forked from minhngoc25a/freetype2
[sdf] Added functions to solve polynomial equations.
This commit is contained in:
Anuj Verma
anujverma
parent
92a5312036
commit
f1de833a9d
|
|
@ -1,3 +1,12 @@
|
|||
2020-06-30 Anuj Verma <anujv@iitbhilai.ac.in>
|
||||
|
||||
[sdf] Added functions to solve polynomial equations.
|
||||
|
||||
* src/sdf/ftsdf.c (solve_quadratic_equation,
|
||||
solve_cubic_equation): Added functions to solve
|
||||
quadratic and cubic equations. These will be used for
|
||||
conic bezier curves only.
|
||||
|
||||
2020-06-30 Anuj Verma <anujv@iitbhilai.ac.in>
|
||||
|
||||
* src/sdf/ftsdf.c (square_root, cube_root, arc_cos):
|
||||
|
|
|
|||
171
src/sdf/ftsdf.c
171
src/sdf/ftsdf.c
|
|
@ -625,6 +625,10 @@
|
|||
return q;
|
||||
}
|
||||
|
||||
/* [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 */
|
||||
/* cube root of a fixed point integer. */
|
||||
static FT_16D16
|
||||
|
|
@ -678,6 +682,173 @@
|
|||
return FT_Atan2( b, p );
|
||||
}
|
||||
|
||||
/* The function compute the roots of a quadratic */
|
||||
/* polynomial, assigns it to `out' and returns the */
|
||||
/* number of real roots of the equation. */
|
||||
/* 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;
|
||||
}
|
||||
}
|
||||
|
||||
/* The function compute the roots of a cubic polynomial */
|
||||
/* assigns it to `out' and returns the umber of real */
|
||||
/* roots of the equation. */
|
||||
/* The procedure can be found at: */
|
||||
/* https://mathworld.wolfram.com/QuadraticFormula.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 */
|
||||
if ( a == 0 || FT_ABS( a ) < 16 )
|
||||
{
|
||||
/* quadratic equation */
|
||||
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 causes 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;
|
||||
}
|
||||
}
|
||||
|
||||
/*************************************************************************/
|
||||
/*************************************************************************/
|
||||
/** **/
|
||||
|
|
|
|||
Loading…
Reference in New Issue