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         extern std::pair<Real,Real> BezierTurningPoints(const Real & p0, const Real & p1, const Real & p2, const Real & p3);
  15
  16         extern std::vector<Real> SolveQuadratic(const Real & a, const Real & b, const Real & c, const Real & min = 0, const Real & max = 1);
  17
  18         extern std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d, const Real & min = 0, const Real & max = 1, const Real & delta = 1e-4);
  19
  20         /** A _cubic_ bezier. **/
  21         struct Bezier
  22         {
  23                 Real x0; Real y0;
  24                 Real x1; Real y1;
  25                 Real x2; Real y2;
  26                 Real x3; Real y3;
  27
  28                 typedef enum {UNKNOWN, LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
  29                 Type type;
  30
  31                 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  32                 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), type(UNKNOWN)
  33                 {
  34
  35                 }
  36
  37                 const Type & GetType()
  38                 {
  39                         if (type != Bezier::UNKNOWN)
  40                                 return type;
  41                         // From Loop-Blinn 2005, with w0 == w1 == w2 == w3 = 1
  42                         // Transformed control points: (a0 = x0, b0 = y0)
  43                         Real a1 = (x1-x0)*3;
  44                         Real a2 = (x0- x1*2 +x2)*3;
  45                         Real a3 = (x3 - x0 + (x1 - x2)*3);
  46
  47                         Real b1 = (y1-y0)*3;
  48                         Real b2 = (y0- y1*2 +y2)*3;
  49                         Real b3 = (y3 - y0 + (y1 - y2)*3);
  50
  51                         // d vector (d0 = 0 since all w = 1)
  52                         Real d1 = a2*b3 - a3*b2;
  53                         Real d2 = a3*b1 - a1*b3;
  54                         Real d3 = a1*b2 - a2*b1;
  55
  56                         if (d1 == d2 && d2 == d3 && d3 == 0)
  57                         {
  58                                 type = LINE;
  59                                 //Debug("LINE %s", Str().c_str());
  60                                 return type;
  61                         }
  62
  63                         Real delta1 = -d1*d1;
  64                         Real delta2 = d1*d2;
  65                         Real delta3 = d1*d3 -d2*d2;
  66                         if (delta1 == delta2 && delta2 == delta3 && delta3 == 0)
  67                         {
  68                                 type = QUADRATIC;
  69
  70                                 //Debug("QUADRATIC %s", Str().c_str());
  71                                 return type;
  72                         }
  73
  74                         Real discriminant = d1*d3*4 -d2*d2;
  75                         if (discriminant == 0)
  76                         {
  77                                 type = CUSP;
  78                                 //Debug("CUSP %s", Str().c_str());
  79                         }
  80                         else if (discriminant > 0)
  81                         {
  82                                 type = SERPENTINE;
  83                                 //Debug("SERPENTINE %s", Str().c_str());
  84                         }
  85                         else
  86                         {
  87                                 type = LOOP;
  88                                 //Debug("LOOP %s", Str().c_str());
  89                         }
  90                         return type;
  91                 }
  92
  93
  94                 std::string Str() const
  95                 {
  96                         std::stringstream s;
  97                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  98                         return s.str();
  99                 }
 100
 101                 /**
 102                  * Construct absolute control points using relative control points to a bounding rectangle
 103                  * ie: If cpy is relative to bounds rectangle, this will be absolute
 104                  */
 105                 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), type(UNKNOWN)
 106                 {
 107                         x0 *= t.w;
 108                         y0 *= t.h;
 109                         x1 *= t.w;
 110                         y1 *= t.h;
 111                         x2 *= t.w;
 112                         y2 *= t.h;
 113                         x3 *= t.w;
 114                         y3 *= t.h;
 115                         x0 += t.x;
 116                         y0 += t.y;
 117                         x1 += t.x;
 118                         y1 += t.y;
 119                         x2 += t.x;
 120                         y2 += t.y;
 121                         x3 += t.x;
 122                         y3 += t.y;
 123                 }
 124
 125                 Rect SolveBounds() const;
 126
 127                 std::pair<Real,Real> GetTop() const;
 128                 std::pair<Real,Real> GetBottom() const;
 129                 std::pair<Real,Real> GetLeft() const;
 130                 std::pair<Real,Real> GetRight() const;
 131
 132                 Bezier ToAbsolute(const Rect & bounds) const
 133                 {
 134                         return Bezier(*this, bounds);
 135                 }
 136
 137                 /** Convert absolute control points to control points relative to bounds
 138                  * (This basically does the opposite of the Copy constructor)
 139                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 140                  */
 141                 Bezier ToRelative(const Rect & bounds) const
 142                 {
 143                         // x' <- (x - x0)/w etc
 144                         // special cases when w or h = 0
 145                         // (So can't just use the Copy constructor on the inverse of bounds)
 146                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 147                         Bezier result;
 148                         if (bounds.w == 0)
 149                         {
 150                                 result.x0 = 0;
 151                                 result.x1 = 0;
 152                                 result.x2 = 0;
 153                                 result.x3 = 0;
 154                         }
 155                         else
 156                         {
 157                                 result.x0 = (x0 - bounds.x)/bounds.w;
 158                                 result.x1 = (x1 - bounds.x)/bounds.w;
 159                                 result.x2 = (x2 - bounds.x)/bounds.w;
 160                                 result.x3 = (x3 - bounds.x)/bounds.w;
 161                         }
 162
 163                         if (bounds.h == 0)
 164                         {
 165                                 result.y0 = 0;
 166                                 result.y1 = 0;
 167                                 result.y2 = 0;
 168                                 result.y3 = 0;
 169                         }
 170                         else
 171                         {
 172                                 result.y0 = (y0 - bounds.y)/bounds.h;
 173                                 result.y1 = (y1 - bounds.y)/bounds.h;
 174                                 result.y2 = (y2 - bounds.y)/bounds.h;
 175                                 result.y3 = (y3 - bounds.y)/bounds.h;
 176                         }
 177                         return result;
 178                 }
 179
 180                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 181                 Bezier DeCasteljauSubdivideRight(const Real& t)
 182                 {
 183                         Real one_minus_t = Real(1) - t;
 184
 185                         // X Coordinates
 186                         Real x01 = x0*t + x1*one_minus_t;
 187                         Real x12 = x1*t + x2*one_minus_t;
 188                         Real x23 = x2*t + x3*one_minus_t;
 189
 190                         Real x012 = x01*t + x12*one_minus_t;
 191                         Real x123 = x12*t + x23*one_minus_t;
 192
 193                         Real x0123 = x012*t + x123*one_minus_t;
 194
 195                         // Y Coordinates
 196                         Real y01 = y0*t + y1*one_minus_t;
 197                         Real y12 = y1*t + y2*one_minus_t;
 198                         Real y23 = y2*t + y3*one_minus_t;
 199
 200                         Real y012 = y01*t + y12*one_minus_t;
 201                         Real y123 = y12*t + y23*one_minus_t;
 202
 203                         Real y0123 = y012*t + y123*one_minus_t;
 204
 205                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 206                 }
 207                 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
 208                 Bezier DeCasteljauSubdivideLeft(const Real& t)
 209                 {
 210                         Real one_minus_t = Real(1) - t;
 211
 212                         // X Coordinates
 213                         Real x01 = x0*t + x1*one_minus_t;
 214                         Real x12 = x1*t + x2*one_minus_t;
 215                         Real x23 = x2*t + x3*one_minus_t;
 216
 217                         Real x012 = x01*t + x12*one_minus_t;
 218                         Real x123 = x12*t + x23*one_minus_t;
 219
 220                         Real x0123 = x012*t + x123*one_minus_t;
 221
 222                         // Y Coordinates
 223                         Real y01 = y0*t + y1*one_minus_t;
 224                         Real y12 = y1*t + y2*one_minus_t;
 225                         Real y23 = y2*t + y3*one_minus_t;
 226
 227                         Real y012 = y01*t + y12*one_minus_t;
 228                         Real y123 = y12*t + y23*one_minus_t;
 229
 230                         Real y0123 = y012*t + y123*one_minus_t;
 231
 232                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 233                 }
 234
 235                 Bezier ReParametrise(const Real& t0, const Real& t1)
 236                 {
 237                         Debug("Reparametrise: %f -> %f",t0,t1);
 238                         Bezier new_bezier;
 239                         // Subdivide to get from [0,t1]
 240                         new_bezier = DeCasteljauSubdivideLeft(t1);
 241                         // Convert t0 from [0,1] range to [0, t1]
 242                         Real new_t0 = t0 / t1;
 243                         Debug("New t0 = %f", new_t0);
 244                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 245
 246                         Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 247                         return new_bezier;
 248                 }
 249
 250                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 251                 {
 252                         // Find points of intersection with the rectangle.
 253                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 254
 255                         // Convert bezier coefficients -> cubic coefficients
 256                         Real xd = x0 - r.x;
 257                         Real xc = Real(3)*(x1 - x0);
 258                         Real xb = Real(3)*(x2 - x1) - xc;
 259                         Real xa = x3 - x0 - xc - xb;
 260
 261                         // Find its roots.
 262                         std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
 263
 264                         // And for the other side.
 265                         xd = x0 - r.x - r.w;
 266
 267                         std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
 268                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 269
 270                         // Similarly for y-coordinates.
 271                         // Convert bezier coefficients -> cubic coefficients
 272                         Real yd = y0 - r.y;
 273                         Real yc = Real(3)*(y1 - y0);
 274                         Real yb = Real(3)*(y2 - y1) - yc;
 275                         Real ya = y3 - y0 - yc - yb;
 276
 277                         // Find its roots.
 278                         std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
 279
 280                         // And for the other side.
 281                         yd = y0 - r.y - r.h;
 282
 283                         std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
 284                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 285
 286                         // Merge and sort.
 287                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 288                         x_intersection.push_back(Real(0));
 289                         x_intersection.push_back(Real(1));
 290                         std::sort(x_intersection.begin(), x_intersection.end());
 291
 292                         Debug("Found %d intersections.\n", x_intersection.size());
 293
 294                         std::vector<Bezier> all_beziers;
 295                         if (x_intersection.size() <= 2)
 296                         {
 297                                 all_beziers.push_back(*this);
 298                                 return all_beziers;
 299                         }
 300                         Real t0 = *(x_intersection.begin());
 301                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 302                         {
 303                                 Real t1 = *it;
 304                                 if (t1 == t0) continue;
 305                                 Debug(" -- t0: %f to t1: %f", t0, t1);
 306                                 Real ptx, pty;
 307                                 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
 308                                 if (true || r.PointIn(ptx, pty))
 309                                 {
 310                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 311                                 }
 312                                 else
 313                                 {
 314                                         Debug("Segment removed (point at %f, %f)", ptx, pty);
 315                                 }
 316                                 t0 = t1;
 317                         }
 318                         return all_beziers;
 319                 }
 320
 321                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 322                 void Evaluate(Real & x, Real & y, const Real & u) const
 323                 {
 324                         Real coeff[4];
 325                         for (unsigned i = 0; i < 4; ++i)
 326                                 coeff[i] = Bernstein(i,3,u);
 327                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 328                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 329                 }
 330                 std::vector<Vec2> Evaluate(const std::vector<Real> & u) const;
 331
 332                 std::vector<Real> SolveXParam(const Real & x) const;
 333                 std::vector<Real> SolveYParam(const Real & x) const;
 334
 335                 // Get points with same X
 336                 inline std::vector<Vec2> SolveX(const Real & x) const
 337                 {
 338                         return Evaluate(SolveXParam(x));
 339                 }
 340                 // Get points with same Y
 341                 inline std::vector<Vec2> SolveY(const Real & y) const
 342                 {
 343                         return Evaluate(SolveYParam(y));
 344                 }
 345
 346                 bool operator==(const Bezier & equ) const
 347                 {
 348                         return (x0 == equ.x0 && y0 == equ.y0
 349                                 &&  x1 == equ.x1 && y1 == equ.y1
 350                                 &&      x2 == equ.x2 && y2 == equ.y2
 351                                 &&      x3 == equ.x3 && y3 == equ.y3);
 352                 }
 353                 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}
 354
 355         };
 356
 357
 358
 359 }
 360
 361 #endif //_BEZIER_H