[sdf] Add essential math functions.

* src/sdf/ftsdf.c (cube_root, arc_cos) [!USE_NEWTON_FOR_CONIC]: New
auxiliary functions.

* src/sdf/ftsdf.c (solve_quadratic_equation, solve_cubic_equation)
[!USE_NEWTON_FOR_CONIC]: New functions.
This commit is contained in:
Anuj Verma 2020-08-18 10:17:46 +05:30 committed by Werner Lemberg
parent cd4138458a
commit a255125fe4
2 changed files with 254 additions and 0 deletions

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@ -1,3 +1,13 @@
2020-08-18 Anuj Verma <anujv@iitbhilai.ac.in>
[sdf] Add essential math functions.
* src/sdf/ftsdf.c (cube_root, arc_cos) [!USE_NEWTON_FOR_CONIC]: New
auxiliary functions.
* src/sdf/ftsdf.c (solve_quadratic_equation, solve_cubic_equation)
[!USE_NEWTON_FOR_CONIC]: New functions.
2020-08-18 Anuj Verma <anujv@iitbhilai.ac.in>
[sdf] Add utility functions for contours.

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@ -1312,4 +1312,248 @@
return error;
}
/**************************************************************************
*
* math functions
*
*/
#if !USE_NEWTON_FOR_CONIC
/* [NOTE]: All the functions below down until rasterizer */
/* can be avoided if we decide to subdivide the */
/* curve into lines. */
/* This function uses Newton's iteration to find */
/* the cube root of a fixed-point integer. */
static FT_16D16
cube_root( FT_16D16 val )
{
/* [IMPORTANT]: This function is not good as it may */
/* not break, so use a lookup table instead. Or we */
/* can use an algorithm similar to `square_root`. */
FT_Int v, g, c;
if ( val == 0 ||
val == -FT_INT_16D16( 1 ) ||
val == FT_INT_16D16( 1 ) )
return val;
v = val < 0 ? -val : val;
g = square_root( v );
c = 0;
while ( 1 )
{
c = FT_MulFix( FT_MulFix( g, g ), g ) - v;
c = FT_DivFix( c, 3 * FT_MulFix( g, g ) );
g -= c;
if ( ( c < 0 ? -c : c ) < 30 )
break;
}
return val < 0 ? -g : g;
}
/* Calculate the perpendicular by using '1 - base^2'. */
/* Then use arctan to compute the angle. */
static FT_16D16
arc_cos( FT_16D16 val )
{
FT_16D16 p;
FT_16D16 b = val;
FT_16D16 one = FT_INT_16D16( 1 );
if ( b > one )
b = one;
if ( b < -one )
b = -one;
p = one - FT_MulFix( b, b );
p = square_root( p );
return FT_Atan2( b, p );
}
/* Compute roots of a quadratic polynomial, assign them to `out`, */
/* and return number of real roots. */
/* */
/* The procedure can be found at */
/* */
/* https://mathworld.wolfram.com/QuadraticFormula.html */
static FT_UShort
solve_quadratic_equation( FT_26D6 a,
FT_26D6 b,
FT_26D6 c,
FT_16D16 out[2] )
{
FT_16D16 discriminant = 0;
a = FT_26D6_16D16( a );
b = FT_26D6_16D16( b );
c = FT_26D6_16D16( c );
if ( a == 0 )
{
if ( b == 0 )
return 0;
else
{
out[0] = FT_DivFix( -c, b );
return 1;
}
}
discriminant = FT_MulFix( b, b ) - 4 * FT_MulFix( a, c );
if ( discriminant < 0 )
return 0;
else if ( discriminant == 0 )
{
out[0] = FT_DivFix( -b, 2 * a );
return 1;
}
else
{
discriminant = square_root( discriminant );
out[0] = FT_DivFix( -b + discriminant, 2 * a );
out[1] = FT_DivFix( -b - discriminant, 2 * a );
return 2;
}
}
/* Compute roots of a cubic polynomial, assign them to `out`, */
/* and return number of real roots. */
/* */
/* The procedure can be found at */
/* */
/* https://mathworld.wolfram.com/CubicFormula.html */
static FT_UShort
solve_cubic_equation( FT_26D6 a,
FT_26D6 b,
FT_26D6 c,
FT_26D6 d,
FT_16D16 out[3] )
{
FT_16D16 q = 0; /* intermediate */
FT_16D16 r = 0; /* intermediate */
FT_16D16 a2 = b; /* x^2 coefficients */
FT_16D16 a1 = c; /* x coefficients */
FT_16D16 a0 = d; /* constant */
FT_16D16 q3 = 0;
FT_16D16 r2 = 0;
FT_16D16 a23 = 0;
FT_16D16 a22 = 0;
FT_16D16 a1x2 = 0;
/* cutoff value for `a` to be a cubic, otherwise solve quadratic */
if ( a == 0 || FT_ABS( a ) < 16 )
return solve_quadratic_equation( b, c, d, out );
if ( d == 0 )
{
out[0] = 0;
return solve_quadratic_equation( a, b, c, out + 1 ) + 1;
}
/* normalize the coefficients; this also makes them 16.16 */
a2 = FT_DivFix( a2, a );
a1 = FT_DivFix( a1, a );
a0 = FT_DivFix( a0, a );
/* compute intermediates */
a1x2 = FT_MulFix( a1, a2 );
a22 = FT_MulFix( a2, a2 );
a23 = FT_MulFix( a22, a2 );
q = ( 3 * a1 - a22 ) / 9;
r = ( 9 * a1x2 - 27 * a0 - 2 * a23 ) / 54;
/* [BUG]: `q3` and `r2` still cause underflow. */
q3 = FT_MulFix( q, q );
q3 = FT_MulFix( q3, q );
r2 = FT_MulFix( r, r );
if ( q3 < 0 && r2 < -q3 )
{
FT_16D16 t = 0;
q3 = square_root( -q3 );
t = FT_DivFix( r, q3 );
if ( t > ( 1 << 16 ) )
t = ( 1 << 16 );
if ( t < -( 1 << 16 ) )
t = -( 1 << 16 );
t = arc_cos( t );
a2 /= 3;
q = 2 * square_root( -q );
out[0] = FT_MulFix( q, FT_Cos( t / 3 ) ) - a2;
out[1] = FT_MulFix( q, FT_Cos( ( t + FT_ANGLE_PI * 2 ) / 3 ) ) - a2;
out[2] = FT_MulFix( q, FT_Cos( ( t + FT_ANGLE_PI * 4 ) / 3 ) ) - a2;
return 3;
}
else if ( r2 == -q3 )
{
FT_16D16 s = 0;
s = cube_root( r );
a2 /= -3;
out[0] = a2 + ( 2 * s );
out[1] = a2 - s;
return 2;
}
else
{
FT_16D16 s = 0;
FT_16D16 t = 0;
FT_16D16 dis = 0;
if ( q3 == 0 )
dis = FT_ABS( r );
else
dis = square_root( q3 + r2 );
s = cube_root( r + dis );
t = cube_root( r - dis );
a2 /= -3;
out[0] = ( a2 + ( s + t ) );
return 1;
}
}
#endif /* !USE_NEWTON_FOR_CONIC */
/* END */