src/bezier.h

   1 #ifndef _BEZIER_H
   2 #define _BEZIER_H
   3
   4 #include "real.h"
   5 #include "rect.h"
   6 namespace IPDF
   7 {
   8         extern int Factorial(int n);
   9         extern int BinomialCoeff(int n, int k);
  10         extern Real Bernstein(int k, int n, const Real & u);
  11
  12         inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
  13         {
  14                 Real x0((-b + Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  15                 Real x1((-b - Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  16                 return std::pair<Real,Real>(x0,x1);
  17         }
  18
  19         inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
  20         {
  21                 // This is going to be a big one...
  22                 // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
  23
  24                 // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
  25                 /*
  26                 Real discriminant = Real(18) * a * b * c * d - Real(4) * (b * b * b) * d
  27                                 + (b * b) * (c * c) - Real(4) * a * (c * c * c)
  28                                 - Real(27) * (a * a) * (d * d);
  29                 */
  30                 // discriminant > 0 => 3 distinct, real roots.
  31                 // discriminant = 0 => a multiple root (1 or 2 real roots)
  32                 // discriminant < 0 => 1 real root, 2 complex conjugate roots
  33
  34                 ////HACK: We know any roots we care about will be between 0 and 1, so...
  35                 Real maxi(100);
  36                 Real prevRes(d);
  37                 std::vector<Real> roots;
  38                 for(int i = 0; i <= 100; ++i)
  39                 {
  40                         Real x(i);
  41                         x /= maxi;
  42                         Real y = a*(x*x*x) + b*(x*x) + c*x + d;
  43                         if (y == Real(0) || (y < Real(0) && prevRes > Real(0)) || (y > Real(0) && prevRes < Real(0)))
  44                         {
  45                                 roots.push_back(x);
  46                         }
  47                 }
  48                 return roots;
  49
  50         }
  51
  52
  53         /** A _cubic_ bezier. **/
  54         struct Bezier
  55         {
  56                 Real x0; Real y0;
  57                 Real x1; Real y1;
  58                 Real x2; Real y2;
  59                 Real x3; Real y3;
  60                 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  61                 Bezier(Real _x0, Real _y0, Real _x1, Real _y1, Real _x2, Real _y2, Real _x3, Real _y3) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x3), y3(_y3)
  62                 {
  63
  64                 }
  65
  66                 Bezier(Real _x0, Real _y0, Real _x1, Real _y1, Real _x2, Real _y2) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x2), y3(_y2) {}
  67
  68                 std::string Str() const
  69                 {
  70                         std::stringstream s;
  71                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  72                         return s.str();
  73                 }
  74
  75                 /**
  76                  * Construct absolute control points using relative control points to a bounding rectangle
  77                  * ie: If cpy is relative to bounds rectangle, this will be absolute
  78                  */
  79                 Bezier(const Bezier & cpy, const Rect & t = Rect(0,0,1,1)) : x0(cpy.x0), y0(cpy.y0), x1(cpy.x1), y1(cpy.y1), x2(cpy.x2),y2(cpy.y2), x3(cpy.x3), y3(cpy.y3)
  80                 {
  81                         x0 *= t.w;
  82                         y0 *= t.h;
  83                         x1 *= t.w;
  84                         y1 *= t.h;
  85                         x2 *= t.w;
  86                         y2 *= t.h;
  87                         x3 *= t.w;
  88                         y3 *= t.h;
  89                         x0 += t.x;
  90                         y0 += t.y;
  91                         x1 += t.x;
  92                         y1 += t.y;
  93                         x2 += t.x;
  94                         y2 += t.y;
  95                         x3 += t.x;
  96                         y3 += t.y;
  97                 }
  98
  99                 Rect SolveBounds() const;
 100
 101                 Bezier ToAbsolute(const Rect & bounds) const
 102                 {
 103                         return Bezier(*this, bounds);
 104                 }
 105
 106                 /** Convert absolute control points to control points relative to bounds
 107                  * (This basically does the opposite of the Copy constructor)
 108                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 109                  */
 110                 Bezier ToRelative(const Rect & bounds) const
 111                 {
 112                         // x' <- (x - x0)/w etc
 113                         // special cases when w or h = 0
 114                         // (So can't just use the Copy constructor on the inverse of bounds)
 115                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 116                         Bezier result;
 117                         if (bounds.w == 0)
 118                         {
 119                                 result.x0 = 0;
 120                                 result.x1 = 0;
 121                                 result.x2 = 0;
 122                                 result.x3 = 0;
 123                         }
 124                         else
 125                         {
 126                                 result.x0 = (x0 - bounds.x)/bounds.w;
 127                                 result.x1 = (x1 - bounds.x)/bounds.w;
 128                                 result.x2 = (x2 - bounds.x)/bounds.w;
 129                                 result.x3 = (x3 - bounds.x)/bounds.w;
 130                         }
 131
 132                         if (bounds.h == 0)
 133                         {
 134                                 result.y0 = 0;
 135                                 result.y1 = 0;
 136                                 result.y2 = 0;
 137                                 result.y3 = 0;
 138                         }
 139                         else
 140                         {
 141                                 result.y0 = (y0 - bounds.y)/bounds.h;
 142                                 result.y1 = (y1 - bounds.y)/bounds.h;
 143                                 result.y2 = (y2 - bounds.y)/bounds.h;
 144                                 result.y3 = (y3 - bounds.y)/bounds.h;
 145                         }
 146                         return result;
 147                 }
 148
 149
 150                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 151                 void Evaluate(Real & x, Real & y, const Real & u) const
 152                 {
 153                         Real coeff[4];
 154                         for (unsigned i = 0; i < 4; ++i)
 155                                 coeff[i] = Bernstein(i,3,u);
 156                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 157                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 158                 }
 159
 160         };
 161
 162
 163
 164 }
 165
 166 #endif //_BEZIER_H