src/bezier.h

   1 #ifndef _BEZIER_H
   2 #define _BEZIER_H
   3
   4 #include <vector>
   5 #include <algorithm>
   6
   7 #include "real.h"
   8 #include "rect.h"
   9 namespace IPDF
  10 {
  11         extern int Factorial(int n);
  12         extern int BinomialCoeff(int n, int k);
  13         extern Real Bernstein(int k, int n, const Real & u);
  14
  15         inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
  16         {
  17                 Real x0((-b + Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  18                 Real x1((-b - Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  19                 return std::pair<Real,Real>(x0,x1);
  20         }
  21
  22         inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
  23         {
  24                 // This is going to be a big one...
  25                 // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
  26
  27                 // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
  28                 /*
  29                 Real discriminant = Real(18) * a * b * c * d - Real(4) * (b * b * b) * d
  30                                 + (b * b) * (c * c) - Real(4) * a * (c * c * c)
  31                                 - Real(27) * (a * a) * (d * d);
  32                 */
  33                 // discriminant > 0 => 3 distinct, real roots.
  34                 // discriminant = 0 => a multiple root (1 or 2 real roots)
  35                 // discriminant < 0 => 1 real root, 2 complex conjugate roots
  36
  37                 ////HACK: We know any roots we care about will be between 0 and 1, so...
  38                 Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f", a,b,c,d);
  39                 Real maxi(100);
  40                 Real prevRes(d);
  41                 std::vector<Real> roots;
  42                 for(int i = -1; i <= 100; ++i)
  43                 {
  44                         Real x(i);
  45                         x /= maxi;
  46                         Real y = a*(x*x*x) + b*(x*x) + c*x + d;
  47                         if ( ((y < Real(0)) && (prevRes > Real(0))) || ((y > Real(0)) && (prevRes < Real(0))))
  48                         {
  49                                 Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y);
  50                                 roots.push_back(x);
  51                         }
  52                         prevRes = y;
  53                 }
  54                 return roots;
  55
  56         }
  57
  58         /** A _cubic_ bezier. **/
  59         struct Bezier
  60         {
  61                 Real x0; Real y0;
  62                 Real x1; Real y1;
  63                 Real x2; Real y2;
  64                 Real x3; Real y3;
  65                 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  66                 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)
  67                 {
  68
  69                 }
  70
  71                 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) {}
  72
  73                 std::string Str() const
  74                 {
  75                         std::stringstream s;
  76                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  77                         return s.str();
  78                 }
  79
  80                 /**
  81                  * Construct absolute control points using relative control points to a bounding rectangle
  82                  * ie: If cpy is relative to bounds rectangle, this will be absolute
  83                  */
  84                 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)
  85                 {
  86                         x0 *= t.w;
  87                         y0 *= t.h;
  88                         x1 *= t.w;
  89                         y1 *= t.h;
  90                         x2 *= t.w;
  91                         y2 *= t.h;
  92                         x3 *= t.w;
  93                         y3 *= t.h;
  94                         x0 += t.x;
  95                         y0 += t.y;
  96                         x1 += t.x;
  97                         y1 += t.y;
  98                         x2 += t.x;
  99                         y2 += t.y;
 100                         x3 += t.x;
 101                         y3 += t.y;
 102                 }
 103
 104                 Rect SolveBounds() const;
 105
 106                 Bezier ToAbsolute(const Rect & bounds) const
 107                 {
 108                         return Bezier(*this, bounds);
 109                 }
 110
 111                 /** Convert absolute control points to control points relative to bounds
 112                  * (This basically does the opposite of the Copy constructor)
 113                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 114                  */
 115                 Bezier ToRelative(const Rect & bounds) const
 116                 {
 117                         // x' <- (x - x0)/w etc
 118                         // special cases when w or h = 0
 119                         // (So can't just use the Copy constructor on the inverse of bounds)
 120                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 121                         Bezier result;
 122                         if (bounds.w == 0)
 123                         {
 124                                 result.x0 = 0;
 125                                 result.x1 = 0;
 126                                 result.x2 = 0;
 127                                 result.x3 = 0;
 128                         }
 129                         else
 130                         {
 131                                 result.x0 = (x0 - bounds.x)/bounds.w;
 132                                 result.x1 = (x1 - bounds.x)/bounds.w;
 133                                 result.x2 = (x2 - bounds.x)/bounds.w;
 134                                 result.x3 = (x3 - bounds.x)/bounds.w;
 135                         }
 136
 137                         if (bounds.h == 0)
 138                         {
 139                                 result.y0 = 0;
 140                                 result.y1 = 0;
 141                                 result.y2 = 0;
 142                                 result.y3 = 0;
 143                         }
 144                         else
 145                         {
 146                                 result.y0 = (y0 - bounds.y)/bounds.h;
 147                                 result.y1 = (y1 - bounds.y)/bounds.h;
 148                                 result.y2 = (y2 - bounds.y)/bounds.h;
 149                                 result.y3 = (y3 - bounds.y)/bounds.h;
 150                         }
 151                         return result;
 152                 }
 153
 154                 Bezier ReParametrise(const Real& t0, const Real& t1)
 155                 {
 156                         // This function is very, very ugly, but with luck my derivation is correct (even if it isn't optimal, performance wise)
 157                         // (Very) rough working for the derivation is at: http://davidgow.net/stuff/cubic_bezier_reparam.pdf
 158                         Debug("Reparametrise: %f -> %f",t0,t1);
 159                         Bezier new_bezier;
 160                         Real tdiff = t1 - t0;
 161                         Real tdiff_squared = tdiff*tdiff;
 162                         Real tdiff_cubed = tdiff*tdiff_squared;
 163
 164                         Real t0_squared = t0*t0;
 165                         Real t0_cubed = t0*t0_squared;
 166
 167                         // X coordinates
 168                         Real Dx0 = x0 / tdiff_cubed;
 169                         Real Dx1 = x1 / (tdiff_squared - tdiff_cubed);
 170                         Real Dx2 = x2 / (tdiff - Real(2)*tdiff_squared + tdiff_cubed);
 171                         Real Dx3 = x3 / (Real(1) - Real(3)*tdiff + Real(3)*tdiff_squared - tdiff_cubed);
 172
 173                         new_bezier.x3 = Dx3*t0_cubed + Real(3)*Dx3*t0_squared + Real(3)*Dx3*t0 + Dx3 - Dx2*t0_cubed - Real(2)*Dx2*t0_squared - Dx2*t0 + Dx1*t0_cubed + Dx1*t0_squared - Dx0*t0_cubed;
 174                         new_bezier.x2 = Real(3)*Dx0*t0_squared - Real(2)*Dx1*t0 - Real(3)*Dx1*t0_squared + Dx2 + Real(4)*Dx2*t0 + Real(3)*Dx2*t0_squared - Real(3)*Dx3 - Real(6)*Dx3*t0 - Real(3)*Dx3*t0_squared + Real(3)*new_bezier.x3;
 175                         new_bezier.x1 = Real(-3)*Dx0*t0 + Real(3)*Dx1*t0 + Dx1 - Real(2)*Dx2 - Real(3)*Dx2*t0 + Real(3)*Dx3 + Real(3)*Dx3*t0 + Real(2)*new_bezier.x2 - Real(3)*new_bezier.x3;
 176                         new_bezier.x0 = Dx0 - Dx1 + Dx2 - Dx3 + new_bezier.x1 - new_bezier.x2 + new_bezier.x3;
 177
 178                         // Y coordinates
 179                         Real Dy0 = y0 / tdiff_cubed;
 180                         Real Dy1 = y1 / (tdiff_squared - tdiff_cubed);
 181                         Real Dy2 = y2 / (tdiff - Real(2)*tdiff_squared + tdiff_cubed);
 182                         Real Dy3 = y3 / (Real(1) - Real(3)*tdiff + Real(3)*tdiff_squared - tdiff_cubed);
 183
 184                         new_bezier.y3 = Dy3*t0_cubed + Real(3)*Dy3*t0_squared + Real(3)*Dy3*t0 + Dy3 - Dy2*t0_cubed - Real(2)*Dy2*t0_squared - Dy2*t0 + Dy1*t0_cubed + Dy1*t0_squared - Dy0*t0_cubed;
 185                         new_bezier.y2 = Real(3)*Dy0*t0_squared - Real(2)*Dy1*t0 - Real(3)*Dy1*t0_squared + Dy2 + Real(4)*Dy2*t0 + Real(3)*Dy2*t0_squared - Real(3)*Dy3 - Real(6)*Dy3*t0 - Real(3)*Dy3*t0_squared + Real(3)*new_bezier.y3;
 186                         new_bezier.y1 = Real(-3)*Dy0*t0 + Real(3)*Dy1*t0 + Dy1 - Real(2)*Dy2 - Real(3)*Dy2*t0 + Real(3)*Dy3 + Real(3)*Dy3*t0 + Real(2)*new_bezier.y2 - Real(3)*new_bezier.y3;
 187                         new_bezier.y0 = Dy0 - Dy1 + Dy2 - Dy3 + new_bezier.y1 - new_bezier.y2 + new_bezier.y3;
 188
 189
 190                         Debug("(%f,%f),(%f,%f),(%f,%f),(%f,%f) -> (%f,%f),(%f,%f),(%f,%f),(%f,%f)", x0, y0, x1, y1, x2, y2, x3, y3, new_bezier.x0, new_bezier.y0, new_bezier.x1, new_bezier.y1, new_bezier.x2, new_bezier.y2, new_bezier.x3, new_bezier.y3);
 191                         return new_bezier;
 192                 }
 193
 194                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 195                 {
 196                         // Find points of intersection with the rectangle.
 197                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 198
 199                         // Convert bezier coefficients -> cubic coefficients
 200                         Real xa = x0-x1+x2-x3;
 201                         Real xb = x1 - Real(2)*x2 + Real(3)*x3;
 202                         Real xc = x2 - Real(3)*x3;
 203                         Real xd = x3 - r.x;
 204
 205                         // Find its roots.
 206                         std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
 207
 208                         // And for the other side.
 209                         xd = x3 - r.x - r.w;
 210
 211                         std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
 212                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 213
 214                         // Similarly for y-coordinates.
 215                         // Convert bezier coefficients -> cubic coefficients
 216                         Real ya = y0-y1+y2-y3;
 217                         Real yb = y1 - Real(2)*y2 + Real(3)*y3;
 218                         Real yc = y2 - Real(3)*y3;
 219                         Real yd = y3 - r.y;
 220
 221                         // Find its roots.
 222                         std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
 223
 224                         // And for the other side.
 225                         yd = y3 - r.y - r.h;
 226
 227                         std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
 228                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 229
 230                         // Merge and sort.
 231                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 232                         std::sort(x_intersection.begin(), x_intersection.end());
 233
 234                         Debug("Found %d intersections.\n", x_intersection.size());
 235
 236                         std::vector<Bezier> all_beziers;
 237                         if (x_intersection.empty())
 238                         {
 239                                 all_beziers.push_back(*this);
 240                                 return all_beziers;
 241                         }
 242                         Real t0 = *(x_intersection.begin());
 243                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 244                         {
 245                                 Real t1 = *it;
 246                                 Debug(" -- t0: %f to t1: %f", t0, t1);
 247                                 all_beziers.push_back(this->ReParametrise(t0, t1));
 248                                 t0 = t1;
 249                         }
 250                         return all_beziers;
 251                 }
 252
 253                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 254                 void Evaluate(Real & x, Real & y, const Real & u) const
 255                 {
 256                         Real coeff[4];
 257                         for (unsigned i = 0; i < 4; ++i)
 258                                 coeff[i] = Bernstein(i,3,u);
 259                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 260                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 261                 }
 262
 263         };
 264
 265
 266
 267 }
 268
 269 #endif //_BEZIER_H