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
  10
  11 namespace IPDF
  12 {
  13         extern int Factorial(int n);
  14         extern int BinomialCoeff(int n, int k);
  15         extern Real Bernstein(int k, int n, const Real & u);
  16         extern std::pair<Real,Real> BezierTurningPoints(const Real & p0, const Real & p1, const Real & p2, const Real & p3);
  17
  18         extern std::vector<Real> SolveQuadratic(const Real & a, const Real & b, const Real & c, const Real & min = 0, const Real & max = 1);
  19
  20         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);
  21
  22         /** A _cubic_ bezier. **/
  23         struct Bezier
  24         {
  25                 Real x0; Real y0;
  26                 Real x1; Real y1;
  27                 Real x2; Real y2;
  28                 Real x3; Real y3;
  29
  30                 typedef enum {UNKNOWN, LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
  31                 Type type;
  32
  33                 //Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  34                 Bezier(Real _x0=0, Real _y0=0, Real _x1=0, Real _y1=0, Real _x2=0, Real _y2=0, Real _x3=0, Real _y3=0) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x3), y3(_y3), type(UNKNOWN)
  35                 {
  36
  37                 }
  38
  39                 Type GetType()
  40                 {
  41                         if (type != Bezier::UNKNOWN)
  42                                 return type;
  43                         // From Loop-Blinn 2005, with w0 == w1 == w2 == w3 = 1
  44                         // Transformed control points: (a0 = x0, b0 = y0)
  45                         Real a1 = (x1-x0)*Real(3);
  46                         Real a2 = (x0- x1*Real(2) +x2)*Real(3);
  47                         Real a3 = (x3 - x0 + (x1 - x2)*Real(3));
  48
  49                         Real b1 = (y1-y0)*Real(3);
  50                         Real b2 = (y0- y1*Real(2) +y2)*Real(3);
  51                         Real b3 = (y3 - y0 + (y1 - y2)*Real(3));
  52
  53                         // d vector (d0 = 0 since all w = 1)
  54                         Real d1 = a2*b3 - a3*b2;
  55                         Real d2 = a3*b1 - a1*b3;
  56                         Real d3 = a1*b2 - a2*b1;
  57
  58                         if (Abs(d1+d2+d3) < Real(1e-6))
  59                         {
  60                                 type = LINE;
  61                                 //Debug("LINE %s", Str().c_str());
  62                                 return type;
  63                         }
  64
  65                         Real delta1 = -(d1*d1);
  66                         Real delta2 = d1*d2;
  67                         Real delta3 = d1*d3 -(d2*d2);
  68                         if (Abs(delta1+delta2+delta3) < Real(1e-6))
  69                         {
  70                                 type = QUADRATIC;
  71
  72                                 //Debug("QUADRATIC %s", Str().c_str());
  73                                 return type;
  74                         }
  75
  76                         Real discriminant = d1*d3*Real(4) -d2*d2;
  77                         if (Abs(discriminant) < Real(1e-6))
  78                         {
  79                                 type = CUSP;
  80                                 //Debug("CUSP %s", Str().c_str());
  81                         }
  82                         else if (discriminant > Real(0))
  83                         {
  84                                 type = SERPENTINE;
  85                                 //Debug("SERPENTINE %s", Str().c_str());
  86                         }
  87                         else
  88                         {
  89                                 type = LOOP;
  90                                 //Debug("LOOP %s", Str().c_str());
  91                         }
  92                         //Debug("disc %.30f", discriminant);
  93                         return type;
  94                 }
  95
  96
  97                 std::string Str() const
  98                 {
  99                         std::stringstream s;
 100                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
 101                         return s.str();
 102                 }
 103
 104                 /**
 105                  * Construct absolute control points using relative control points to a bounding rectangle
 106                  * ie: If cpy is relative to bounds rectangle, this will be absolute
 107                  */
 108                 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)
 109                 {
 110                         x0 *= t.w;
 111                         y0 *= t.h;
 112                         x1 *= t.w;
 113                         y1 *= t.h;
 114                         x2 *= t.w;
 115                         y2 *= t.h;
 116                         x3 *= t.w;
 117                         y3 *= t.h;
 118                         x0 += t.x;
 119                         y0 += t.y;
 120                         x1 += t.x;
 121                         y1 += t.y;
 122                         x2 += t.x;
 123                         y2 += t.y;
 124                         x3 += t.x;
 125                         y3 += t.y;
 126                 }
 127
 128                 Rect SolveBounds() const;
 129
 130                 std::pair<Real,Real> GetTop() const;
 131                 std::pair<Real,Real> GetBottom() const;
 132                 std::pair<Real,Real> GetLeft() const;
 133                 std::pair<Real,Real> GetRight() const;
 134
 135                 Bezier ToAbsolute(const Rect & bounds) const
 136                 {
 137                         return Bezier(*this, bounds);
 138                 }
 139
 140                 /** Convert absolute control points to control points relative to bounds
 141                  * (This basically does the opposite of the Copy constructor)
 142                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 143                  */
 144                 Bezier ToRelative(const Rect & bounds) const
 145                 {
 146                         // x' <- (x - x0)/w etc
 147                         // special cases when w or h = 0
 148                         // (So can't just use the Copy constructor on the inverse of bounds)
 149                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 150                         Bezier result;
 151                         if (bounds.w == Real(0))
 152                         {
 153                                 result.x0 = 0;
 154                                 result.x1 = 0;
 155                                 result.x2 = 0;
 156                                 result.x3 = 0;
 157                         }
 158                         else
 159                         {
 160                                 result.x0 = (x0 - bounds.x)/bounds.w;
 161                                 result.x1 = (x1 - bounds.x)/bounds.w;
 162                                 result.x2 = (x2 - bounds.x)/bounds.w;
 163                                 result.x3 = (x3 - bounds.x)/bounds.w;
 164                         }
 165
 166                         if (bounds.h == Real(0))
 167                         {
 168                                 result.y0 = 0;
 169                                 result.y1 = 0;
 170                                 result.y2 = 0;
 171                                 result.y3 = 0;
 172                         }
 173                         else
 174                         {
 175                                 result.y0 = (y0 - bounds.y)/bounds.h;
 176                                 result.y1 = (y1 - bounds.y)/bounds.h;
 177                                 result.y2 = (y2 - bounds.y)/bounds.h;
 178                                 result.y3 = (y3 - bounds.y)/bounds.h;
 179                         }
 180                         return result;
 181                 }
 182
 183                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 184                 Bezier DeCasteljauSubdivideLeft(const Real& t)
 185                 {
 186                         Real one_minus_t = Real(1) - t;
 187
 188                         // X Coordinates
 189                         Real x01 = x1*t + x0*one_minus_t;
 190                         Real x12 = x2*t + x1*one_minus_t;
 191                         Real x23 = x3*t + x2*one_minus_t;
 192
 193                         Real x012 = x12*t + x01*one_minus_t;
 194                         Real x123 = x23*t + x12*one_minus_t;
 195
 196                         Real x0123 = x123*t + x012*one_minus_t;
 197
 198                         // Y Coordinates
 199                         Real y01 = y1*t + y0*one_minus_t;
 200                         Real y12 = y2*t + y1*one_minus_t;
 201                         Real y23 = y3*t + y2*one_minus_t;
 202
 203                         Real y012 = y12*t + y01*one_minus_t;
 204                         Real y123 = y23*t + y12*one_minus_t;
 205
 206                         Real y0123 = y123*t + y012*one_minus_t;
 207
 208                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 209                 }
 210                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 211                 Bezier DeCasteljauSubdivideRight(const Real& t)
 212                 {
 213                         Real one_minus_t = Real(1) - t;
 214
 215                         // X Coordinates
 216                         Real x01 = x1*t + x0*one_minus_t;
 217                         Real x12 = x2*t + x1*one_minus_t;
 218                         Real x23 = x3*t + x2*one_minus_t;
 219
 220                         Real x012 = x12*t + x01*one_minus_t;
 221                         Real x123 = x23*t + x12*one_minus_t;
 222
 223                         Real x0123 = x123*t + x012*one_minus_t;
 224
 225                         // Y Coordinates
 226                         Real y01 = y1*t + y0*one_minus_t;
 227                         Real y12 = y2*t + y1*one_minus_t;
 228                         Real y23 = y3*t + y2*one_minus_t;
 229
 230                         Real y012 = y12*t + y01*one_minus_t;
 231                         Real y123 = y23*t + y12*one_minus_t;
 232
 233                         Real y0123 = y123*t + y012*one_minus_t;
 234
 235                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 236                 }
 237
 238                 Bezier ReParametrise(const Real& t0, const Real& t1)
 239                 {
 240                         Debug("Reparametrise: %f -> %f",Double(t0),Double(t1));
 241                         Bezier new_bezier;
 242                         // Subdivide to get from [0,t1]
 243                         new_bezier = DeCasteljauSubdivideLeft(t1);
 244                         // Convert t0 from [0,1] range to [0, t1]
 245                         Real new_t0 = t0 / t1;
 246                         Debug("New t0 = %f", Double(new_t0));
 247                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 248
 249                         Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 250                         return new_bezier;
 251                 }
 252
 253                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 254                 {
 255                         // Find points of intersection with the rectangle.
 256                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 257
 258
 259                         // Find its roots.
 260                         std::vector<Real> x_intersection = SolveXParam(r.x);
 261                         Debug("Found %d intersections on left edge", x_intersection.size());
 262
 263                         // And for the other side.
 264
 265                         std::vector<Real> x_intersection_pt2 = SolveXParam(r.x + r.w);
 266                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 267                         Debug("Found %d intersections on right edge (total x: %d)", x_intersection_pt2.size(), x_intersection.size());
 268
 269                         // Find its roots.
 270                         std::vector<Real> y_intersection = SolveYParam(r.y);
 271                         Debug("Found %d intersections on top edge", y_intersection.size());
 272
 273                         std::vector<Real> y_intersection_pt2 = SolveYParam(r.y+r.h);
 274                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 275                         Debug("Found %d intersections on bottom edge (total y: %d)", y_intersection_pt2.size(), y_intersection.size());
 276
 277                         // Merge and sort.
 278                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 279                         x_intersection.push_back(Real(0));
 280                         x_intersection.push_back(Real(1));
 281                         std::sort(x_intersection.begin(), x_intersection.end());
 282
 283                         Debug("Found %d intersections.\n", x_intersection.size());
 284                         for(auto t : x_intersection)
 285                         {
 286                                 Real ptx, pty;
 287                                 Evaluate(ptx, pty, t);
 288                                 Debug("Root: t = %f, (%f,%f)", Double(t), Double(ptx), Double(pty));
 289                         }
 290
 291                         std::vector<Bezier> all_beziers;
 292                         if (x_intersection.size() <= 2)
 293                         {
 294                                 all_beziers.push_back(*this);
 295                                 return all_beziers;
 296                         }
 297                         Real t0 = *(x_intersection.begin());
 298                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 299                         {
 300                                 Real t1 = *it;
 301                                 if (t1 == t0) continue;
 302                                 Debug(" -- t0: %f to t1: %f: %f", Double(t0), Double(t1), Double((t1 + t0)/Real(2)));
 303                                 Real ptx, pty;
 304                                 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
 305                                 if (r.PointIn(ptx, pty))
 306                                 {
 307                                         Debug("Adding segment: (point at %f, %f)", Double(ptx), Double(pty));
 308                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 309                                 }
 310                                 else
 311                                 {
 312                                         Debug("Segment removed (point at %f, %f)", Double(ptx), Double(pty));
 313                                 }
 314                                 t0 = t1;
 315                         }
 316                         return all_beziers;
 317                 }
 318
 319                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 320                 void Evaluate(Real & x, Real & y, const Real & u) const
 321                 {
 322                         Real coeff[4];
 323                         for (unsigned i = 0; i < 4; ++i)
 324                                 coeff[i] = Bernstein(i,3,u);
 325                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 326                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 327                 }
 328                 std::vector<Vec2> Evaluate(const std::vector<Real> & u) const;
 329
 330                 std::vector<Real> SolveXParam(const Real & x) const;
 331                 std::vector<Real> SolveYParam(const Real & x) const;
 332
 333                 // Get points with same X
 334                 inline std::vector<Vec2> SolveX(const Real & x) const
 335                 {
 336                         return Evaluate(SolveXParam(x));
 337                 }
 338                 // Get points with same Y
 339                 inline std::vector<Vec2> SolveY(const Real & y) const
 340                 {
 341                         return Evaluate(SolveYParam(y));
 342                 }
 343
 344                 bool operator==(const Bezier & equ) const
 345                 {
 346                         return (x0 == equ.x0 && y0 == equ.y0
 347                                 &&  x1 == equ.x1 && y1 == equ.y1
 348                                 &&      x2 == equ.x2 && y2 == equ.y2
 349                                 &&      x3 == equ.x3 && y3 == equ.y3);
 350                 }
 351                 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}
 352
 353         };
 354
 355
 356
 357 }
 358
 359 #endif //_BEZIER_H