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
  15         inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
  16         {
  17                 Real x0((-b + Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  18                 Real x1((-b - Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
  19                 return std::pair<Real,Real>(x0,x1);
  20         }
  21
  22         inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
  23         {
  24                 // This is going to be a big one...
  25                 // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
  26
  27                 // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
  28                 /*
  29                 Real discriminant = Real(18) * a * b * c * d - Real(4) * (b * b * b) * d
  30                                 + (b * b) * (c * c) - Real(4) * a * (c * c * c)
  31                                 - Real(27) * (a * a) * (d * d);
  32                 */
  33                 // discriminant > 0 => 3 distinct, real roots.
  34                 // discriminant = 0 => a multiple root (1 or 2 real roots)
  35                 // discriminant < 0 => 1 real root, 2 complex conjugate roots
  36
  37                 ////HACK: We know any roots we care about will be between 0 and 1, so...
  38                 Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f", a,b,c,d);
  39                 Real maxi(100);
  40                 Real prevRes(d);
  41                 std::vector<Real> roots;
  42                 for(int i = -1; i <= 100; ++i)
  43                 {
  44                         Real x(i);
  45                         x /= maxi;
  46                         Real y = a*(x*x*x) + b*(x*x) + c*x + d;
  47                         if ( ((y < Real(0)) && (prevRes > Real(0))) || ((y > Real(0)) && (prevRes < Real(0))))
  48                         {
  49                                 Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y);
  50                                 roots.push_back(x);
  51                         }
  52                         prevRes = y;
  53                 }
  54                 return roots;
  55
  56         }
  57
  58         /** A _cubic_ bezier. **/
  59         struct Bezier
  60         {
  61                 Real x0; Real y0;
  62                 Real x1; Real y1;
  63                 Real x2; Real y2;
  64                 Real x3; Real y3;
  65                 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
  66                 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)
  67                 {
  68
  69                 }
  70
  71                 Bezier(Real _x0, Real _y0, Real _x1, Real _y1, Real _x2, Real _y2) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x2), y3(_y2) {}
  72
  73                 std::string Str() const
  74                 {
  75                         std::stringstream s;
  76                         s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
  77                         return s.str();
  78                 }
  79
  80                 /**
  81                  * Construct absolute control points using relative control points to a bounding rectangle
  82                  * ie: If cpy is relative to bounds rectangle, this will be absolute
  83                  */
  84                 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)
  85                 {
  86                         x0 *= t.w;
  87                         y0 *= t.h;
  88                         x1 *= t.w;
  89                         y1 *= t.h;
  90                         x2 *= t.w;
  91                         y2 *= t.h;
  92                         x3 *= t.w;
  93                         y3 *= t.h;
  94                         x0 += t.x;
  95                         y0 += t.y;
  96                         x1 += t.x;
  97                         y1 += t.y;
  98                         x2 += t.x;
  99                         y2 += t.y;
 100                         x3 += t.x;
 101                         y3 += t.y;
 102                 }
 103
 104                 Rect SolveBounds() const;
 105
 106                 Bezier ToAbsolute(const Rect & bounds) const
 107                 {
 108                         return Bezier(*this, bounds);
 109                 }
 110
 111                 /** Convert absolute control points to control points relative to bounds
 112                  * (This basically does the opposite of the Copy constructor)
 113                  * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
 114                  */
 115                 Bezier ToRelative(const Rect & bounds) const
 116                 {
 117                         // x' <- (x - x0)/w etc
 118                         // special cases when w or h = 0
 119                         // (So can't just use the Copy constructor on the inverse of bounds)
 120                         // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
 121                         Bezier result;
 122                         if (bounds.w == 0)
 123                         {
 124                                 result.x0 = 0;
 125                                 result.x1 = 0;
 126                                 result.x2 = 0;
 127                                 result.x3 = 0;
 128                         }
 129                         else
 130                         {
 131                                 result.x0 = (x0 - bounds.x)/bounds.w;
 132                                 result.x1 = (x1 - bounds.x)/bounds.w;
 133                                 result.x2 = (x2 - bounds.x)/bounds.w;
 134                                 result.x3 = (x3 - bounds.x)/bounds.w;
 135                         }
 136
 137                         if (bounds.h == 0)
 138                         {
 139                                 result.y0 = 0;
 140                                 result.y1 = 0;
 141                                 result.y2 = 0;
 142                                 result.y3 = 0;
 143                         }
 144                         else
 145                         {
 146                                 result.y0 = (y0 - bounds.y)/bounds.h;
 147                                 result.y1 = (y1 - bounds.y)/bounds.h;
 148                                 result.y2 = (y2 - bounds.y)/bounds.h;
 149                                 result.y3 = (y3 - bounds.y)/bounds.h;
 150                         }
 151                         return result;
 152                 }
 153
 154                 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
 155                 Bezier DeCasteljauSubdivideRight(const Real& t)
 156                 {
 157                         Real one_minus_t = Real(1) - t;
 158
 159                         // X Coordinates
 160                         Real x01 = x0*t + x1*one_minus_t;
 161                         Real x12 = x1*t + x2*one_minus_t;
 162                         Real x23 = x2*t + x3*one_minus_t;
 163
 164                         Real x012 = x01*t + x12*one_minus_t;
 165                         Real x123 = x12*t + x23*one_minus_t;
 166
 167                         Real x0123 = x012*t + x123*one_minus_t;
 168
 169                         // Y Coordinates
 170                         Real y01 = y0*t + y1*one_minus_t;
 171                         Real y12 = y1*t + y2*one_minus_t;
 172                         Real y23 = y2*t + y3*one_minus_t;
 173
 174                         Real y012 = y01*t + y12*one_minus_t;
 175                         Real y123 = y12*t + y23*one_minus_t;
 176
 177                         Real y0123 = y012*t + y123*one_minus_t;
 178
 179                         return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
 180                 }
 181                 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
 182                 Bezier DeCasteljauSubdivideLeft(const Real& t)
 183                 {
 184                         Real one_minus_t = Real(1) - t;
 185
 186                         // X Coordinates
 187                         Real x01 = x0*t + x1*one_minus_t;
 188                         Real x12 = x1*t + x2*one_minus_t;
 189                         Real x23 = x2*t + x3*one_minus_t;
 190
 191                         Real x012 = x01*t + x12*one_minus_t;
 192                         Real x123 = x12*t + x23*one_minus_t;
 193
 194                         Real x0123 = x012*t + x123*one_minus_t;
 195
 196                         // Y Coordinates
 197                         Real y01 = y0*t + y1*one_minus_t;
 198                         Real y12 = y1*t + y2*one_minus_t;
 199                         Real y23 = y2*t + y3*one_minus_t;
 200
 201                         Real y012 = y01*t + y12*one_minus_t;
 202                         Real y123 = y12*t + y23*one_minus_t;
 203
 204                         Real y0123 = y012*t + y123*one_minus_t;
 205
 206                         return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
 207                 }
 208
 209                 Bezier ReParametrise(const Real& t0, const Real& t1)
 210                 {
 211                         Debug("Reparametrise: %f -> %f",t0,t1);
 212                         Bezier new_bezier;
 213                         // Subdivide to get from [0,t1]
 214                         new_bezier = DeCasteljauSubdivideLeft(t1);
 215                         // Convert t0 from [0,1] range to [0, t1]
 216                         Real new_t0 = t0 / t1;
 217                         Debug("New t0 = %f", new_t0);
 218                         new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
 219
 220                         Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
 221                         return new_bezier;
 222                 }
 223
 224                 std::vector<Bezier> ClipToRectangle(const Rect& r)
 225                 {
 226                         // Find points of intersection with the rectangle.
 227                         Debug("Clipping Bezier to Rect %s", r.Str().c_str());
 228
 229                         // Convert bezier coefficients -> cubic coefficients
 230                         Real xa = x0-x1+x2-x3;
 231                         Real xb = x1 - Real(2)*x2 + Real(3)*x3;
 232                         Real xc = x2 - Real(3)*x3;
 233                         Real xd = x3 - r.x;
 234
 235                         // Find its roots.
 236                         std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
 237
 238                         // And for the other side.
 239                         xd = x3 - r.x - r.w;
 240
 241                         std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
 242                         x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
 243
 244                         // Similarly for y-coordinates.
 245                         // Convert bezier coefficients -> cubic coefficients
 246                         Real ya = y0-y1+y2-y3;
 247                         Real yb = y1 - Real(2)*y2 + Real(3)*y3;
 248                         Real yc = y2 - Real(3)*y3;
 249                         Real yd = y3 - r.y;
 250
 251                         // Find its roots.
 252                         std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
 253
 254                         // And for the other side.
 255                         yd = y3 - r.y - r.h;
 256
 257                         std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
 258                         y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
 259
 260                         // Merge and sort.
 261                         x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
 262                         x_intersection.push_back(Real(0));
 263                         x_intersection.push_back(Real(1));
 264                         std::sort(x_intersection.begin(), x_intersection.end());
 265
 266                         Debug("Found %d intersections.\n", x_intersection.size());
 267
 268                         std::vector<Bezier> all_beziers;
 269                         if (x_intersection.empty())
 270                         {
 271                                 all_beziers.push_back(*this);
 272                                 return all_beziers;
 273                         }
 274                         Real t0 = *(x_intersection.begin());
 275                         for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
 276                         {
 277                                 Real t1 = *it;
 278                                 if (t1 == t0) continue;
 279                                 Debug(" -- t0: %f to t1: %f", t0, t1);
 280                                 Real ptx, pty;
 281                                 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
 282                                 if (r.PointIn(ptx, pty))
 283                                 {
 284                                         all_beziers.push_back(this->ReParametrise(t0, t1));
 285                                 }
 286                                 t0 = t1;
 287                         }
 288                         return all_beziers;
 289                 }
 290
 291                 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
 292                 void Evaluate(Real & x, Real & y, const Real & u) const
 293                 {
 294                         Real coeff[4];
 295                         for (unsigned i = 0; i < 4; ++i)
 296                                 coeff[i] = Bernstein(i,3,u);
 297                         x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
 298                         y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
 299                 }
 300
 301         };
 302
 303
 304
 305 }
 306
 307 #endif //_BEZIER_H