[freetype2] anuj-distance-field 0a12f0e 22/95: [sdf] Added functions to solve polynomial equations.

[Anuj Verma]

branch: anuj-distance-field
commit 0a12f0e9301eea1ee40f7c9ed0ae6953f9859003
Author: Anuj Verma <anujv@iitbhilai.ac.in>
Commit: Anuj Verma <anujv@iitbhilai.ac.in>

    [sdf] Added functions to solve polynomial equations.
---
 [GSoC]ChangeLog |   9 +++
 src/sdf/ftsdf.c | 171 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 180 insertions(+)

diff --git a/[GSoC]ChangeLog b/[GSoC]ChangeLog
index 47509fb..74fa792 100644
--- a/[GSoC]ChangeLog
+++ b/[GSoC]ChangeLog
@@ -1,5 +1,14 @@
 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):
          Use FT_16D16 instead of FT_Fixed to avoid confusion.
 
diff --git a/src/sdf/ftsdf.c b/src/sdf/ftsdf.c
index ed20df9..5691643 100644
--- a/src/sdf/ftsdf.c
+++ b/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;
+    }
+  }
+
   /*************************************************************************/
   /*************************************************************************/
   /**                                                                     **/