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 {UNKNOWN, 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), type(UNKNOWN)
37 const Type & GetType()
39 if (type != Bezier::UNKNOWN)
41 // From Loop-Blinn 2005, with w0 == w1 == w2 == w3 = 1
42 // Transformed control points: (a0 = x0, b0 = y0)
44 Real a2 = (x0- x1*2 +x2)*3;
45 Real a3 = (x3 - x0 + (x1 - x2)*3);
48 Real b2 = (y0- y1*2 +y2)*3;
49 Real b3 = (y3 - y0 + (y1 - y2)*3);
51 // d vector (d0 = 0 since all w = 1)
52 Real d1 = a2*b3 - a3*b2;
53 Real d2 = a3*b1 - a1*b3;
54 Real d3 = a1*b2 - a2*b1;
56 if (d1 == d2 && d2 == d3 && d3 == 0)
59 //Debug("LINE %s", Str().c_str());
65 Real delta3 = d1*d3 -d2*d2;
66 if (delta1 == delta2 && delta2 == delta3 && delta3 == 0)
70 //Debug("QUADRATIC %s", Str().c_str());
74 Real discriminant = d1*d3*4 -d2*d2;
75 if (discriminant == 0)
78 //Debug("CUSP %s", Str().c_str());
80 else if (discriminant > 0)
83 //Debug("SERPENTINE %s", Str().c_str());
88 //Debug("LOOP %s", Str().c_str());
94 std::string Str() const
97 s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
102 * Construct absolute control points using relative control points to a bounding rectangle
103 * ie: If cpy is relative to bounds rectangle, this will be absolute
105 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(UNKNOWN)
125 Rect SolveBounds() const;
127 std::pair<Real,Real> GetTop() const;
128 std::pair<Real,Real> GetBottom() const;
129 std::pair<Real,Real> GetLeft() const;
130 std::pair<Real,Real> GetRight() const;
132 Bezier ToAbsolute(const Rect & bounds) const
134 return Bezier(*this, bounds);
137 /** Convert absolute control points to control points relative to bounds
138 * (This basically does the opposite of the Copy constructor)
139 * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
141 Bezier ToRelative(const Rect & bounds) const
143 // x' <- (x - x0)/w etc
144 // special cases when w or h = 0
145 // (So can't just use the Copy constructor on the inverse of bounds)
146 // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
157 result.x0 = (x0 - bounds.x)/bounds.w;
158 result.x1 = (x1 - bounds.x)/bounds.w;
159 result.x2 = (x2 - bounds.x)/bounds.w;
160 result.x3 = (x3 - bounds.x)/bounds.w;
172 result.y0 = (y0 - bounds.y)/bounds.h;
173 result.y1 = (y1 - bounds.y)/bounds.h;
174 result.y2 = (y2 - bounds.y)/bounds.h;
175 result.y3 = (y3 - bounds.y)/bounds.h;
180 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
181 Bezier DeCasteljauSubdivideRight(const Real& t)
183 Real one_minus_t = Real(1) - t;
186 Real x01 = x0*t + x1*one_minus_t;
187 Real x12 = x1*t + x2*one_minus_t;
188 Real x23 = x2*t + x3*one_minus_t;
190 Real x012 = x01*t + x12*one_minus_t;
191 Real x123 = x12*t + x23*one_minus_t;
193 Real x0123 = x012*t + x123*one_minus_t;
196 Real y01 = y0*t + y1*one_minus_t;
197 Real y12 = y1*t + y2*one_minus_t;
198 Real y23 = y2*t + y3*one_minus_t;
200 Real y012 = y01*t + y12*one_minus_t;
201 Real y123 = y12*t + y23*one_minus_t;
203 Real y0123 = y012*t + y123*one_minus_t;
205 return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
207 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
208 Bezier DeCasteljauSubdivideLeft(const Real& t)
210 Real one_minus_t = Real(1) - t;
213 Real x01 = x0*t + x1*one_minus_t;
214 Real x12 = x1*t + x2*one_minus_t;
215 Real x23 = x2*t + x3*one_minus_t;
217 Real x012 = x01*t + x12*one_minus_t;
218 Real x123 = x12*t + x23*one_minus_t;
220 Real x0123 = x012*t + x123*one_minus_t;
223 Real y01 = y0*t + y1*one_minus_t;
224 Real y12 = y1*t + y2*one_minus_t;
225 Real y23 = y2*t + y3*one_minus_t;
227 Real y012 = y01*t + y12*one_minus_t;
228 Real y123 = y12*t + y23*one_minus_t;
230 Real y0123 = y012*t + y123*one_minus_t;
232 return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
235 Bezier ReParametrise(const Real& t0, const Real& t1)
237 Debug("Reparametrise: %f -> %f",t0,t1);
239 // Subdivide to get from [0,t1]
240 new_bezier = DeCasteljauSubdivideLeft(t1);
241 // Convert t0 from [0,1] range to [0, t1]
242 Real new_t0 = t0 / t1;
243 Debug("New t0 = %f", new_t0);
244 new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
246 Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
250 std::vector<Bezier> ClipToRectangle(const Rect& r)
252 // Find points of intersection with the rectangle.
253 Debug("Clipping Bezier to Rect %s", r.Str().c_str());
255 // Convert bezier coefficients -> cubic coefficients
257 Real xc = Real(3)*(x1 - x0);
258 Real xb = Real(3)*(x2 - x1) - xc;
259 Real xa = x3 - x0 - xc - xb;
262 std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
264 // And for the other side.
267 std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
268 x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
270 // Similarly for y-coordinates.
271 // Convert bezier coefficients -> cubic coefficients
273 Real yc = Real(3)*(y1 - y0);
274 Real yb = Real(3)*(y2 - y1) - yc;
275 Real ya = y3 - y0 - yc - yb;
278 std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
280 // And for the other side.
283 std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
284 y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
287 x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
288 x_intersection.push_back(Real(0));
289 x_intersection.push_back(Real(1));
290 std::sort(x_intersection.begin(), x_intersection.end());
292 Debug("Found %d intersections.\n", x_intersection.size());
294 std::vector<Bezier> all_beziers;
295 if (x_intersection.size() <= 2)
297 all_beziers.push_back(*this);
300 Real t0 = *(x_intersection.begin());
301 for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
304 if (t1 == t0) continue;
305 Debug(" -- t0: %f to t1: %f", t0, t1);
307 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
308 if (true || r.PointIn(ptx, pty))
310 all_beziers.push_back(this->ReParametrise(t0, t1));
314 Debug("Segment removed (point at %f, %f)", ptx, pty);
321 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
322 void Evaluate(Real & x, Real & y, const Real & u) const
325 for (unsigned i = 0; i < 4; ++i)
326 coeff[i] = Bernstein(i,3,u);
327 x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
328 y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
330 std::vector<Vec2> Evaluate(const std::vector<Real> & u) const;
332 std::vector<Real> SolveXParam(const Real & x) const;
333 std::vector<Real> SolveYParam(const Real & x) const;
335 // Get points with same X
336 inline std::vector<Vec2> SolveX(const Real & x) const
338 return Evaluate(SolveXParam(x));
340 // Get points with same Y
341 inline std::vector<Vec2> SolveY(const Real & y) const
343 return Evaluate(SolveYParam(y));
346 bool operator==(const Bezier & equ) const
348 return (x0 == equ.x0 && y0 == equ.y0
349 && x1 == equ.x1 && y1 == equ.y1
350 && x2 == equ.x2 && y2 == equ.y2
351 && x3 == equ.x3 && y3 == equ.y3);
353 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}