/***************************************************************************/
/* */
/* ftcalc.c */
/* */
/* Arithmetic computations (body). */
/* */
/* Copyright 1996-2016 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. */
/* */
/***************************************************************************/
/*************************************************************************/
/* */
/* Support for 1-complement arithmetic has been totally dropped in this */
/* release. You can still write your own code if you need it. */
/* */
/*************************************************************************/
/*************************************************************************/
/* */
/* Implementing basic computation routines. */
/* */
/* FT_MulDiv(), FT_MulFix(), FT_DivFix(), FT_RoundFix(), FT_CeilFix(), */
/* and FT_FloorFix() are declared in freetype.h. */
/* */
/*************************************************************************/
#include <ft2build.h>
#include FT_GLYPH_H
#include FT_TRIGONOMETRY_H
#include FT_INTERNAL_CALC_H
#include FT_INTERNAL_DEBUG_H
#include FT_INTERNAL_OBJECTS_H
#ifdef FT_MULFIX_ASSEMBLER
#undef FT_MulFix
#endif
/* we need to emulate a 64-bit data type if a real one isn't available */
#ifndef FT_LONG64
typedef struct FT_Int64_
{
FT_UInt32 lo;
FT_UInt32 hi;
} FT_Int64;
#endif /* !FT_LONG64 */
/*************************************************************************/
/* */
/* The macro FT_COMPONENT is used in trace mode. It is an implicit */
/* parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log */
/* messages during execution. */
/* */
#undef FT_COMPONENT
#define FT_COMPONENT trace_calc
/* transfer sign leaving a positive number */
#define FT_MOVE_SIGN( x, s ) \
FT_BEGIN_STMNT \
if ( x < 0 ) \
{ \
x = -x; \
s = -s; \
} \
FT_END_STMNT
/* The following three functions are available regardless of whether */
/* FT_LONG64 is defined. */
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Fixed )
FT_RoundFix( FT_Fixed a )
{
return ( a + 0x8000L - ( a < 0 ) ) & ~0xFFFFL;
}
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Fixed )
FT_CeilFix( FT_Fixed a )
{
return ( a + 0xFFFFL ) & ~0xFFFFL;
}
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Fixed )
FT_FloorFix( FT_Fixed a )
{
return a & ~0xFFFFL;
}
#ifndef FT_MSB
FT_BASE_DEF ( FT_Int )
FT_MSB( FT_UInt32 z )
{
FT_Int shift = 0;
/* determine msb bit index in `shift' */
if ( z & 0xFFFF0000UL )
{
z >>= 16;
shift += 16;
}
if ( z & 0x0000FF00UL )
{
z >>= 8;
shift += 8;
}
if ( z & 0x000000F0UL )
{
z >>= 4;
shift += 4;
}
if ( z & 0x0000000CUL )
{
z >>= 2;
shift += 2;
}
if ( z & 0x00000002UL )
{
/* z >>= 1; */
shift += 1;
}
return shift;
}
#endif /* !FT_MSB */
/* documentation is in ftcalc.h */
FT_BASE_DEF( FT_Fixed )
FT_Hypot( FT_Fixed x,
FT_Fixed y )
{
FT_Vector v;
v.x = x;
v.y = y;
return FT_Vector_Length( &v );
}
#ifdef FT_LONG64
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Long )
FT_MulDiv( FT_Long a_,
FT_Long b_,
FT_Long c_ )
{
FT_Int s = 1;
FT_UInt64 a, b, c, d;
FT_Long d_;
FT_MOVE_SIGN( a_, s );
FT_MOVE_SIGN( b_, s );
FT_MOVE_SIGN( c_, s );
a = (FT_UInt64)a_;
b = (FT_UInt64)b_;
c = (FT_UInt64)c_;
d = c > 0 ? ( a * b + ( c >> 1 ) ) / c
: 0x7FFFFFFFUL;
d_ = (FT_Long)d;
return s < 0 ? -d_ : d_;
}
/* documentation is in ftcalc.h */
FT_BASE_DEF( FT_Long )
FT_MulDiv_No_Round( FT_Long a_,
FT_Long b_,
FT_Long c_ )
{
FT_Int s = 1;
FT_UInt64 a, b, c, d;
FT_Long d_;
FT_MOVE_SIGN( a_, s );
FT_MOVE_SIGN( b_, s );
FT_MOVE_SIGN( c_, s );
a = (FT_UInt64)a_;
b = (FT_UInt64)b_;
c = (FT_UInt64)c_;
d = c > 0 ? a * b / c
: 0x7FFFFFFFUL;
d_ = (FT_Long)d;
return s < 0 ? -d_ : d_;
}
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Long )
FT_MulFix( FT_Long a_,
FT_Long b_ )
{
#ifdef FT_MULFIX_ASSEMBLER
return FT_MULFIX_ASSEMBLER( (FT_Int32)a_, (FT_Int32)b_ );
#else
FT_Int64 ab = (FT_Int64)a_ * (FT_Int64)b_;
/* this requires arithmetic right shift of signed numbers */
return (FT_Long)( ( ab + 0x8000L - ( ab < 0 ) ) >> 16 );
#endif /* FT_MULFIX_ASSEMBLER */
}
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Long )
FT_DivFix( FT_Long a_,
FT_Long b_ )
{
FT_Int s = 1;
FT_UInt64 a, b, q;
FT_Long q_;
FT_MOVE_SIGN( a_, s );
FT_MOVE_SIGN( b_, s );
a = (FT_UInt64)a_;
b = (FT_UInt64)b_;
q = b > 0 ? ( ( a << 16 ) + ( b >> 1 ) ) / b
: 0x7FFFFFFFUL;
q_ = (FT_Long)q;
return s < 0 ? -q_ : q_;
}
#else /* !FT_LONG64 */
static void
ft_multo64( FT_UInt32 x,
FT_UInt32 y,
FT_Int64 *z )
{
FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2;
lo1 = x & 0x0000FFFFU; hi1 = x >> 16;
lo2 = y & 0x0000FFFFU; hi2 = y >> 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 );
z->lo = lo;
z->hi = hi;
}
static FT_UInt32
ft_div64by32( FT_UInt32 hi,
FT_UInt32 lo,
FT_UInt32 y )
{
FT_UInt32 r, q;
FT_Int i;
if ( hi >= y )
return (FT_UInt32)0x7FFFFFFFL;
/* We shift as many bits as we can into the high register, perform */
/* 32-bit division with modulo there, then work through the remaining */
/* bits with long division. This optimization is especially noticeable */
/* for smaller dividends that barely use the high register. */
i = 31 - FT_MSB( hi );
r = ( hi << i ) | ( lo >> ( 32 - i ) ); lo <<= i; /* left 64-bit shift */
q = r / y;
r -= q * y; /* remainder */
i = 32 - i; /* bits remaining in low register */
do
{
q <<= 1;
r = ( r << 1 ) | ( lo >> 31 ); lo <<= 1;
if ( r >= y )
{
r -= y;
q |= 1;
}
} while ( --i );
return q;
}
static void
FT_Add64( FT_Int64* x,
FT_Int64* y,
FT_Int64 *z )
{
FT_UInt32 lo, hi;
lo = x->lo + y->lo;
hi = x->hi + y->hi + ( lo < x->lo );
z->lo = lo;
z->hi = hi;
}
/* The FT_MulDiv function has been optimized thanks to ideas from */
/* Graham Asher and Alexei Podtelezhnikov. The trick is to optimize */
/* a rather common case when everything fits within 32-bits. */
/* */
/* We compute 'a*b+c/2', then divide it by 'c' (all positive values). */
/* */
/* The product of two positive numbers never exceeds the square of */
/* its mean values. Therefore, we always avoid the overflow by */
/* imposing */
/* */
/* (a + b) / 2 <= sqrt(X - c/2) , */
/* */
/* where X = 2^32 - 1, the maximum unsigned 32-bit value, and using */
/* unsigned arithmetic. Now we replace `sqrt' with a linear function */
/* that is smaller or equal for all values of c in the interval */
/* [0;X/2]; it should be equal to sqrt(X) and sqrt(3X/4) at the */
/* endpoints. Substituting the linear solution and explicit numbers */
/* we get */
/* */
/* a + b <= 131071.99 - c / 122291.84 . */
/* */
/* In practice, we should use a faster and even stronger inequality */
/* */
/* a + b <= 131071 - (c >> 16) */
/* */
/* or, alternatively, */
/* */
/* a + b <= 129894 - (c >> 17) . */
/* */
/* FT_MulFix, on the other hand, is optimized for a small value of */
/* the first argument, when the second argument can be much larger. */
/* This can be achieved by scaling the second argument and the limit */
/* in the above inequalities. For example, */
/* */
/* a + (b >> 8) <= (131071 >> 4) */
/* */
/* covers the practical range of use. The actual test below is a bit */
/* tighter to avoid the border case overflows. */
/* */
/* In the case of FT_DivFix, the exact overflow check */
/* */
/* a << 16 <= X - c/2 */
/* */
/* is scaled down by 2^16 and we use */
/* */
/* a <= 65535 - (c >> 17) . */
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Long )
FT_MulDiv( FT_Long a_,
FT_Long b_,
FT_Long c_ )
{
FT_Int s = 1;
FT_UInt32 a, b, c;
/* XXX: this function does not allow 64-bit arguments */
FT_MOVE_SIGN( a_, s );
FT_MOVE_SIGN( b_, s );
FT_MOVE_SIGN( c_, s );
a = (FT_UInt32)a_;
b = (FT_UInt32)b_;
c = (FT_UInt32)c_;
if ( c == 0 )
a = 0x7FFFFFFFUL;
else if ( a + b <= 129894UL - ( c >> 17 ) )
a = ( a * b + ( c >> 1 ) ) / c;
else
{
FT_Int64 temp, temp2;
ft_multo64( a, b, &temp );
temp2.hi = 0;
temp2.lo = c >> 1;
FT_Add64( &temp, &temp2, &temp );
/* last attempt to ditch long division */
a = temp.hi == 0 ? temp.lo / c
: ft_div64by32( temp.hi, temp.lo, c );
}
a_ = (FT_Long)a;
return s < 0 ? -a_ : a_;
}
FT_BASE_DEF( FT_Long )
FT_MulDiv_No_Round( FT_Long a_,
FT_Long b_,
FT_Long c_ )
{
FT_Int s = 1;
FT_UInt32 a, b, c;
/* XXX: this function does not allow 64-bit arguments */
FT_MOVE_SIGN( a_, s );
FT_MOVE_SIGN( b_, s );
FT_MOVE_SIGN( c_, s );
a = (FT_UInt32)a_;
b = (FT_UInt32)b_;
c = (FT_UInt32)c_;
if ( c == 0 )
a = 0x7FFFFFFFUL;
else if ( a + b <= 131071UL )
a = a * b / c;
else
{
FT_Int64 temp;
ft_multo64( a, b, &temp );
/* last attempt to ditch long division */
a = temp.hi == 0 ? temp.lo / c
: ft_div64by32( temp.hi, temp.lo, c );
}
a_ = (FT_Long)a;
return s < 0 ? -a_ : a_;
}
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Long )
FT_MulFix( FT_Long a_,
FT_Long b_ )
{
#ifdef FT_MULFIX_ASSEMBLER
return FT_MULFIX_ASSEMBLER( a_, b_ );
#elif 0
/*
* This code is nonportable. See comment below.
*
* However, on a platform where right-shift of a signed quantity fills
* the leftmost bits by copying the sign bit, it might be faster.
*/
FT_Long sa, sb;
FT_UInt32 a, b;
/*
* This is a clever way of converting a signed number `a' into its
* absolute value (stored back into `a') and its sign. The sign is
* stored in `sa'; 0 means `a' was positive or zero, and -1 means `a'
* was negative. (Similarly for `b' and `sb').
*
* Unfortunately, it doesn't work (at least not portably).
*
* It makes the assumption that right-shift on a negative signed value
* fills the leftmost bits by copying the sign bit. This is wrong.
* According to K&R 2nd ed, section `A7.8 Shift Operators' on page 206,
* the result of right-shift of a negative signed value is
* implementation-defined. At least one implementation fills the
* leftmost bits with 0s (i.e., it is exactly the same as an unsigned
* right shift). This means that when `a' is negative, `sa' ends up
* with the value 1 rather than -1. After that, everything else goes
* wrong.
*/
sa = ( a_ >> ( sizeof ( a_ ) * 8 - 1 ) );
a = ( a_ ^ sa ) - sa;
sb = ( b_ >> ( sizeof ( b_ ) * 8 - 1 ) );
b = ( b_ ^ sb ) - sb;
a = (FT_UInt32)a_;
b = (FT_UInt32)b_;
if ( a + ( b >> 8 ) <= 8190UL )
a = ( a * b + 0x8000U ) >> 16;
else
{
FT_UInt32 al = a & 0xFFFFUL;
a = ( a >> 16 ) * b + al * ( b >> 16 ) +
( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 );
}
sa ^= sb;
a = ( a ^ sa ) - sa;
return (FT_Long)a;
#else /* 0 */
FT_Int s = 1;
FT_UInt32 a, b;
/* XXX: this function does not allow 64-bit arguments */
FT_MOVE_SIGN( a_, s );
FT_MOVE_SIGN( b_, s );
a = (FT_UInt32)a_;
b = (FT_UInt32)b_;
if ( a + ( b >> 8 ) <= 8190UL )
a = ( a * b + 0x8000UL ) >> 16;
else
{
FT_UInt32 al = a & 0xFFFFUL;
a = ( a >> 16 ) * b + al * ( b >> 16 ) +
( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 );
}
a_ = (FT_Long)a;
return s < 0 ? -a_ : a_;
#endif /* 0 */
}
/* documentation is in freetype.h */
FT_EXPORT_DEF( FT_Long )
FT_DivFix( FT_Long a_,
FT_Long b_ )
{
FT_Int s = 1;
FT_UInt32 a, b, q;
FT_Long q_;
/* XXX: this function does not allow 64-bit arguments */
FT_MOVE_SIGN( a_, s );
FT_MOVE_SIGN( b_, s );
a = (FT_UInt32)a_;
b = (FT_UInt32)b_;
if ( b == 0 )
{
/* check for division by 0 */
q = 0x7FFFFFFFUL;
}
else if ( a <= 65535UL - ( b >> 17 ) )
{
/* compute result directly */
q = ( ( a << 16 ) + ( b >> 1 ) ) / b;
}
else
{
/* we need more bits; we have to do it by hand */
FT_Int64 temp, temp2;
temp.hi = a >> 16;
temp.lo = a << 16;
temp2.hi = 0;
temp2.lo = b >> 1;
FT_Add64( &temp, &temp2, &temp );
q = ft_div64by32( temp.hi, temp.lo, b );
}
q_ = (FT_Long)q;
return s < 0 ? -q_ : q_;
}
#endif /* !FT_LONG64 */
/* documentation is in ftglyph.h */
FT_EXPORT_DEF( void )
FT_Matrix_Multiply( const FT_Matrix* a,
FT_Matrix *b )
{
FT_Fixed xx, xy, yx, yy;
if ( !a || !b )
return;
xx = FT_MulFix( a->xx, b->xx ) + FT_MulFix( a->xy, b->yx );
xy = FT_MulFix( a->xx, b->xy ) + FT_MulFix( a->xy, b->yy );
yx = FT_MulFix( a->yx, b->xx ) + FT_MulFix( a->yy, b->yx );
yy = FT_MulFix( a->yx, b->xy ) + FT_MulFix( a->yy, b->yy );
b->xx = xx; b->xy = xy;
b->yx = yx; b->yy = yy;
}
/* documentation is in ftglyph.h */
FT_EXPORT_DEF( FT_Error )
FT_Matrix_Invert( FT_Matrix* matrix )
{
FT_Pos delta, xx, yy;
if ( !matrix )
return FT_THROW( Invalid_Argument );
/* compute discriminant */
delta = FT_MulFix( matrix->xx, matrix->yy ) -
FT_MulFix( matrix->xy, matrix->yx );
if ( !delta )
return FT_THROW( Invalid_Argument ); /* matrix can't be inverted */
matrix->xy = - FT_DivFix( matrix->xy, delta );
matrix->yx = - FT_DivFix( matrix->yx, delta );
xx = matrix->xx;
yy = matrix->yy;
matrix->xx = FT_DivFix( yy, delta );
matrix->yy = FT_DivFix( xx, delta );
return FT_Err_Ok;
}
/* documentation is in ftcalc.h */
FT_BASE_DEF( void )
FT_Matrix_Multiply_Scaled( const FT_Matrix* a,
FT_Matrix *b,
FT_Long scaling )
{
FT_Fixed xx, xy, yx, yy;
FT_Long val = 0x10000L * scaling;
if ( !a || !b )
return;
xx = FT_MulDiv( a->xx, b->xx, val ) + FT_MulDiv( a->xy, b->yx, val );
xy = FT_MulDiv( a->xx, b->xy, val ) + FT_MulDiv( a->xy, b->yy, val );
yx = FT_MulDiv( a->yx, b->xx, val ) + FT_MulDiv( a->yy, b->yx, val );
yy = FT_MulDiv( a->yx, b->xy, val ) + FT_MulDiv( a->yy, b->yy, val );
b->xx = xx; b->xy = xy;
b->yx = yx; b->yy = yy;
}
/* documentation is in ftcalc.h */
FT_BASE_DEF( void )
FT_Vector_Transform_Scaled( FT_Vector* vector,
const FT_Matrix* matrix,
FT_Long scaling )
{
FT_Pos xz, yz;
FT_Long val = 0x10000L * scaling;
if ( !vector || !matrix )
return;
xz = FT_MulDiv( vector->x, matrix->xx, val ) +
FT_MulDiv( vector->y, matrix->xy, val );
yz = FT_MulDiv( vector->x, matrix->yx, val ) +
FT_MulDiv( vector->y, matrix->yy, val );
vector->x = xz;
vector->y = yz;
}
/* documentation is in ftcalc.h */
FT_BASE_DEF( FT_UInt32 )
FT_Vector_NormLen( FT_Vector* vector )
{
FT_Int32 x_ = vector->x;
FT_Int32 y_ = vector->y;
FT_Int32 b, z;
FT_UInt32 x, y, u, v, l;
FT_Int sx = 1, sy = 1, shift;
FT_MOVE_SIGN( x_, sx );
FT_MOVE_SIGN( y_, sy );
x = (FT_UInt32)x_;
y = (FT_UInt32)y_;
/* trivial cases */
if ( x == 0 )
{
if ( y > 0 )
vector->y = sy * 0x10000;
return y;
}
else if ( y == 0 )
{
if ( x > 0 )
vector->x = sx * 0x10000;
return x;
}
/* Estimate length and prenormalize by shifting so that */
/* the new approximate length is between 2/3 and 4/3. */
/* The magic constant 0xAAAAAAAAUL (2/3 of 2^32) helps */
/* achieve this in 16.16 fixed-point representation. */
l = x > y ? x + ( y >> 1 )
: y + ( x >> 1 );
shift = 31 - FT_MSB( l );
shift -= 15 + ( l >= ( 0xAAAAAAAAUL >> shift ) );
if ( shift > 0 )
{
x <<= shift;
y <<= shift;
/* re-estimate length for tiny vectors */
l = x > y ? x + ( y >> 1 )
: y + ( x >> 1 );
}
else
{
x >>= -shift;
y >>= -shift;
l >>= -shift;
}
/* lower linear approximation for reciprocal length minus one */
b = 0x10000 - (FT_Int32)l;
x_ = (FT_Int32)x;
y_ = (FT_Int32)y;
/* Newton's iterations */
do
{
u = (FT_UInt32)( x_ + ( x_ * b >> 16 ) );
v = (FT_UInt32)( y_ + ( y_ * b >> 16 ) );
/* Normalized squared length in the parentheses approaches 2^32. */
/* On two's complement systems, converting to signed gives the */
/* difference with 2^32 even if the expression wraps around. */
z = -(FT_Int32)( u * u + v * v ) / 0x200;
z = z * ( ( 0x10000 + b ) >> 8 ) / 0x10000;
b += z;
} while ( z > 0 );
vector->x = sx < 0 ? -(FT_Pos)u : (FT_Pos)u;
vector->y = sy < 0 ? -(FT_Pos)v : (FT_Pos)v;
/* Conversion to signed helps to recover from likely wrap around */
/* in calculating the prenormalized length, because it gives the */
/* correct difference with 2^32 on two's complement systems. */
l = (FT_UInt32)( 0x10000 + (FT_Int32)( u * x + v * y ) / 0x10000 );
if ( shift > 0 )
l = ( l + ( 1 << ( shift - 1 ) ) ) >> shift;
else
l <<= -shift;
return l;
}
#if 0
/* documentation is in ftcalc.h */
FT_BASE_DEF( FT_Int32 )
FT_SqrtFixed( FT_Int32 x )
{
FT_UInt32 root, rem_hi, rem_lo, test_div;
FT_Int count;
root = 0;
if ( x > 0 )
{
rem_hi = 0;
rem_lo = (FT_UInt32)x;
count = 24;
do
{
rem_hi = ( rem_hi << 2 ) | ( rem_lo >> 30 );
rem_lo <<= 2;
root <<= 1;
test_div = ( root << 1 ) + 1;
if ( rem_hi >= test_div )
{
rem_hi -= test_div;
root += 1;
}
} while ( --count );
}
return (FT_Int32)root;
}
#endif /* 0 */
/* documentation is in ftcalc.h */
FT_BASE_DEF( FT_Int )
ft_corner_orientation( FT_Pos in_x,
FT_Pos in_y,
FT_Pos out_x,
FT_Pos out_y )
{
#ifdef FT_LONG64
FT_Int64 delta = (FT_Int64)in_x * out_y - (FT_Int64)in_y * out_x;
return ( delta > 0 ) - ( delta < 0 );
#else
FT_Int result;
if ( (FT_ULong)FT_ABS( in_x ) + (FT_ULong)FT_ABS( out_y ) <= 131071UL &&
(FT_ULong)FT_ABS( in_y ) + (FT_ULong)FT_ABS( out_x ) <= 131071UL )
{
FT_Long z1 = in_x * out_y;
FT_Long z2 = in_y * out_x;
if ( z1 > z2 )
result = +1;
else if ( z1 < z2 )
result = -1;
else
result = 0;
}
else /* products might overflow 32 bits */
{
FT_Int64 z1, z2;
/* XXX: this function does not allow 64-bit arguments */
ft_multo64( (FT_UInt32)in_x, (FT_UInt32)out_y, &z1 );
ft_multo64( (FT_UInt32)in_y, (FT_UInt32)out_x, &z2 );
if ( z1.hi > z2.hi )
result = +1;
else if ( z1.hi < z2.hi )
result = -1;
else if ( z1.lo > z2.lo )
result = +1;
else if ( z1.lo < z2.lo )
result = -1;
else
result = 0;
}
/* XXX: only the sign of return value, +1/0/-1 must be used */
return result;
#endif
}
/* documentation is in ftcalc.h */
FT_BASE_DEF( FT_Int )
ft_corner_is_flat( FT_Pos in_x,
FT_Pos in_y,
FT_Pos out_x,
FT_Pos out_y )
{
FT_Pos ax = in_x + out_x;
FT_Pos ay = in_y + out_y;
FT_Pos d_in, d_out, d_hypot;
/* The idea of this function is to compare the length of the */
/* hypotenuse with the `in' and `out' length. The `corner' */
/* represented by `in' and `out' is flat if the hypotenuse's */
/* length isn't too large. */
/* */
/* This approach has the advantage that the angle between */
/* `in' and `out' is not checked. In case one of the two */
/* vectors is `dominant', this is, much larger than the */
/* other vector, we thus always have a flat corner. */
/* */
/* hypotenuse */
/* x---------------------------x */
/* \ / */
/* \ / */
/* in \ / out */
/* \ / */
/* o */
/* Point */
d_in = FT_HYPOT( in_x, in_y );
d_out = FT_HYPOT( out_x, out_y );
d_hypot = FT_HYPOT( ax, ay );
/* now do a simple length comparison: */
/* */
/* d_in + d_out < 17/16 d_hypot */
return ( d_in + d_out - d_hypot ) < ( d_hypot >> 4 );
}
/* END */
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