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 // discriminant > 0 => 3 distinct, real roots.
34 // discriminant = 0 => a multiple root (1 or 2 real roots)
35 // discriminant < 0 => 1 real root, 2 complex conjugate roots
37 ////HACK: We know any roots we care about will be between 0 and 1, so...
38 Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f", a,b,c,d);
41 std::vector<Real> roots;
42 for(int i = -1; i <= 100; ++i)
46 Real y = a*(x*x*x) + b*(x*x) + c*x + d;
47 if ( ((y < Real(0)) && (prevRes > Real(0))) || ((y > Real(0)) && (prevRes < Real(0))))
49 Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y);
58 /** A _cubic_ bezier. **/
65 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
66 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)
71 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) {}
73 std::string Str() const
76 s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
81 * Construct absolute control points using relative control points to a bounding rectangle
82 * ie: If cpy is relative to bounds rectangle, this will be absolute
84 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)
104 Rect SolveBounds() const;
106 Bezier ToAbsolute(const Rect & bounds) const
108 return Bezier(*this, bounds);
111 /** Convert absolute control points to control points relative to bounds
112 * (This basically does the opposite of the Copy constructor)
113 * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
115 Bezier ToRelative(const Rect & bounds) const
117 // x' <- (x - x0)/w etc
118 // special cases when w or h = 0
119 // (So can't just use the Copy constructor on the inverse of bounds)
120 // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
131 result.x0 = (x0 - bounds.x)/bounds.w;
132 result.x1 = (x1 - bounds.x)/bounds.w;
133 result.x2 = (x2 - bounds.x)/bounds.w;
134 result.x3 = (x3 - bounds.x)/bounds.w;
146 result.y0 = (y0 - bounds.y)/bounds.h;
147 result.y1 = (y1 - bounds.y)/bounds.h;
148 result.y2 = (y2 - bounds.y)/bounds.h;
149 result.y3 = (y3 - bounds.y)/bounds.h;
154 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
155 Bezier DeCasteljauSubdivideRight(const Real& t)
157 Real one_minus_t = Real(1) - t;
160 Real x01 = x0*t + x1*one_minus_t;
161 Real x12 = x1*t + x2*one_minus_t;
162 Real x23 = x2*t + x3*one_minus_t;
164 Real x012 = x01*t + x12*one_minus_t;
165 Real x123 = x12*t + x23*one_minus_t;
167 Real x0123 = x012*t + x123*one_minus_t;
170 Real y01 = y0*t + y1*one_minus_t;
171 Real y12 = y1*t + y2*one_minus_t;
172 Real y23 = y2*t + y3*one_minus_t;
174 Real y012 = y01*t + y12*one_minus_t;
175 Real y123 = y12*t + y23*one_minus_t;
177 Real y0123 = y012*t + y123*one_minus_t;
179 return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
181 // Performs one round of De Casteljau subdivision and returns the [0,t] part.
182 Bezier DeCasteljauSubdivideLeft(const Real& t)
184 Real one_minus_t = Real(1) - t;
187 Real x01 = x0*t + x1*one_minus_t;
188 Real x12 = x1*t + x2*one_minus_t;
189 Real x23 = x2*t + x3*one_minus_t;
191 Real x012 = x01*t + x12*one_minus_t;
192 Real x123 = x12*t + x23*one_minus_t;
194 Real x0123 = x012*t + x123*one_minus_t;
197 Real y01 = y0*t + y1*one_minus_t;
198 Real y12 = y1*t + y2*one_minus_t;
199 Real y23 = y2*t + y3*one_minus_t;
201 Real y012 = y01*t + y12*one_minus_t;
202 Real y123 = y12*t + y23*one_minus_t;
204 Real y0123 = y012*t + y123*one_minus_t;
206 return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
209 Bezier ReParametrise(const Real& t0, const Real& t1)
211 Debug("Reparametrise: %f -> %f",t0,t1);
213 // Subdivide to get from [0,t1]
214 new_bezier = DeCasteljauSubdivideLeft(t1);
215 // Convert t0 from [0,1] range to [0, t1]
216 Real new_t0 = t0 / t1;
217 Debug("New t0 = %f", new_t0);
218 new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
220 Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
224 std::vector<Bezier> ClipToRectangle(const Rect& r)
226 // Find points of intersection with the rectangle.
227 Debug("Clipping Bezier to Rect %s", r.Str().c_str());
229 // Convert bezier coefficients -> cubic coefficients
230 Real xa = x0-x1+x2-x3;
231 Real xb = x1 - Real(2)*x2 + Real(3)*x3;
232 Real xc = x2 - Real(3)*x3;
236 std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
238 // And for the other side.
241 std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
242 x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
244 // Similarly for y-coordinates.
245 // Convert bezier coefficients -> cubic coefficients
246 Real ya = y0-y1+y2-y3;
247 Real yb = y1 - Real(2)*y2 + Real(3)*y3;
248 Real yc = y2 - Real(3)*y3;
252 std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
254 // And for the other side.
257 std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
258 y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
261 x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
262 x_intersection.push_back(Real(0));
263 x_intersection.push_back(Real(1));
264 std::sort(x_intersection.begin(), x_intersection.end());
266 Debug("Found %d intersections.\n", x_intersection.size());
268 std::vector<Bezier> all_beziers;
269 if (x_intersection.empty())
271 all_beziers.push_back(*this);
274 Real t0 = *(x_intersection.begin());
275 for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
278 if (t1 == t0) continue;
279 Debug(" -- t0: %f to t1: %f", t0, t1);
281 Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
282 if (r.PointIn(ptx, pty))
284 all_beziers.push_back(this->ReParametrise(t0, t1));
291 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
292 void Evaluate(Real & x, Real & y, const Real & u) const
295 for (unsigned i = 0; i < 4; ++i)
296 coeff[i] = Bernstein(i,3,u);
297 x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
298 y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];