src/bezier.h

   1 #ifndef _BEZIER_H
   2 #define _BEZIER_H
   3
   4 #include <vector>
   5 #include <algorithm>
   6 #include "rect.h"
   7 #include "real.h"
   8
   9
  10
  11 namespace IPDF
  12 {
  13         typedef Real BReal;
  14         typedef TRect<BReal> BRect;
  15
  16         extern int Factorial(int n);
  17         extern int BinomialCoeff(int n, int k);
  18         extern BReal Bernstein(int k, int n, const BReal & u);
  19         extern std::pair<BReal,BReal> BezierTurningPoints(const BReal & p0, const BReal & p1, const BReal & p2, const BReal & p3);
  20
  21         extern std::vector<BReal> SolveQuadratic(const BReal & a, const BReal & b, const BReal & c, const BReal & min = 0, const BReal & max = 1);
  22
  23         extern std::vector<BReal> SolveCubic(const BReal & a, const BReal & b, const BReal & c, const BReal & d, const BReal & min = 0, const BReal & max = 1, const BReal & delta = 1e-9);
  24
  25         /** A _cubic_ bezier. **/
  26         struct Bezier
  27         {
  28                 BReal x0; BReal y0;
  29                 BReal x1; BReal y1;
  30                 BReal x2; BReal y2;
  31                 BReal x3; BReal y3;
  32
  33                 typedef enum {UNKNOWN, LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
  34                 Type type;
  35
  36                 //Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  37                 Bezier(BReal _x0=0, BReal _y0=0, BReal _x1=0, BReal _y1=0, BReal _x2=0, BReal _y2=0, BReal _x3=0, BReal _y3=0) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x3), y3(_y3), type(UNKNOWN)
  38                 {
  39
  40                 }
  41
  42                 Type GetType()
  43                 {
  44                         if (type != Bezier::UNKNOWN)
  45                                 return type;
  46                         // From Loop-Blinn 2005, with w0 == w1 == w2 == w3 = 1
  47                         // Transformed control points: (a0 = x0, b0 = y0)
  48                         BReal a1 = (x1-x0)*BReal(3);
  49                         BReal a2 = (x0- x1*BReal(2) +x2)*BReal(3);
  50                         BReal a3 = (x3 - x0 + (x1 - x2)*BReal(3));
  51
  52                         BReal b1 = (y1-y0)*BReal(3);
  53                         BReal b2 = (y0- y1*BReal(2) +y2)*BReal(3);
  54                         BReal b3 = (y3 - y0 + (y1 - y2)*BReal(3));
  55
  56                         // d vector (d0 = 0 since all w = 1)
  57                         BReal d1 = a2*b3 - a3*b2;
  58                         BReal d2 = a3*b1 - a1*b3;
  59                         BReal d3 = a1*b2 - a2*b1;
  60
  61                         if (Abs(d1+d2+d3) < BReal(1e-6))
  62                         {
  63                                 type = LINE;
  64                                 //Debug("LINE %s", Str().c_str());
  65                                 return type;
  66                         }
  67
  68                         BReal delta1 = -(d1*d1);
  69                         BReal delta2 = d1*d2;
  70                         BReal delta3 = d1*d3 -(d2*d2);
  71                         if (Abs(delta1+delta2+delta3) < BReal(1e-6))
  72                         {
  73                                 type = QUADRATIC;
  74
  75                                 //Debug("QUADRATIC %s", Str().c_str());
  76                                 return type;
  77                         }
  78
  79                         BReal discriminant = d1*d3*BReal(4) -d2*d2;
  80                         if (Abs(discriminant) < BReal(1e-6))
  81                         {
  82                                 type = CUSP;
  83                                 //Debug("CUSP %s", Str().c_str());
  84                         }
  85                         else if (discriminant > BReal(0))
  86                         {
  87                                 type = SERPENTINE;
  88                                 //Debug("SERPENTINE %s", Str().c_str());
  89                         }
  90                         else
  91                         {
  92                                 type = LOOP;
  93                                 //Debug("LOOP %s", Str().c_str());
  94                         }
  95                         //Debug("disc %.30f", discriminant);
  96                         return type;
  97                 }
  98
  99
 100                 std::string Str() const
 101                 {
 102                         std::stringstream s;
 103                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
 104                         return s.str();
 105                 }
 106
 107                 /**
 108                  * Construct absolute control points using relative control points to a bounding rectangle
 109                  * ie: If cpy is relative to bounds rectangle, this will be absolute
 110                  */
 111                 Bezier(const Bezier & cpy, const BRect & t = BRect(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)
 112                 {
 113                         x0 *= t.w;
 114                         y0 *= t.h;
 115                         x1 *= t.w;
 116                         y1 *= t.h;
 117                         x2 *= t.w;
 118                         y2 *= t.h;
 119                         x3 *= t.w;
 120                         y3 *= t.h;
 121                         x0 += t.x;
 122                         y0 += t.y;
 123                         x1 += t.x;
 124                         y1 += t.y;
 125                         x2 += t.x;
 126                         y2 += t.y;
 127                         x3 += t.x;
 128                         y3 += t.y;
 129                 }
 130
 131                 BRect SolveBounds() const;
 132
 133                 std::pair<BReal,BReal> GetTop() const;
 134                 std::pair<BReal,BReal> GetBottom() const;
 135                 std::pair<BReal,BReal> GetLeft() const;
 136                 std::pair<BReal,BReal> GetRight() const;
 137
 138                 Bezier ToAbsolute(const BRect & bounds) const
 139                 {
 140                         return Bezier(*this, bounds);
 141                 }
 142
 143                 /** Convert absolute control points to control points relative to bounds
 144                  * (This basically does the opposite of the Copy constructor)
 145                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 146                  */
 147                 Bezier ToRelative(const BRect & bounds) const
 148                 {
 149                         // x' <- (x - x0)/w etc
 150                         // special cases when w or h = 0
 151                         // (So can't just use the Copy constructor on the inverse of bounds)
 152                         // BRect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, BReal(1)/bounds.w, BReal(1)/bounds.h};
 153                         Bezier result;
 154                         if (bounds.w == 0)
 155                         {
 156                                 result.x0 = 0;
 157                                 result.x1 = 0;
 158                                 result.x2 = 0;
 159                                 result.x3 = 0;
 160                         }
 161                         else
 162                         {
 163                                 result.x0 = (x0 - bounds.x)/bounds.w;
 164                                 result.x1 = (x1 - bounds.x)/bounds.w;
 165                                 result.x2 = (x2 - bounds.x)/bounds.w;
 166                                 result.x3 = (x3 - bounds.x)/bounds.w;
 167                         }
 168
 169                         if (bounds.h == 0)
 170                         {
 171                                 result.y0 = 0;
 172                                 result.y1 = 0;
 173                                 result.y2 = 0;
 174                                 result.y3 = 0;
 175                         }
 176                         else
 177                         {
 178                                 result.y0 = (y0 - bounds.y)/bounds.h;
 179                                 result.y1 = (y1 - bounds.y)/bounds.h;
 180                                 result.y2 = (y2 - bounds.y)/bounds.h;
 181                                 result.y3 = (y3 - bounds.y)/bounds.h;
 182                         }
 183                         return result;
 184                 }
 185
 186                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 187                 Bezier DeCasteljauSubdivideLeft(const BReal& t)
 188                 {
 189                         BReal one_minus_t = BReal(1) - t;
 190
 191                         // X Coordinates
 192                         BReal x01 = x1*t + x0*one_minus_t;
 193                         BReal x12 = x2*t + x1*one_minus_t;
 194                         BReal x23 = x3*t + x2*one_minus_t;
 195
 196                         BReal x012 = x12*t + x01*one_minus_t;
 197                         BReal x123 = x23*t + x12*one_minus_t;
 198
 199                         BReal x0123 = x123*t + x012*one_minus_t;
 200
 201                         // Y Coordinates
 202                         BReal y01 = y1*t + y0*one_minus_t;
 203                         BReal y12 = y2*t + y1*one_minus_t;
 204                         BReal y23 = y3*t + y2*one_minus_t;
 205
 206                         BReal y012 = y12*t + y01*one_minus_t;
 207                         BReal y123 = y23*t + y12*one_minus_t;
 208
 209                         BReal y0123 = y123*t + y012*one_minus_t;
 210
 211                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 212                 }
 213                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 214                 Bezier DeCasteljauSubdivideRight(const BReal& t)
 215                 {
 216                         BReal one_minus_t = BReal(1) - t;
 217
 218                         // X Coordinates
 219                         BReal x01 = x1*t + x0*one_minus_t;
 220                         BReal x12 = x2*t + x1*one_minus_t;
 221                         BReal x23 = x3*t + x2*one_minus_t;
 222
 223                         BReal x012 = x12*t + x01*one_minus_t;
 224                         BReal x123 = x23*t + x12*one_minus_t;
 225
 226                         BReal x0123 = x123*t + x012*one_minus_t;
 227
 228                         // Y Coordinates
 229                         BReal y01 = y1*t + y0*one_minus_t;
 230                         BReal y12 = y2*t + y1*one_minus_t;
 231                         BReal y23 = y3*t + y2*one_minus_t;
 232
 233                         BReal y012 = y12*t + y01*one_minus_t;
 234                         BReal y123 = y23*t + y12*one_minus_t;
 235
 236                         BReal y0123 = y123*t + y012*one_minus_t;
 237
 238                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 239                 }
 240
 241                 Bezier ReParametrise(const BReal& t0, const BReal& t1)
 242                 {
 243                         //Debug("Reparametrise: %f -> %f",Double(t0),Double(t1));
 244                         Bezier new_bezier;
 245                         // Subdivide to get from [0,t1]
 246                         new_bezier = DeCasteljauSubdivideLeft(t1);
 247                         // Convert t0 from [0,1] range to [0, t1]
 248                         BReal new_t0 = t0 / t1;
 249                         //Debug("New t0 = %f", Double(new_t0));
 250                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 251
 252                         //Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 253                         return new_bezier;
 254                 }
 255
 256                 std::vector<Bezier> ClipToRectangle(const BRect & r)
 257                 {
 258                         // Find points of intersection with the rectangle.
 259                         Debug("Clipping Bezier to BRect %s", r.Str().c_str());
 260
 261
 262                         // Find its roots.
 263                         std::vector<BReal> x_intersection = SolveXParam(r.x);
 264                         //Debug("Found %d intersections on left edge", x_intersection.size());
 265
 266                         // And for the other side.
 267
 268                         std::vector<BReal> x_intersection_pt2 = SolveXParam(r.x + r.w);
 269                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 270                         //Debug("Found %d intersections on right edge (total x: %d)", x_intersection_pt2.size(), x_intersection.size());
 271
 272                         // Find its roots.
 273                         std::vector<BReal> y_intersection = SolveYParam(r.y);
 274                         //Debug("Found %d intersections on top edge", y_intersection.size());
 275
 276                         std::vector<BReal> y_intersection_pt2 = SolveYParam(r.y+r.h);
 277                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 278                         //Debug("Found %d intersections on bottom edge (total y: %d)", y_intersection_pt2.size(), y_intersection.size());
 279
 280                         // Merge and sort.
 281                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 282                         x_intersection.push_back(BReal(0));
 283                         x_intersection.push_back(BReal(1));
 284                         std::sort(x_intersection.begin(), x_intersection.end());
 285
 286                         //Debug("Found %d intersections.\n", x_intersection.size());
 287                         /*for(auto t : x_intersection)
 288                         {
 289                                 BReal ptx, pty;
 290                                 Evaluate(ptx, pty, t);
 291                                 Debug("Root: t = %f, (%f,%f)", Double(t), Double(ptx), Double(pty));
 292                         }*/
 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                         BReal t0 = *(x_intersection.begin());
 301                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 302                         {
 303                                 BReal t1 = *it;
 304                                 if (t1 == t0) continue;
 305                                 //Debug(" -- t0: %f to t1: %f: %f", Double(t0), Double(t1), Double((t1 + t0)/BReal(2)));
 306                                 BReal ptx, pty;
 307                                 Evaluate(ptx, pty, ((t1 + t0) / BReal(2)));
 308                                 if (r.PointIn(ptx, pty))
 309                                 {
 310                                         //Debug("Adding segment: (point at %f, %f)", Double(ptx), Double(pty));
 311                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 312                                 }
 313                                 else
 314                                 {
 315                                         //Debug("Segment removed (point at %f, %f)", Double(ptx), Double(pty));
 316                                 }
 317                                 t0 = t1;
 318                         }
 319                         return all_beziers;
 320                 }
 321
 322                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 323                 void Evaluate(BReal & x, BReal & y, const BReal & u) const
 324                 {
 325                         BReal coeff[4];
 326                         for (unsigned i = 0; i < 4; ++i)
 327                                 coeff[i] = Bernstein(i,3,u);
 328                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 329                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 330                 }
 331                 std::vector<Vec2> Evaluate(const std::vector<BReal> & u) const;
 332
 333                 std::vector<BReal> SolveXParam(const BReal & x) const;
 334                 std::vector<BReal> SolveYParam(const BReal & x) const;
 335
 336                 // Get points with same X
 337                 inline std::vector<Vec2> SolveX(const BReal & x) const
 338                 {
 339                         return Evaluate(SolveXParam(x));
 340                 }
 341                 // Get points with same Y
 342                 inline std::vector<Vec2> SolveY(const BReal & y) const
 343                 {
 344                         return Evaluate(SolveYParam(y));
 345                 }
 346
 347                 bool operator==(const Bezier & equ) const
 348                 {
 349                         return (x0 == equ.x0 && y0 == equ.y0
 350                                 &&  x1 == equ.x1 && y1 == equ.y1
 351                                 &&      x2 == equ.x2 && y2 == equ.y2
 352                                 &&      x3 == equ.x3 && y3 == equ.y3);
 353                 }
 354                 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}
 355
 356         };
 357
 358
 359
 360 }
 361
 362 #endif //_BEZIER_H