[sdf] Added essential math functions.

* src/sdf/ftsdf.c (cube_root, arc_cos): Added functions to compute cube root and
  cosine inverse of a 16.16 fixed point number.

* src/sdf/ftsdf.c (solve_quadratic_equation, solve_cubic_equation): Added
  functions to find roots of quadratic and cubic polynomial equations.

src/sdf/ftsdf.c

@@ -1251,4 +1251,228 @@
     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 */
+  /* 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 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;
+  }
+
+  /* The function calculate the perpendicular */
+  /* using 1 - ( base ^ 2 ) and then use arc  */
+  /* tan to compute the angle.                */
+  static FT_16D16
+  arc_cos( FT_16D16  val )
+  {
+    FT_16D16  p, 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 );
+  }
+
+  /* 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 number of real   */
+  /* roots of the equation.                               */
+  /* 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 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;
+    }
+  }
+
+#endif
+
 /* END */