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);
15 inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
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);
22 inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
24 // This is going to be a big one...
25 // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
27 // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
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);
33 Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f (Discriminant: %f)", a,b,c,d, discriminant);
34 // discriminant > 0 => 3 distinct, real roots.
35 // discriminant = 0 => a multiple root (1 or 2 real roots)
36 // discriminant < 0 => 1 real root, 2 complex conjugate roots
38 Real delta0 = (b*b) - Real(3) * a * c;
39 Real delta1 = Real(2) * (b * b * b) - Real(9) * a * b * c + Real(27) * (a * a) * d;
41 std::vector<Real> roots;
43 Real C = pow((delta1 + Sqrt((delta1 * delta1) - 4 * (delta0 * delta0 * delta0)) ) / Real(2), 1/3);
45 if (false && discriminant < 0)
47 Real real_root = (Real(-1) / (Real(3) * a)) * (b + C + delta0 / C);
49 roots.push_back(real_root);
55 ////HACK: We know any roots we care about will be between 0 and 1, so...
58 for(int i = -1; i <= 100; ++i)
62 Real y = a*(x*x*x) + b*(x*x) + c*x + d;
63 if ( ((y < Real(0)) && (prevRes > Real(0))) || ((y > Real(0)) && (prevRes < Real(0))))
65 Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y);
74 /** A _cubic_ bezier. **/
81 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
82 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)
87 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) {}
89 std::string Str() const
92 s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
97 * Construct absolute control points using relative control points to a bounding rectangle
98 * ie: If cpy is relative to bounds rectangle, this will be absolute
100 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)
120 Rect SolveBounds() const;
122 Bezier ToAbsolute(const Rect & bounds) const
124 return Bezier(*this, bounds);
127 /** Convert absolute control points to control points relative to bounds
128 * (This basically does the opposite of the Copy constructor)
129 * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
131 Bezier ToRelative(const Rect & bounds) const
133 // x' <- (x - x0)/w etc
134 // special cases when w or h = 0
135 // (So can't just use the Copy constructor on the inverse of bounds)
136 // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
147 result.x0 = (x0 - bounds.x)/bounds.w;
148 result.x1 = (x1 - bounds.x)/bounds.w;
149 result.x2 = (x2 - bounds.x)/bounds.w;
150 result.x3 = (x3 - bounds.x)/bounds.w;
162 result.y0 = (y0 - bounds.y)/bounds.h;
163 result.y1 = (y1 - bounds.y)/bounds.h;
164 result.y2 = (y2 - bounds.y)/bounds.h;
165 result.y3 = (y3 - bounds.y)/bounds.h;
170 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
171 Bezier DeCasteljauSubdivideRight(const Real& t)
173 Real one_minus_t = Real(1) - t;
176 Real x01 = x0*t + x1*one_minus_t;
177 Real x12 = x1*t + x2*one_minus_t;
178 Real x23 = x2*t + x3*one_minus_t;
180 Real x012 = x01*t + x12*one_minus_t;
181 Real x123 = x12*t + x23*one_minus_t;
183 Real x0123 = x012*t + x123*one_minus_t;
186 Real y01 = y0*t + y1*one_minus_t;
187 Real y12 = y1*t + y2*one_minus_t;
188 Real y23 = y2*t + y3*one_minus_t;
190 Real y012 = y01*t + y12*one_minus_t;
191 Real y123 = y12*t + y23*one_minus_t;
193 Real y0123 = y012*t + y123*one_minus_t;
195 return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
197 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
198 Bezier DeCasteljauSubdivideLeft(const Real& t)
200 Real one_minus_t = Real(1) - t;
203 Real x01 = x0*t + x1*one_minus_t;
204 Real x12 = x1*t + x2*one_minus_t;
205 Real x23 = x2*t + x3*one_minus_t;
207 Real x012 = x01*t + x12*one_minus_t;
208 Real x123 = x12*t + x23*one_minus_t;
210 Real x0123 = x012*t + x123*one_minus_t;
213 Real y01 = y0*t + y1*one_minus_t;
214 Real y12 = y1*t + y2*one_minus_t;
215 Real y23 = y2*t + y3*one_minus_t;
217 Real y012 = y01*t + y12*one_minus_t;
218 Real y123 = y12*t + y23*one_minus_t;
220 Real y0123 = y012*t + y123*one_minus_t;
222 return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
225 Bezier ReParametrise(const Real& t0, const Real& t1)
227 Debug("Reparametrise: %f -> %f",t0,t1);
229 // Subdivide to get from [0,t1]
230 new_bezier = DeCasteljauSubdivideLeft(t1);
231 // Convert t0 from [0,1] range to [0, t1]
232 Real new_t0 = t0 / t1;
233 Debug("New t0 = %f", new_t0);
234 new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
236 Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
240 std::vector<Bezier> ClipToRectangle(const Rect& r)
242 // Find points of intersection with the rectangle.
243 Debug("Clipping Bezier to Rect %s", r.Str().c_str());
245 // Convert bezier coefficients -> cubic coefficients
247 Real xc = Real(3)*(x1 - x0);
248 Real xb = Real(3)*(x2 - x1) - xc;
249 Real xa = x3 - x0 - xc - xb;
252 std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
254 // And for the other side.
257 std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
258 x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
260 // Similarly for y-coordinates.
261 // Convert bezier coefficients -> cubic coefficients
263 Real yc = Real(3)*(y1 - y0);
264 Real yb = Real(3)*(y2 - y1) - yc;
265 Real ya = y3 - y0 - yc - yb;
268 std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
270 // And for the other side.
273 std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
274 y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
277 x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
278 x_intersection.push_back(Real(0));
279 x_intersection.push_back(Real(1));
280 std::sort(x_intersection.begin(), x_intersection.end());
282 Debug("Found %d intersections.\n", x_intersection.size());
284 std::vector<Bezier> all_beziers;
285 if (x_intersection.empty())
287 all_beziers.push_back(*this);
290 Real t0 = *(x_intersection.begin());
291 for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
294 if (t1 == t0) continue;
295 Debug(" -- t0: %f to t1: %f", t0, t1);
297 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
298 if (r.PointIn(ptx, pty))
300 all_beziers.push_back(this->ReParametrise(t0, t1));
304 Debug("Segment removed (point at %f, %f)", ptx, pty);
311 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
312 void Evaluate(Real & x, Real & y, const Real & u) const
315 for (unsigned i = 0; i < 4; ++i)
316 coeff[i] = Bernstein(i,3,u);
317 x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
318 y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];