[sdf] Added functions to solve polynomial equations.
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Anuj Verma
anujverma
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@ -1,3 +1,12 @@
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2020-06-30 Anuj Verma <anujv@iitbhilai.ac.in>
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[sdf] Added functions to solve polynomial equations.
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* src/sdf/ftsdf.c (solve_quadratic_equation,
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solve_cubic_equation): Added functions to solve
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quadratic and cubic equations. These will be used for
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conic bezier curves only.
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2020-06-30 Anuj Verma <anujv@iitbhilai.ac.in>
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* src/sdf/ftsdf.c (square_root, cube_root, arc_cos):
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171
src/sdf/ftsdf.c
171
src/sdf/ftsdf.c
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@ -625,6 +625,10 @@
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return q;
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}
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/* [NOTE]: All the functions below down until rasterizer */
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/* can be avoided if we decide to subdivide the */
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/* curve into lines. */
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/* This function uses newton's iteration to find */
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/* cube root of a fixed point integer. */
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static FT_16D16
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@ -678,6 +682,173 @@
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return FT_Atan2( b, p );
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}
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/* The function compute the roots of a quadratic */
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/* polynomial, assigns it to `out' and returns the */
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/* number of real roots of the equation. */
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/* The procedure can be found at: */
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/* https://mathworld.wolfram.com/QuadraticFormula.html */
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static FT_UShort
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solve_quadratic_equation( FT_26D6 a,
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FT_26D6 b,
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FT_26D6 c,
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FT_16D16 out[2] )
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{
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FT_16D16 discriminant = 0;
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a = FT_26D6_16D16( a );
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b = FT_26D6_16D16( b );
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c = FT_26D6_16D16( c );
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if ( a == 0 )
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{
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if ( b == 0 )
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{
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return 0;
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}
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else
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{
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out[0] = FT_DivFix( -c, b );
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return 1;
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}
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}
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discriminant = FT_MulFix( b, b ) - 4 * FT_MulFix( a, c );
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if ( discriminant < 0 )
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{
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return 0;
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}
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else if ( discriminant == 0 )
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{
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out[0] = FT_DivFix( -b, 2 * a );
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return 1;
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}
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else
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{
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discriminant = square_root( discriminant );
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out[0] = FT_DivFix( -b + discriminant, 2 * a );
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out[1] = FT_DivFix( -b - discriminant, 2 * a );
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return 2;
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}
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}
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/* The function compute the roots of a cubic polynomial */
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/* assigns it to `out' and returns the umber of real */
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/* roots of the equation. */
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/* The procedure can be found at: */
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/* https://mathworld.wolfram.com/QuadraticFormula.html */
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static FT_UShort
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solve_cubic_equation( FT_26D6 a,
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FT_26D6 b,
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FT_26D6 c,
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FT_26D6 d,
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FT_16D16 out[3] )
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{
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FT_16D16 q = 0; /* intermediate */
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FT_16D16 r = 0; /* intermediate */
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FT_16D16 a2 = b; /* x^2 coefficients */
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FT_16D16 a1 = c; /* x coefficients */
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FT_16D16 a0 = d; /* constant */
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FT_16D16 q3 = 0;
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FT_16D16 r2 = 0;
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FT_16D16 a23 = 0;
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FT_16D16 a22 = 0;
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FT_16D16 a1x2 = 0;
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/* cutoff value for `a' to be a cubic */
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if ( a == 0 || FT_ABS( a ) < 16 )
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{
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/* quadratic equation */
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return solve_quadratic_equation( b, c, d, out );
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}
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if ( d == 0 )
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{
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out[0] = 0;
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return solve_quadratic_equation( a, b, c, out + 1 ) + 1;
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}
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/* normalize the coefficients, this also makes them 16.16 */
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a2 = FT_DivFix( a2, a );
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a1 = FT_DivFix( a1, a );
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a0 = FT_DivFix( a0, a );
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/* compute intermediates */
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a1x2 = FT_MulFix( a1, a2 );
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a22 = FT_MulFix( a2, a2 );
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a23 = FT_MulFix( a22, a2 );
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q = ( 3 * a1 - a22 ) / 9;
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r = ( 9 * a1x2 - 27 * a0 - 2 * a23 ) / 54;
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/* [BUG]: `q3' and `r2' still causes underflow. */
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q3 = FT_MulFix( q, q );
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q3 = FT_MulFix( q3, q );
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r2 = FT_MulFix( r, r );
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if ( q3 < 0 && r2 < -q3 )
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{
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FT_16D16 t = 0;
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q3 = square_root( -q3 );
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t = FT_DivFix( r, q3 );
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if ( t > ( 1 << 16 ) ) t = ( 1 << 16 );
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if ( t < -( 1 << 16 ) ) t = -( 1 << 16 );
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t = arc_cos( t );
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a2 /= 3;
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q = 2 * square_root( -q );
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out[0] = FT_MulFix( q, FT_Cos( t / 3 ) ) - a2;
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out[1] = FT_MulFix( q, FT_Cos( ( t + FT_ANGLE_PI * 2 ) / 3 ) ) - a2;
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out[2] = FT_MulFix( q, FT_Cos( ( t + FT_ANGLE_PI * 4 ) / 3 ) ) - a2;
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return 3;
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}
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else if ( r2 == -q3 )
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{
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FT_16D16 s = 0;
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s = cube_root( r );
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a2 /= -3;
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out[0] = a2 + ( 2 * s );
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out[1] = a2 - s;
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return 2;
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}
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else
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{
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FT_16D16 s = 0;
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FT_16D16 t = 0;
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FT_16D16 dis = 0;
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if ( q3 == 0 )
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{
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dis = FT_ABS( r );
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}
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else
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{
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dis = square_root( q3 + r2 );
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}
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s = cube_root( r + dis );
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t = cube_root( r - dis );
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a2 /= -3;
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out[0] = ( a2 + ( s + t ) );
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return 1;
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}
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}
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/*************************************************************************/
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/*************************************************************************/
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/** **/
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