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         inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
  17         {
  18                 Real x0((-b + Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  19                 Real x1((-b - Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  20                 return std::pair<Real,Real>(x0,x1);
  21         }
  22
  23         inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
  24         {
  25                 // This is going to be a big one...
  26                 // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
  27
  28                 // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
  29
  30                 Real discriminant = Real(18) * a * b * c * d - Real(4) * (b * b * b) * d
  31                                 + (b * b) * (c * c) - Real(4) * a * (c * c * c)
  32                                 - Real(27) * (a * a) * (d * d);
  33
  34                 Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f (Discriminant: %f)", a,b,c,d, discriminant);
  35                 // discriminant > 0 => 3 distinct, real roots.
  36                 // discriminant = 0 => a multiple root (1 or 2 real roots)
  37                 // discriminant < 0 => 1 real root, 2 complex conjugate roots
  38
  39                 Real delta0 = (b*b) - Real(3) * a * c;
  40                 Real delta1 = Real(2) * (b * b * b) - Real(9) * a * b * c + Real(27) * (a * a) * d;
  41
  42                 std::vector<Real> roots;
  43
  44                 Real C = pow((delta1 + Sqrt((delta1 * delta1) - 4 * (delta0 * delta0 * delta0)) ) / Real(2), 1/3);
  45
  46                 if (false && discriminant < 0)
  47                 {
  48                         Real real_root = (Real(-1) / (Real(3) * a)) * (b + C + delta0 / C);
  49
  50                         roots.push_back(real_root);
  51
  52                         return roots;
  53
  54                 }
  55
  56                 ////HACK: We know any roots we care about will be between 0 and 1, so...
  57                 Real maxi(100);
  58                 Real prevRes(d);
  59                 for(int i = -1; i <= 100; ++i)
  60                 {
  61                         Real x(i);
  62                         x /= maxi;
  63                         Real y = a*(x*x*x) + b*(x*x) + c*x + d;
  64                         if ( ((y < Real(0)) && (prevRes > Real(0))) || ((y > Real(0)) && (prevRes < Real(0))))
  65                         {
  66                                 Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y);
  67                                 roots.push_back(x);
  68                         }
  69                         prevRes = y;
  70                 }
  71                 return roots;
  72
  73         }
  74
  75         /** A _cubic_ bezier. **/
  76         struct Bezier
  77         {
  78                 Real x0; Real y0;
  79                 Real x1; Real y1;
  80                 Real x2; Real y2;
  81                 Real x3; Real y3;
  82
  83                 typedef enum {LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
  84                 Type type;
  85
  86                 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  87                 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)
  88                 {
  89                         //TODO: classify the curve
  90                         type = SERPENTINE;
  91                 }
  92
  93                 std::string Str() const
  94                 {
  95                         std::stringstream s;
  96                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  97                         return s.str();
  98                 }
  99
 100                 /**
 101                  * Construct absolute control points using relative control points to a bounding rectangle
 102                  * ie: If cpy is relative to bounds rectangle, this will be absolute
 103                  */
 104                 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)
 105                 {
 106                         x0 *= t.w;
 107                         y0 *= t.h;
 108                         x1 *= t.w;
 109                         y1 *= t.h;
 110                         x2 *= t.w;
 111                         y2 *= t.h;
 112                         x3 *= t.w;
 113                         y3 *= t.h;
 114                         x0 += t.x;
 115                         y0 += t.y;
 116                         x1 += t.x;
 117                         y1 += t.y;
 118                         x2 += t.x;
 119                         y2 += t.y;
 120                         x3 += t.x;
 121                         y3 += t.y;
 122                 }
 123
 124                 Rect SolveBounds() const;
 125
 126                 std::pair<Real,Real> GetTop() const;
 127                 std::pair<Real,Real> GetBottom() const;
 128                 std::pair<Real,Real> GetLeft() const;
 129                 std::pair<Real,Real> GetRight() const;
 130
 131                 Bezier ToAbsolute(const Rect & bounds) const
 132                 {
 133                         return Bezier(*this, bounds);
 134                 }
 135
 136                 /** Convert absolute control points to control points relative to bounds
 137                  * (This basically does the opposite of the Copy constructor)
 138                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 139                  */
 140                 Bezier ToRelative(const Rect & bounds) const
 141                 {
 142                         // x' <- (x - x0)/w etc
 143                         // special cases when w or h = 0
 144                         // (So can't just use the Copy constructor on the inverse of bounds)
 145                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 146                         Bezier result;
 147                         if (bounds.w == 0)
 148                         {
 149                                 result.x0 = 0;
 150                                 result.x1 = 0;
 151                                 result.x2 = 0;
 152                                 result.x3 = 0;
 153                         }
 154                         else
 155                         {
 156                                 result.x0 = (x0 - bounds.x)/bounds.w;
 157                                 result.x1 = (x1 - bounds.x)/bounds.w;
 158                                 result.x2 = (x2 - bounds.x)/bounds.w;
 159                                 result.x3 = (x3 - bounds.x)/bounds.w;
 160                         }
 161
 162                         if (bounds.h == 0)
 163                         {
 164                                 result.y0 = 0;
 165                                 result.y1 = 0;
 166                                 result.y2 = 0;
 167                                 result.y3 = 0;
 168                         }
 169                         else
 170                         {
 171                                 result.y0 = (y0 - bounds.y)/bounds.h;
 172                                 result.y1 = (y1 - bounds.y)/bounds.h;
 173                                 result.y2 = (y2 - bounds.y)/bounds.h;
 174                                 result.y3 = (y3 - bounds.y)/bounds.h;
 175                         }
 176                         return result;
 177                 }
 178
 179                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 180                 Bezier DeCasteljauSubdivideRight(const Real& t)
 181                 {
 182                         Real one_minus_t = Real(1) - t;
 183
 184                         // X Coordinates
 185                         Real x01 = x0*t + x1*one_minus_t;
 186                         Real x12 = x1*t + x2*one_minus_t;
 187                         Real x23 = x2*t + x3*one_minus_t;
 188
 189                         Real x012 = x01*t + x12*one_minus_t;
 190                         Real x123 = x12*t + x23*one_minus_t;
 191
 192                         Real x0123 = x012*t + x123*one_minus_t;
 193
 194                         // Y Coordinates
 195                         Real y01 = y0*t + y1*one_minus_t;
 196                         Real y12 = y1*t + y2*one_minus_t;
 197                         Real y23 = y2*t + y3*one_minus_t;
 198
 199                         Real y012 = y01*t + y12*one_minus_t;
 200                         Real y123 = y12*t + y23*one_minus_t;
 201
 202                         Real y0123 = y012*t + y123*one_minus_t;
 203
 204                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 205                 }
 206                 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
 207                 Bezier DeCasteljauSubdivideLeft(const Real& t)
 208                 {
 209                         Real one_minus_t = Real(1) - t;
 210
 211                         // X Coordinates
 212                         Real x01 = x0*t + x1*one_minus_t;
 213                         Real x12 = x1*t + x2*one_minus_t;
 214                         Real x23 = x2*t + x3*one_minus_t;
 215
 216                         Real x012 = x01*t + x12*one_minus_t;
 217                         Real x123 = x12*t + x23*one_minus_t;
 218
 219                         Real x0123 = x012*t + x123*one_minus_t;
 220
 221                         // Y Coordinates
 222                         Real y01 = y0*t + y1*one_minus_t;
 223                         Real y12 = y1*t + y2*one_minus_t;
 224                         Real y23 = y2*t + y3*one_minus_t;
 225
 226                         Real y012 = y01*t + y12*one_minus_t;
 227                         Real y123 = y12*t + y23*one_minus_t;
 228
 229                         Real y0123 = y012*t + y123*one_minus_t;
 230
 231                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 232                 }
 233
 234                 Bezier ReParametrise(const Real& t0, const Real& t1)
 235                 {
 236                         Debug("Reparametrise: %f -> %f",t0,t1);
 237                         Bezier new_bezier;
 238                         // Subdivide to get from [0,t1]
 239                         new_bezier = DeCasteljauSubdivideLeft(t1);
 240                         // Convert t0 from [0,1] range to [0, t1]
 241                         Real new_t0 = t0 / t1;
 242                         Debug("New t0 = %f", new_t0);
 243                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 244
 245                         Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 246                         return new_bezier;
 247                 }
 248
 249                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 250                 {
 251                         // Find points of intersection with the rectangle.
 252                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 253
 254                         // Convert bezier coefficients -> cubic coefficients
 255                         Real xd = x0 - r.x;
 256                         Real xc = Real(3)*(x1 - x0);
 257                         Real xb = Real(3)*(x2 - x1) - xc;
 258                         Real xa = x3 - x0 - xc - xb;
 259
 260                         // Find its roots.
 261                         std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
 262
 263                         // And for the other side.
 264                         xd = x0 - r.x - r.w;
 265
 266                         std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
 267                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 268
 269                         // Similarly for y-coordinates.
 270                         // Convert bezier coefficients -> cubic coefficients
 271                         Real yd = y0 - r.y;
 272                         Real yc = Real(3)*(y1 - y0);
 273                         Real yb = Real(3)*(y2 - y1) - yc;
 274                         Real ya = y3 - y0 - yc - yb;
 275
 276                         // Find its roots.
 277                         std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
 278
 279                         // And for the other side.
 280                         yd = y0 - r.y - r.h;
 281
 282                         std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
 283                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 284
 285                         // Merge and sort.
 286                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 287                         x_intersection.push_back(Real(0));
 288                         x_intersection.push_back(Real(1));
 289                         std::sort(x_intersection.begin(), x_intersection.end());
 290
 291                         Debug("Found %d intersections.\n", x_intersection.size());
 292
 293                         std::vector<Bezier> all_beziers;
 294                         if (x_intersection.size() <= 2)
 295                         {
 296                                 all_beziers.push_back(*this);
 297                                 return all_beziers;
 298                         }
 299                         Real t0 = *(x_intersection.begin());
 300                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 301                         {
 302                                 Real t1 = *it;
 303                                 if (t1 == t0) continue;
 304                                 Debug(" -- t0: %f to t1: %f", t0, t1);
 305                                 Real ptx, pty;
 306                                 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
 307                                 if (true || r.PointIn(ptx, pty))
 308                                 {
 309                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 310                                 }
 311                                 else
 312                                 {
 313                                         Debug("Segment removed (point at %f, %f)", ptx, pty);
 314                                 }
 315                                 t0 = t1;
 316                         }
 317                         return all_beziers;
 318                 }
 319
 320                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 321                 void Evaluate(Real & x, Real & y, const Real & u) const
 322                 {
 323                         Real coeff[4];
 324                         for (unsigned i = 0; i < 4; ++i)
 325                                 coeff[i] = Bernstein(i,3,u);
 326                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 327                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 328                 }
 329
 330                 bool operator==(const Bezier & equ) const
 331                 {
 332                         return (x0 == equ.x0 && y0 == equ.y0
 333                                 &&  x1 == equ.x1 && y1 == equ.y1
 334                                 &&      x2 == equ.x2 && y2 == equ.y2
 335                                 &&      x3 == equ.x3 && y3 == equ.y3);
 336                 }
 337                 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}
 338
 339         };
 340
 341
 342
 343 }
 344
 345 #endif //_BEZIER_H