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);
16 extern std::vector<Real> SolveQuadratic(const Real & a, const Real & b, const Real & c, const Real & min = 0, const Real & max = 1);
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);
20 /** A _cubic_ bezier. **/
28 typedef enum {LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
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)
34 //TODO: classify the curve
38 std::string Str() const
41 s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
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
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)
69 Rect SolveBounds() const;
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;
76 Bezier ToAbsolute(const Rect & bounds) const
78 return Bezier(*this, bounds);
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
85 Bezier ToRelative(const Rect & bounds) const
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};
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;
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;
124 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
125 Bezier DeCasteljauSubdivideRight(const Real& t)
127 Real one_minus_t = Real(1) - t;
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;
134 Real x012 = x01*t + x12*one_minus_t;
135 Real x123 = x12*t + x23*one_minus_t;
137 Real x0123 = x012*t + x123*one_minus_t;
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;
144 Real y012 = y01*t + y12*one_minus_t;
145 Real y123 = y12*t + y23*one_minus_t;
147 Real y0123 = y012*t + y123*one_minus_t;
149 return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
151 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
152 Bezier DeCasteljauSubdivideLeft(const Real& t)
154 Real one_minus_t = Real(1) - t;
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;
161 Real x012 = x01*t + x12*one_minus_t;
162 Real x123 = x12*t + x23*one_minus_t;
164 Real x0123 = x012*t + x123*one_minus_t;
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;
171 Real y012 = y01*t + y12*one_minus_t;
172 Real y123 = y12*t + y23*one_minus_t;
174 Real y0123 = y012*t + y123*one_minus_t;
176 return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
179 Bezier ReParametrise(const Real& t0, const Real& t1)
181 Debug("Reparametrise: %f -> %f",t0,t1);
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);
190 Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
194 std::vector<Bezier> ClipToRectangle(const Rect& r)
196 // Find points of intersection with the rectangle.
197 Debug("Clipping Bezier to Rect %s", r.Str().c_str());
199 // Convert bezier coefficients -> cubic coefficients
201 Real xc = Real(3)*(x1 - x0);
202 Real xb = Real(3)*(x2 - x1) - xc;
203 Real xa = x3 - x0 - xc - xb;
206 std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
208 // And for the other side.
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());
214 // Similarly for y-coordinates.
215 // Convert bezier coefficients -> cubic coefficients
217 Real yc = Real(3)*(y1 - y0);
218 Real yb = Real(3)*(y2 - y1) - yc;
219 Real ya = y3 - y0 - yc - yb;
222 std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
224 // And for the other side.
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());
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());
236 Debug("Found %d intersections.\n", x_intersection.size());
238 std::vector<Bezier> all_beziers;
239 if (x_intersection.size() <= 2)
241 all_beziers.push_back(*this);
244 Real t0 = *(x_intersection.begin());
245 for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
248 if (t1 == t0) continue;
249 Debug(" -- t0: %f to t1: %f", t0, t1);
251 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
252 if (true || r.PointIn(ptx, pty))
254 all_beziers.push_back(this->ReParametrise(t0, t1));
258 Debug("Segment removed (point at %f, %f)", ptx, pty);
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
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];
274 std::vector<Vec2> Evaluate(const std::vector<Real> & u) const;
276 std::vector<Real> SolveXParam(const Real & x) const;
277 std::vector<Real> SolveYParam(const Real & x) const;
279 // Get points with same X
280 inline std::vector<Vec2> SolveX(const Real & x) const
282 return Evaluate(SolveXParam(x));
284 // Get points with same Y
285 inline std::vector<Vec2> SolveY(const Real & y) const
287 return Evaluate(SolveYParam(y));
290 bool operator==(const Bezier & equ) const
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);
297 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}