x, y, xtemp, b;
- const FT_Angle *arctanptr;
-
-
- x = vec->x;
- y = vec->y;
-
- /* Get the vector into [-PI/4,PI/4] sector */
- if ( y > x )
- {
- if ( y > -x )
- {
- theta = FT_ANGLE_PI2;
- xtemp = y;
- y = -x;
- x = xtemp;
- }
- else
- {
- theta = y > 0 ? FT_ANGLE_PI : -FT_ANGLE_PI;
- x = -x;
- y = -y;
- }
- }
- else
- {
- if ( y < -x )
- {
- theta = -FT_ANGLE_PI2;
- xtemp = -y;
- y = x;
- x = xtemp;
- }
- else
- {
- theta = 0;
- }
- }
-
- arctanptr = ft_trig_arctan_table;
-
- /* Pseudorotations, with right shifts */
- for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
- {
- if ( y > 0 )
- {
- xtemp = x + ( ( y + b ) >> i );
- y = y - ( ( x + b ) >> i );
- x = xtemp;
- theta += *arctanptr++;
- }
- else
- {
- xtemp = x - ( ( y + b ) >> i );
- y = y + ( ( x + b ) >> i );
- x = xtemp;
- theta -= *arctanptr++;
- }
- }
-
- /* round theta to acknowledge its error that mostly comes */
- /* from accumulated rounding errors in the arctan table */
- if ( theta >= 0 )
- theta = FT_PAD_ROUND( theta, 16 );
- else
- theta = -FT_PAD_ROUND( -theta, 16 );
-
- vec->x = x;
- vec->y = theta;
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( FT_Fixed )
- FT_Cos( FT_Angle angle )
- {
- FT_Vector v;
-
-
- FT_Vector_Unit( &v, angle );
-
- return v.x;
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( FT_Fixed )
- FT_Sin( FT_Angle angle )
- {
- FT_Vector v;
-
-
- FT_Vector_Unit( &v, angle );
-
- return v.y;
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( FT_Fixed )
- FT_Tan( FT_Angle angle )
- {
- FT_Vector v;
-
-
- FT_Vector_Unit( &v, angle );
-
- return FT_DivFix( v.y, v.x );
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( FT_Angle )
- FT_Atan2( FT_Fixed dx,
- FT_Fixed dy )
- {
- FT_Vector v;
-
-
- if ( dx == 0 && dy == 0 )
- return 0;
-
- v.x = dx;
- v.y = dy;
- ft_trig_prenorm( &v );
- ft_trig_pseudo_polarize( &v );
-
- return v.y;
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( void )
- FT_Vector_Unit( FT_Vector* vec,
- FT_Angle angle )
- {
- if ( !vec )
- return;
-
- vec->x = FT_TRIG_SCALE >> 8;
- vec->y = 0;
- ft_trig_pseudo_rotate( vec, angle );
- vec->x = ( vec->x + 0x80L ) >> 8;
- vec->y = ( vec->y + 0x80L ) >> 8;
- }
-
-
- /* these macros return 0 for positive numbers,
- and -1 for negative ones */
-#define FT_SIGN_LONG( x ) ( (x) >> ( FT_SIZEOF_LONG * 8 - 1 ) )
-#define FT_SIGN_INT( x ) ( (x) >> ( FT_SIZEOF_INT * 8 - 1 ) )
-#define FT_SIGN_INT32( x ) ( (x) >> 31 )
-#define FT_SIGN_INT16( x ) ( (x) >> 15 )
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( void )
- FT_Vector_Rotate( FT_Vector* vec,
- FT_Angle angle )
- {
- FT_Int shift;
- FT_Vector v;
-
-
- if ( !vec || !angle )
- return;
-
- v = *vec;
-
- if ( v.x == 0 && v.y == 0 )
- return;
-
- shift = ft_trig_prenorm( &v );
- ft_trig_pseudo_rotate( &v, angle );
- v.x = ft_trig_downscale( v.x );
- v.y = ft_trig_downscale( v.y );
-
- if ( shift > 0 )
- {
- FT_Int32 half = (FT_Int32)1L << ( shift - 1 );
-
-
- vec->x = ( v.x + half + FT_SIGN_LONG( v.x ) ) >> shift;
- vec->y = ( v.y + half + FT_SIGN_LONG( v.y ) ) >> shift;
- }
- else
- {
- shift = -shift;
- vec->x = (FT_Pos)( (FT_ULong)v.x << shift );
- vec->y = (FT_Pos)( (FT_ULong)v.y << shift );
- }
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( FT_Fixed )
- FT_Vector_Length( FT_Vector* vec )
- {
- FT_Int shift;
- FT_Vector v;
-
-
- if ( !vec )
- return 0;
-
- v = *vec;
-
- /* handle trivial cases */
- if ( v.x == 0 )
- {
- return FT_ABS( v.y );
- }
- else if ( v.y == 0 )
- {
- return FT_ABS( v.x );
- }
-
- /* general case */
- shift = ft_trig_prenorm( &v );
- ft_trig_pseudo_polarize( &v );
-
- v.x = ft_trig_downscale( v.x );
-
- if ( shift > 0 )
- return ( v.x + ( 1L << ( shift - 1 ) ) ) >> shift;
-
- return (FT_Fixed)( (FT_UInt32)v.x << -shift );
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( void )
- FT_Vector_Polarize( FT_Vector* vec,
- FT_Fixed *length,
- FT_Angle *angle )
- {
- FT_Int shift;
- FT_Vector v;
-
-
- if ( !vec || !length || !angle )
- return;
-
- v = *vec;
-
- if ( v.x == 0 && v.y == 0 )
- return;
-
- shift = ft_trig_prenorm( &v );
- ft_trig_pseudo_polarize( &v );
-
- v.x = ft_trig_downscale( v.x );
-
- *length = shift >= 0 ? ( v.x >> shift )
- : (FT_Fixed)( (FT_UInt32)v.x << -shift );
- *angle = v.y;
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( void )
- FT_Vector_From_Polar( FT_Vector* vec,
- FT_Fixed length,
- FT_Angle angle )
- {
- if ( !vec )
- return;
-
- vec->x = length;
- vec->y = 0;
-
- FT_Vector_Rotate( vec, angle );
- }
-
-
- /* documentation is in fttrigon.h */
-
- FT_EXPORT_DEF( FT_Angle )
- FT_Angle_Diff( FT_Angle angle1,
- FT_Angle angle2 )
- {
- FT_Angle delta = angle2 - angle1;
-
-
- while ( delta <= -FT_ANGLE_PI )
- delta += FT_ANGLE_2PI;
-
- while ( delta > FT_ANGLE_PI )
- delta -= FT_ANGLE_2PI;
-
- return delta;
- }
-
-
-/* END */
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