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 inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
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);
23 inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
25 // This is going to be a big one...
26 // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
28 // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
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);
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
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;
42 std::vector<Real> roots;
44 Real C = pow((delta1 + Sqrt((delta1 * delta1) - 4 * (delta0 * delta0 * delta0)) ) / Real(2), 1/3);
46 if (false && discriminant < 0)
48 Real real_root = (Real(-1) / (Real(3) * a)) * (b + C + delta0 / C);
50 roots.push_back(real_root);
56 ////HACK: We know any roots we care about will be between 0 and 1, so...
59 for(int i = -1; i <= 100; ++i)
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))))
66 Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y);
75 /** A _cubic_ bezier. **/
83 typedef enum {LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
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)
89 //TODO: classify the curve
93 std::string Str() const
96 s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
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
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)
124 Rect SolveBounds() const;
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;
131 Bezier ToAbsolute(const Rect & bounds) const
133 return Bezier(*this, bounds);
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
140 Bezier ToRelative(const Rect & bounds) const
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};
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;
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;
179 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
180 Bezier DeCasteljauSubdivideRight(const Real& t)
182 Real one_minus_t = Real(1) - t;
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;
189 Real x012 = x01*t + x12*one_minus_t;
190 Real x123 = x12*t + x23*one_minus_t;
192 Real x0123 = x012*t + x123*one_minus_t;
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;
199 Real y012 = y01*t + y12*one_minus_t;
200 Real y123 = y12*t + y23*one_minus_t;
202 Real y0123 = y012*t + y123*one_minus_t;
204 return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
206 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
207 Bezier DeCasteljauSubdivideLeft(const Real& t)
209 Real one_minus_t = Real(1) - t;
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;
216 Real x012 = x01*t + x12*one_minus_t;
217 Real x123 = x12*t + x23*one_minus_t;
219 Real x0123 = x012*t + x123*one_minus_t;
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;
226 Real y012 = y01*t + y12*one_minus_t;
227 Real y123 = y12*t + y23*one_minus_t;
229 Real y0123 = y012*t + y123*one_minus_t;
231 return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
234 Bezier ReParametrise(const Real& t0, const Real& t1)
236 Debug("Reparametrise: %f -> %f",t0,t1);
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);
245 Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
249 std::vector<Bezier> ClipToRectangle(const Rect& r)
251 // Find points of intersection with the rectangle.
252 Debug("Clipping Bezier to Rect %s", r.Str().c_str());
254 // Convert bezier coefficients -> cubic coefficients
256 Real xc = Real(3)*(x1 - x0);
257 Real xb = Real(3)*(x2 - x1) - xc;
258 Real xa = x3 - x0 - xc - xb;
261 std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
263 // And for the other side.
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());
269 // Similarly for y-coordinates.
270 // Convert bezier coefficients -> cubic coefficients
272 Real yc = Real(3)*(y1 - y0);
273 Real yb = Real(3)*(y2 - y1) - yc;
274 Real ya = y3 - y0 - yc - yb;
277 std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
279 // And for the other side.
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());
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());
291 Debug("Found %d intersections.\n", x_intersection.size());
293 std::vector<Bezier> all_beziers;
294 if (x_intersection.size() <= 2)
296 all_beziers.push_back(*this);
299 Real t0 = *(x_intersection.begin());
300 for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
303 if (t1 == t0) continue;
304 Debug(" -- t0: %f to t1: %f", t0, t1);
306 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
307 if (true || r.PointIn(ptx, pty))
309 all_beziers.push_back(this->ReParametrise(t0, t1));
313 Debug("Segment removed (point at %f, %f)", ptx, pty);
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
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];
330 bool operator==(const Bezier & equ) const
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);
337 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}