Infinite Quadtree precision now works with some B├ęziers
[ipdf/code.git] / 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

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