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-9);
  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                 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 (fabs(d1+d2+d3) < 1e-6)
  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 (fabs(delta1+delta2+delta3) < 1e-6)
  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 (fabs(discriminant) < 1e-6)
  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                         //Debug("disc %.30f", discriminant);
  91                         return type;
  92                 }
  93
  94
  95                 std::string Str() const
  96                 {
  97                         std::stringstream s;
  98                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  99                         return s.str();
 100                 }
 101
 102                 /**
 103                  * Construct absolute control points using relative control points to a bounding rectangle
 104                  * ie: If cpy is relative to bounds rectangle, this will be absolute
 105                  */
 106                 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(cpy.type)
 107                 {
 108                         x0 *= t.w;
 109                         y0 *= t.h;
 110                         x1 *= t.w;
 111                         y1 *= t.h;
 112                         x2 *= t.w;
 113                         y2 *= t.h;
 114                         x3 *= t.w;
 115                         y3 *= t.h;
 116                         x0 += t.x;
 117                         y0 += t.y;
 118                         x1 += t.x;
 119                         y1 += t.y;
 120                         x2 += t.x;
 121                         y2 += t.y;
 122                         x3 += t.x;
 123                         y3 += t.y;
 124                 }
 125
 126                 Rect SolveBounds() const;
 127
 128                 std::pair<Real,Real> GetTop() const;
 129                 std::pair<Real,Real> GetBottom() const;
 130                 std::pair<Real,Real> GetLeft() const;
 131                 std::pair<Real,Real> GetRight() const;
 132
 133                 Bezier ToAbsolute(const Rect & bounds) const
 134                 {
 135                         return Bezier(*this, bounds);
 136                 }
 137
 138                 /** Convert absolute control points to control points relative to bounds
 139                  * (This basically does the opposite of the Copy constructor)
 140                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 141                  */
 142                 Bezier ToRelative(const Rect & bounds) const
 143                 {
 144                         // x' <- (x - x0)/w etc
 145                         // special cases when w or h = 0
 146                         // (So can't just use the Copy constructor on the inverse of bounds)
 147                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 148                         Bezier result;
 149                         if (bounds.w == 0)
 150                         {
 151                                 result.x0 = 0;
 152                                 result.x1 = 0;
 153                                 result.x2 = 0;
 154                                 result.x3 = 0;
 155                         }
 156                         else
 157                         {
 158                                 result.x0 = (x0 - bounds.x)/bounds.w;
 159                                 result.x1 = (x1 - bounds.x)/bounds.w;
 160                                 result.x2 = (x2 - bounds.x)/bounds.w;
 161                                 result.x3 = (x3 - bounds.x)/bounds.w;
 162                         }
 163
 164                         if (bounds.h == 0)
 165                         {
 166                                 result.y0 = 0;
 167                                 result.y1 = 0;
 168                                 result.y2 = 0;
 169                                 result.y3 = 0;
 170                         }
 171                         else
 172                         {
 173                                 result.y0 = (y0 - bounds.y)/bounds.h;
 174                                 result.y1 = (y1 - bounds.y)/bounds.h;
 175                                 result.y2 = (y2 - bounds.y)/bounds.h;
 176                                 result.y3 = (y3 - bounds.y)/bounds.h;
 177                         }
 178                         return result;
 179                 }
 180
 181                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 182                 Bezier DeCasteljauSubdivideLeft(const Real& t)
 183                 {
 184                         Real one_minus_t = Real(1) - t;
 185
 186                         // X Coordinates
 187                         Real x01 = x1*t + x0*one_minus_t;
 188                         Real x12 = x2*t + x1*one_minus_t;
 189                         Real x23 = x3*t + x2*one_minus_t;
 190
 191                         Real x012 = x12*t + x01*one_minus_t;
 192                         Real x123 = x23*t + x12*one_minus_t;
 193
 194                         Real x0123 = x123*t + x012*one_minus_t;
 195
 196                         // Y Coordinates
 197                         Real y01 = y1*t + y0*one_minus_t;
 198                         Real y12 = y2*t + y1*one_minus_t;
 199                         Real y23 = y3*t + y2*one_minus_t;
 200
 201                         Real y012 = y12*t + y01*one_minus_t;
 202                         Real y123 = y23*t + y12*one_minus_t;
 203
 204                         Real y0123 = y123*t + y012*one_minus_t;
 205
 206                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 207                 }
 208                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 209                 Bezier DeCasteljauSubdivideRight(const Real& t)
 210                 {
 211                         Real one_minus_t = Real(1) - t;
 212
 213                         // X Coordinates
 214                         Real x01 = x1*t + x0*one_minus_t;
 215                         Real x12 = x2*t + x1*one_minus_t;
 216                         Real x23 = x3*t + x2*one_minus_t;
 217
 218                         Real x012 = x12*t + x01*one_minus_t;
 219                         Real x123 = x23*t + x12*one_minus_t;
 220
 221                         Real x0123 = x123*t + x012*one_minus_t;
 222
 223                         // Y Coordinates
 224                         Real y01 = y1*t + y0*one_minus_t;
 225                         Real y12 = y2*t + y1*one_minus_t;
 226                         Real y23 = y3*t + y2*one_minus_t;
 227
 228                         Real y012 = y12*t + y01*one_minus_t;
 229                         Real y123 = y23*t + y12*one_minus_t;
 230
 231                         Real y0123 = y123*t + y012*one_minus_t;
 232
 233                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 234                 }
 235
 236                 Bezier ReParametrise(const Real& t0, const Real& t1)
 237                 {
 238                         Debug("Reparametrise: %f -> %f",t0,t1);
 239                         Bezier new_bezier;
 240                         // Subdivide to get from [0,t1]
 241                         new_bezier = DeCasteljauSubdivideLeft(t1);
 242                         // Convert t0 from [0,1] range to [0, t1]
 243                         Real new_t0 = t0 / t1;
 244                         Debug("New t0 = %f", new_t0);
 245                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 246
 247                         Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 248                         return new_bezier;
 249                 }
 250
 251                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 252                 {
 253                         // Find points of intersection with the rectangle.
 254                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 255
 256
 257                         // Find its roots.
 258                         std::vector<Real> x_intersection = SolveXParam(r.x);
 259
 260                         // And for the other side.
 261
 262                         std::vector<Real> x_intersection_pt2 = SolveXParam(r.x + r.w);
 263                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 264
 265                         // Find its roots.
 266                         std::vector<Real> y_intersection = SolveYParam(r.y);
 267
 268                         std::vector<Real> y_intersection_pt2 = SolveYParam(r.y+r.h);
 269                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 270
 271                         // Merge and sort.
 272                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 273                         x_intersection.push_back(Real(0));
 274                         x_intersection.push_back(Real(1));
 275                         std::sort(x_intersection.begin(), x_intersection.end());
 276
 277                         Debug("Found %d intersections.\n", x_intersection.size());
 278                         for(auto t : x_intersection)
 279                         {
 280                                 Real ptx, pty;
 281                                 Evaluate(ptx, pty, t);
 282                                 Debug("Root: t = %f, (%f,%f)", t, ptx, pty);
 283                         }
 284
 285                         std::vector<Bezier> all_beziers;
 286                         if (x_intersection.size() <= 2)
 287                         {
 288                                 all_beziers.push_back(*this);
 289                                 return all_beziers;
 290                         }
 291                         Real t0 = *(x_intersection.begin());
 292                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 293                         {
 294                                 Real t1 = *it;
 295                                 if (t1 == t0) continue;
 296                                 Debug(" -- t0: %f to t1: %f", t0, t1);
 297                                 Real ptx, pty;
 298                                 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
 299                                 if (r.PointIn(ptx, pty))
 300                                 {
 301                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 302                                 }
 303                                 else
 304                                 {
 305                                         Debug("Segment removed (point at %f, %f)", ptx, pty);
 306                                 }
 307                                 t0 = t1;
 308                         }
 309                         return all_beziers;
 310                 }
 311
 312                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 313                 void Evaluate(Real & x, Real & y, const Real & u) const
 314                 {
 315                         Real coeff[4];
 316                         for (unsigned i = 0; i < 4; ++i)
 317                                 coeff[i] = Bernstein(i,3,u);
 318                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 319                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 320                 }
 321                 std::vector<Vec2> Evaluate(const std::vector<Real> & u) const;
 322
 323                 std::vector<Real> SolveXParam(const Real & x) const;
 324                 std::vector<Real> SolveYParam(const Real & x) const;
 325
 326                 // Get points with same X
 327                 inline std::vector<Vec2> SolveX(const Real & x) const
 328                 {
 329                         return Evaluate(SolveXParam(x));
 330                 }
 331                 // Get points with same Y
 332                 inline std::vector<Vec2> SolveY(const Real & y) const
 333                 {
 334                         return Evaluate(SolveYParam(y));
 335                 }
 336
 337                 bool operator==(const Bezier & equ) const
 338                 {
 339                         return (x0 == equ.x0 && y0 == equ.y0
 340                                 &&  x1 == equ.x1 && y1 == equ.y1
 341                                 &&      x2 == equ.x2 && y2 == equ.y2
 342                                 &&      x3 == equ.x3 && y3 == equ.y3);
 343                 }
 344                 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}
 345
 346         };
 347
 348
 349
 350 }
 351
 352 #endif //_BEZIER_H