/***************************************************************************/ | |
/* */ | |
/* fttrigon.c */ | |
/* */ | |
/* FreeType trigonometric functions (body). */ | |
/* */ | |
/* Copyright 2001-2015 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_Int64)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 */ |