/***************************************************************************/
/* */
/* 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 */