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                 Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f (Discriminant: %f)", a,b,c,d, discriminant);
  34                 // discriminant > 0 => 3 distinct, real roots.
  35                 // discriminant = 0 => a multiple root (1 or 2 real roots)
  36                 // discriminant < 0 => 1 real root, 2 complex conjugate roots
  37
  38                 Real delta0 = (b*b) - Real(3) * a * c;
  39                 Real delta1 = Real(2) * (b * b * b) - Real(9) * a * b * c + Real(27) * (a * a) * d;
  40
  41                 std::vector<Real> roots;
  42
  43                 Real C = pow((delta1 + Sqrt((delta1 * delta1) - 4 * (delta0 * delta0 * delta0)) ) / Real(2), 1/3);
  44
  45                 if (false && discriminant < 0)
  46                 {
  47                         Real real_root = (Real(-1) / (Real(3) * a)) * (b + C + delta0 / C);
  48
  49                         roots.push_back(real_root);
  50
  51                         return roots;
  52
  53                 }
  54
  55                 ////HACK: We know any roots we care about will be between 0 and 1, so...
  56                 Real maxi(100);
  57                 Real prevRes(d);
  58                 for(int i = -1; i <= 100; ++i)
  59                 {
  60                         Real x(i);
  61                         x /= maxi;
  62                         Real y = a*(x*x*x) + b*(x*x) + c*x + d;
  63                         if ( ((y < Real(0)) && (prevRes > Real(0))) || ((y > Real(0)) && (prevRes < Real(0))))
  64                         {
  65                                 Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y);
  66                                 roots.push_back(x);
  67                         }
  68                         prevRes = y;
  69                 }
  70                 return roots;
  71
  72         }
  73
  74         /** A _cubic_ bezier. **/
  75         struct Bezier
  76         {
  77                 Real x0; Real y0;
  78                 Real x1; Real y1;
  79                 Real x2; Real y2;
  80                 Real x3; Real y3;
  81                 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  82                 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)
  83                 {
  84
  85                 }
  86
  87                 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) {}
  88
  89                 std::string Str() const
  90                 {
  91                         std::stringstream s;
  92                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  93                         return s.str();
  94                 }
  95
  96                 /**
  97                  * Construct absolute control points using relative control points to a bounding rectangle
  98                  * ie: If cpy is relative to bounds rectangle, this will be absolute
  99                  */
 100                 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)
 101                 {
 102                         x0 *= t.w;
 103                         y0 *= t.h;
 104                         x1 *= t.w;
 105                         y1 *= t.h;
 106                         x2 *= t.w;
 107                         y2 *= t.h;
 108                         x3 *= t.w;
 109                         y3 *= t.h;
 110                         x0 += t.x;
 111                         y0 += t.y;
 112                         x1 += t.x;
 113                         y1 += t.y;
 114                         x2 += t.x;
 115                         y2 += t.y;
 116                         x3 += t.x;
 117                         y3 += t.y;
 118                 }
 119
 120                 Rect SolveBounds() const;
 121
 122                 Bezier ToAbsolute(const Rect & bounds) const
 123                 {
 124                         return Bezier(*this, bounds);
 125                 }
 126
 127                 /** Convert absolute control points to control points relative to bounds
 128                  * (This basically does the opposite of the Copy constructor)
 129                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 130                  */
 131                 Bezier ToRelative(const Rect & bounds) const
 132                 {
 133                         // x' <- (x - x0)/w etc
 134                         // special cases when w or h = 0
 135                         // (So can't just use the Copy constructor on the inverse of bounds)
 136                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 137                         Bezier result;
 138                         if (bounds.w == 0)
 139                         {
 140                                 result.x0 = 0;
 141                                 result.x1 = 0;
 142                                 result.x2 = 0;
 143                                 result.x3 = 0;
 144                         }
 145                         else
 146                         {
 147                                 result.x0 = (x0 - bounds.x)/bounds.w;
 148                                 result.x1 = (x1 - bounds.x)/bounds.w;
 149                                 result.x2 = (x2 - bounds.x)/bounds.w;
 150                                 result.x3 = (x3 - bounds.x)/bounds.w;
 151                         }
 152
 153                         if (bounds.h == 0)
 154                         {
 155                                 result.y0 = 0;
 156                                 result.y1 = 0;
 157                                 result.y2 = 0;
 158                                 result.y3 = 0;
 159                         }
 160                         else
 161                         {
 162                                 result.y0 = (y0 - bounds.y)/bounds.h;
 163                                 result.y1 = (y1 - bounds.y)/bounds.h;
 164                                 result.y2 = (y2 - bounds.y)/bounds.h;
 165                                 result.y3 = (y3 - bounds.y)/bounds.h;
 166                         }
 167                         return result;
 168                 }
 169
 170                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 171                 Bezier DeCasteljauSubdivideRight(const Real& t)
 172                 {
 173                         Real one_minus_t = Real(1) - t;
 174
 175                         // X Coordinates
 176                         Real x01 = x0*t + x1*one_minus_t;
 177                         Real x12 = x1*t + x2*one_minus_t;
 178                         Real x23 = x2*t + x3*one_minus_t;
 179
 180                         Real x012 = x01*t + x12*one_minus_t;
 181                         Real x123 = x12*t + x23*one_minus_t;
 182
 183                         Real x0123 = x012*t + x123*one_minus_t;
 184
 185                         // Y Coordinates
 186                         Real y01 = y0*t + y1*one_minus_t;
 187                         Real y12 = y1*t + y2*one_minus_t;
 188                         Real y23 = y2*t + y3*one_minus_t;
 189
 190                         Real y012 = y01*t + y12*one_minus_t;
 191                         Real y123 = y12*t + y23*one_minus_t;
 192
 193                         Real y0123 = y012*t + y123*one_minus_t;
 194
 195                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 196                 }
 197                 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
 198                 Bezier DeCasteljauSubdivideLeft(const Real& t)
 199                 {
 200                         Real one_minus_t = Real(1) - t;
 201
 202                         // X Coordinates
 203                         Real x01 = x0*t + x1*one_minus_t;
 204                         Real x12 = x1*t + x2*one_minus_t;
 205                         Real x23 = x2*t + x3*one_minus_t;
 206
 207                         Real x012 = x01*t + x12*one_minus_t;
 208                         Real x123 = x12*t + x23*one_minus_t;
 209
 210                         Real x0123 = x012*t + x123*one_minus_t;
 211
 212                         // Y Coordinates
 213                         Real y01 = y0*t + y1*one_minus_t;
 214                         Real y12 = y1*t + y2*one_minus_t;
 215                         Real y23 = y2*t + y3*one_minus_t;
 216
 217                         Real y012 = y01*t + y12*one_minus_t;
 218                         Real y123 = y12*t + y23*one_minus_t;
 219
 220                         Real y0123 = y012*t + y123*one_minus_t;
 221
 222                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 223                 }
 224
 225                 Bezier ReParametrise(const Real& t0, const Real& t1)
 226                 {
 227                         Debug("Reparametrise: %f -> %f",t0,t1);
 228                         Bezier new_bezier;
 229                         // Subdivide to get from [0,t1]
 230                         new_bezier = DeCasteljauSubdivideLeft(t1);
 231                         // Convert t0 from [0,1] range to [0, t1]
 232                         Real new_t0 = t0 / t1;
 233                         Debug("New t0 = %f", new_t0);
 234                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 235
 236                         Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 237                         return new_bezier;
 238                 }
 239
 240                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 241                 {
 242                         // Find points of intersection with the rectangle.
 243                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 244
 245                         // Convert bezier coefficients -> cubic coefficients
 246                         Real xd = x0 - r.x;
 247                         Real xc = Real(3)*(x1 - x0);
 248                         Real xb = Real(3)*(x2 - x1) - xc;
 249                         Real xa = x3 - x0 - xc - xb;
 250
 251                         // Find its roots.
 252                         std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
 253
 254                         // And for the other side.
 255                         xd = x0 - r.x - r.w;
 256
 257                         std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
 258                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 259
 260                         // Similarly for y-coordinates.
 261                         // Convert bezier coefficients -> cubic coefficients
 262                         Real yd = y0 - r.y;
 263                         Real yc = Real(3)*(y1 - y0);
 264                         Real yb = Real(3)*(y2 - y1) - yc;
 265                         Real ya = y3 - y0 - yc - yb;
 266
 267                         // Find its roots.
 268                         std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
 269
 270                         // And for the other side.
 271                         yd = y0 - r.y - r.h;
 272
 273                         std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
 274                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 275
 276                         // Merge and sort.
 277                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 278                         x_intersection.push_back(Real(0));
 279                         x_intersection.push_back(Real(1));
 280                         std::sort(x_intersection.begin(), x_intersection.end());
 281
 282                         Debug("Found %d intersections.\n", x_intersection.size());
 283
 284                         std::vector<Bezier> all_beziers;
 285                         if (x_intersection.empty())
 286                         {
 287                                 all_beziers.push_back(*this);
 288                                 return all_beziers;
 289                         }
 290                         Real t0 = *(x_intersection.begin());
 291                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 292                         {
 293                                 Real t1 = *it;
 294                                 if (t1 == t0) continue;
 295                                 Debug(" -- t0: %f to t1: %f", t0, t1);
 296                                 Real ptx, pty;
 297                                 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
 298                                 if (r.PointIn(ptx, pty))
 299                                 {
 300                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 301                                 }
 302                                 else
 303                                 {
 304                                         Debug("Segment removed (point at %f, %f)", ptx, pty);
 305                                 }
 306                                 t0 = t1;
 307                         }
 308                         return all_beziers;
 309                 }
 310
 311                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 312                 void Evaluate(Real & x, Real & y, const Real & u) const
 313                 {
 314                         Real coeff[4];
 315                         for (unsigned i = 0; i < 4; ++i)
 316                                 coeff[i] = Bernstein(i,3,u);
 317                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 318                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 319                 }
 320
 321         };
 322
 323
 324
 325 }
 326
 327 #endif //_BEZIER_H