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 {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)
  33                 {
  34                         //TODO: classify the curve
  35                         type = SERPENTINE;
  36                 }
  37
  38                 std::string Str() const
  39                 {
  40                         std::stringstream s;
  41                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  42                         return s.str();
  43                 }
  44
  45                 /**
  46                  * Construct absolute control points using relative control points to a bounding rectangle
  47                  * ie: If cpy is relative to bounds rectangle, this will be absolute
  48                  */
  49                 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)
  50                 {
  51                         x0 *= t.w;
  52                         y0 *= t.h;
  53                         x1 *= t.w;
  54                         y1 *= t.h;
  55                         x2 *= t.w;
  56                         y2 *= t.h;
  57                         x3 *= t.w;
  58                         y3 *= t.h;
  59                         x0 += t.x;
  60                         y0 += t.y;
  61                         x1 += t.x;
  62                         y1 += t.y;
  63                         x2 += t.x;
  64                         y2 += t.y;
  65                         x3 += t.x;
  66                         y3 += t.y;
  67                 }
  68
  69                 Rect SolveBounds() const;
  70
  71                 std::pair<Real,Real> GetTop() const;
  72                 std::pair<Real,Real> GetBottom() const;
  73                 std::pair<Real,Real> GetLeft() const;
  74                 std::pair<Real,Real> GetRight() const;
  75
  76                 Bezier ToAbsolute(const Rect & bounds) const
  77                 {
  78                         return Bezier(*this, bounds);
  79                 }
  80
  81                 /** Convert absolute control points to control points relative to bounds
  82                  * (This basically does the opposite of the Copy constructor)
  83                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
  84                  */
  85                 Bezier ToRelative(const Rect & bounds) const
  86                 {
  87                         // x' <- (x - x0)/w etc
  88                         // special cases when w or h = 0
  89                         // (So can't just use the Copy constructor on the inverse of bounds)
  90                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
  91                         Bezier result;
  92                         if (bounds.w == 0)
  93                         {
  94                                 result.x0 = 0;
  95                                 result.x1 = 0;
  96                                 result.x2 = 0;
  97                                 result.x3 = 0;
  98                         }
  99                         else
 100                         {
 101                                 result.x0 = (x0 - bounds.x)/bounds.w;
 102                                 result.x1 = (x1 - bounds.x)/bounds.w;
 103                                 result.x2 = (x2 - bounds.x)/bounds.w;
 104                                 result.x3 = (x3 - bounds.x)/bounds.w;
 105                         }
 106
 107                         if (bounds.h == 0)
 108                         {
 109                                 result.y0 = 0;
 110                                 result.y1 = 0;
 111                                 result.y2 = 0;
 112                                 result.y3 = 0;
 113                         }
 114                         else
 115                         {
 116                                 result.y0 = (y0 - bounds.y)/bounds.h;
 117                                 result.y1 = (y1 - bounds.y)/bounds.h;
 118                                 result.y2 = (y2 - bounds.y)/bounds.h;
 119                                 result.y3 = (y3 - bounds.y)/bounds.h;
 120                         }
 121                         return result;
 122                 }
 123
 124                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 125                 Bezier DeCasteljauSubdivideRight(const Real& t)
 126                 {
 127                         Real one_minus_t = Real(1) - t;
 128
 129                         // X Coordinates
 130                         Real x01 = x0*t + x1*one_minus_t;
 131                         Real x12 = x1*t + x2*one_minus_t;
 132                         Real x23 = x2*t + x3*one_minus_t;
 133
 134                         Real x012 = x01*t + x12*one_minus_t;
 135                         Real x123 = x12*t + x23*one_minus_t;
 136
 137                         Real x0123 = x012*t + x123*one_minus_t;
 138
 139                         // Y Coordinates
 140                         Real y01 = y0*t + y1*one_minus_t;
 141                         Real y12 = y1*t + y2*one_minus_t;
 142                         Real y23 = y2*t + y3*one_minus_t;
 143
 144                         Real y012 = y01*t + y12*one_minus_t;
 145                         Real y123 = y12*t + y23*one_minus_t;
 146
 147                         Real y0123 = y012*t + y123*one_minus_t;
 148
 149                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 150                 }
 151                 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
 152                 Bezier DeCasteljauSubdivideLeft(const Real& t)
 153                 {
 154                         Real one_minus_t = Real(1) - t;
 155
 156                         // X Coordinates
 157                         Real x01 = x0*t + x1*one_minus_t;
 158                         Real x12 = x1*t + x2*one_minus_t;
 159                         Real x23 = x2*t + x3*one_minus_t;
 160
 161                         Real x012 = x01*t + x12*one_minus_t;
 162                         Real x123 = x12*t + x23*one_minus_t;
 163
 164                         Real x0123 = x012*t + x123*one_minus_t;
 165
 166                         // Y Coordinates
 167                         Real y01 = y0*t + y1*one_minus_t;
 168                         Real y12 = y1*t + y2*one_minus_t;
 169                         Real y23 = y2*t + y3*one_minus_t;
 170
 171                         Real y012 = y01*t + y12*one_minus_t;
 172                         Real y123 = y12*t + y23*one_minus_t;
 173
 174                         Real y0123 = y012*t + y123*one_minus_t;
 175
 176                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 177                 }
 178
 179                 Bezier ReParametrise(const Real& t0, const Real& t1)
 180                 {
 181                         Debug("Reparametrise: %f -> %f",t0,t1);
 182                         Bezier new_bezier;
 183                         // Subdivide to get from [0,t1]
 184                         new_bezier = DeCasteljauSubdivideLeft(t1);
 185                         // Convert t0 from [0,1] range to [0, t1]
 186                         Real new_t0 = t0 / t1;
 187                         Debug("New t0 = %f", new_t0);
 188                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 189
 190                         Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 191                         return new_bezier;
 192                 }
 193
 194                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 195                 {
 196                         // Find points of intersection with the rectangle.
 197                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 198
 199                         // Convert bezier coefficients -> cubic coefficients
 200                         Real xd = x0 - r.x;
 201                         Real xc = Real(3)*(x1 - x0);
 202                         Real xb = Real(3)*(x2 - x1) - xc;
 203                         Real xa = x3 - x0 - xc - xb;
 204
 205                         // Find its roots.
 206                         std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
 207
 208                         // And for the other side.
 209                         xd = x0 - r.x - r.w;
 210
 211                         std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
 212                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 213
 214                         // Similarly for y-coordinates.
 215                         // Convert bezier coefficients -> cubic coefficients
 216                         Real yd = y0 - r.y;
 217                         Real yc = Real(3)*(y1 - y0);
 218                         Real yb = Real(3)*(y2 - y1) - yc;
 219                         Real ya = y3 - y0 - yc - yb;
 220
 221                         // Find its roots.
 222                         std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
 223
 224                         // And for the other side.
 225                         yd = y0 - r.y - r.h;
 226
 227                         std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
 228                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 229
 230                         // Merge and sort.
 231                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 232                         x_intersection.push_back(Real(0));
 233                         x_intersection.push_back(Real(1));
 234                         std::sort(x_intersection.begin(), x_intersection.end());
 235
 236                         Debug("Found %d intersections.\n", x_intersection.size());
 237
 238                         std::vector<Bezier> all_beziers;
 239                         if (x_intersection.size() <= 2)
 240                         {
 241                                 all_beziers.push_back(*this);
 242                                 return all_beziers;
 243                         }
 244                         Real t0 = *(x_intersection.begin());
 245                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 246                         {
 247                                 Real t1 = *it;
 248                                 if (t1 == t0) continue;
 249                                 Debug(" -- t0: %f to t1: %f", t0, t1);
 250                                 Real ptx, pty;
 251                                 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
 252                                 if (true || r.PointIn(ptx, pty))
 253                                 {
 254                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 255                                 }
 256                                 else
 257                                 {
 258                                         Debug("Segment removed (point at %f, %f)", ptx, pty);
 259                                 }
 260                                 t0 = t1;
 261                         }
 262                         return all_beziers;
 263                 }
 264
 265                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 266                 void Evaluate(Real & x, Real & y, const Real & u) const
 267                 {
 268                         Real coeff[4];
 269                         for (unsigned i = 0; i < 4; ++i)
 270                                 coeff[i] = Bernstein(i,3,u);
 271                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 272                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 273                 }
 274                 std::vector<Vec2> Evaluate(const std::vector<Real> & u) const;
 275
 276                 std::vector<Real> SolveXParam(const Real & x) const;
 277                 std::vector<Real> SolveYParam(const Real & x) const;
 278
 279                 // Get points with same X
 280                 inline std::vector<Vec2> SolveX(const Real & x) const
 281                 {
 282                         return Evaluate(SolveXParam(x));
 283                 }
 284                 // Get points with same Y
 285                 inline std::vector<Vec2> SolveY(const Real & y) const
 286                 {
 287                         return Evaluate(SolveYParam(y));
 288                 }
 289
 290                 bool operator==(const Bezier & equ) const
 291                 {
 292                         return (x0 == equ.x0 && y0 == equ.y0
 293                                 &&  x1 == equ.x1 && y1 == equ.y1
 294                                 &&      x2 == equ.x2 && y2 == equ.y2
 295                                 &&      x3 == equ.x3 && y3 == equ.y3);
 296                 }
 297                 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}
 298
 299         };
 300
 301
 302
 303 }
 304
 305 #endif //_BEZIER_H