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+/***************************************************************************/
+/* */
+/* fttrigon.c */
+/* */
+/* FreeType trigonometric functions (body). */
+/* */
+/* Copyright 2001-2005, 2012-2013 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_Fixed
+ 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_Fixed s;
+ FT_Int64 v;
+
+
+ s = val;
+ val = FT_ABS( val );
+
+ v = ( val * (FT_Int64)FT_TRIG_SCALE ) + 0x100000000UL;
+ val = (FT_Fixed)( v >> 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_Fixed s;
+ FT_UInt32 v1, v2, k1, k2, hi, lo1, lo2, lo3;
+
+
+ s = val;
+ val = FT_ABS( val );
+
+ v1 = (FT_UInt32)val >> 16;
+ v2 = (FT_UInt32)( val & 0xFFFFL );
+
+ k1 = (FT_UInt32)FT_TRIG_SCALE >> 16; /* constant */
+ k2 = (FT_UInt32)( FT_TRIG_SCALE & 0xFFFFL ); /* constant */
+
+ hi = k1 * v1;
+ lo1 = k1 * v2 + k2 * v1; /* can't overflow */
+
+ lo2 = ( k2 * v2 ) >> 16;
+ lo3 = FT_MAX( lo1, lo2 );
+ lo1 += lo2;
+
+ hi += lo1 >> 16;
+ if ( lo1 < lo3 )
+ hi += (FT_UInt32)0x10000UL;
+
+ val = (FT_Fixed)hi;
+
+ return ( s >= 0 ) ? val : -val;
+ }
+
+#endif /* !FT_LONG64 */
+
+
+ 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_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_Fixed *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_Fixed *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 */
+ if ( theta >= 0 )
+ theta = FT_PAD_ROUND( theta, 32 );
+ else
+ theta = -FT_PAD_ROUND( -theta, 32 );
+
+ vec->x = x;
+ vec->y = theta;
+ }
+
+
+ /* documentation is in fttrigon.h */
+
+ FT_EXPORT_DEF( FT_Fixed )
+ FT_Cos( FT_Angle angle )
+ {
+ FT_Vector v;
+
+
+ v.x = FT_TRIG_SCALE >> 8;
+ v.y = 0;
+ ft_trig_pseudo_rotate( &v, angle );
+
+ return ( v.x + 0x80L ) >> 8;
+ }
+
+
+ /* documentation is in fttrigon.h */
+
+ FT_EXPORT_DEF( FT_Fixed )
+ FT_Sin( FT_Angle angle )
+ {
+ return FT_Cos( FT_ANGLE_PI2 - angle );
+ }
+
+
+ /* documentation is in fttrigon.h */
+
+ FT_EXPORT_DEF( FT_Fixed )
+ FT_Tan( FT_Angle angle )
+ {
+ FT_Vector v;
+
+
+ v.x = FT_TRIG_SCALE >> 8;
+ v.y = 0;
+ ft_trig_pseudo_rotate( &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 )
+ {
+ 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;
+
+
+ v.x = vec->x;
+ v.y = vec->y;
+
+ if ( angle && ( v.x != 0 || v.y != 0 ) )
+ {
+ 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;
+
+
+ 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 + ( 1 << ( 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;
+
+
+ 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 )
+ {
+ 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;
+
+
+ delta %= FT_ANGLE_2PI;
+ if ( delta < 0 )
+ delta += FT_ANGLE_2PI;
+
+ if ( delta > FT_ANGLE_PI )
+ delta -= FT_ANGLE_2PI;
+
+ return delta;
+ }
+
+
+/* END */
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