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 Bezier ReParametrise(const Real& t0, const Real& t1)
156 // This function is very, very ugly, but with luck my derivation is correct (even if it isn't optimal, performance wise)
157 // (Very) rough working for the derivation is at: http://davidgow.net/stuff/cubic_bezier_reparam.pdf
158 Debug("Reparametrise: %f -> %f",t0,t1);
160 Real tdiff = t1 - t0;
161 Real tdiff_squared = tdiff*tdiff;
162 Real tdiff_cubed = tdiff*tdiff_squared;
164 Real t0_squared = t0*t0;
165 Real t0_cubed = t0*t0_squared;
168 Real Dx0 = x0 / tdiff_cubed;
169 Real Dx1 = x1 / (tdiff_squared - tdiff_cubed);
170 Real Dx2 = x2 / (tdiff - Real(2)*tdiff_squared + tdiff_cubed);
171 Real Dx3 = x3 / (Real(1) - Real(3)*tdiff + Real(3)*tdiff_squared - tdiff_cubed);
173 new_bezier.x3 = Dx3*t0_cubed + Real(3)*Dx3*t0_squared + Real(3)*Dx3*t0 + Dx3 - Dx2*t0_cubed - Real(2)*Dx2*t0_squared - Dx2*t0 + Dx1*t0_cubed + Dx1*t0_squared - Dx0*t0_cubed;
174 new_bezier.x2 = Real(3)*Dx0*t0_squared - Real(2)*Dx1*t0 - Real(3)*Dx1*t0_squared + Dx2 + Real(4)*Dx2*t0 + Real(3)*Dx2*t0_squared - Real(3)*Dx3 - Real(6)*Dx3*t0 - Real(3)*Dx3*t0_squared + Real(3)*new_bezier.x3;
175 new_bezier.x1 = Real(-3)*Dx0*t0 + Real(3)*Dx1*t0 + Dx1 - Real(2)*Dx2 - Real(3)*Dx2*t0 + Real(3)*Dx3 + Real(3)*Dx3*t0 + Real(2)*new_bezier.x2 - Real(3)*new_bezier.x3;
176 new_bezier.x0 = Dx0 - Dx1 + Dx2 - Dx3 + new_bezier.x1 - new_bezier.x2 + new_bezier.x3;
179 Real Dy0 = y0 / tdiff_cubed;
180 Real Dy1 = y1 / (tdiff_squared - tdiff_cubed);
181 Real Dy2 = y2 / (tdiff - Real(2)*tdiff_squared + tdiff_cubed);
182 Real Dy3 = y3 / (Real(1) - Real(3)*tdiff + Real(3)*tdiff_squared - tdiff_cubed);
184 new_bezier.y3 = Dy3*t0_cubed + Real(3)*Dy3*t0_squared + Real(3)*Dy3*t0 + Dy3 - Dy2*t0_cubed - Real(2)*Dy2*t0_squared - Dy2*t0 + Dy1*t0_cubed + Dy1*t0_squared - Dy0*t0_cubed;
185 new_bezier.y2 = Real(3)*Dy0*t0_squared - Real(2)*Dy1*t0 - Real(3)*Dy1*t0_squared + Dy2 + Real(4)*Dy2*t0 + Real(3)*Dy2*t0_squared - Real(3)*Dy3 - Real(6)*Dy3*t0 - Real(3)*Dy3*t0_squared + Real(3)*new_bezier.y3;
186 new_bezier.y1 = Real(-3)*Dy0*t0 + Real(3)*Dy1*t0 + Dy1 - Real(2)*Dy2 - Real(3)*Dy2*t0 + Real(3)*Dy3 + Real(3)*Dy3*t0 + Real(2)*new_bezier.y2 - Real(3)*new_bezier.y3;
187 new_bezier.y0 = Dy0 - Dy1 + Dy2 - Dy3 + new_bezier.y1 - new_bezier.y2 + new_bezier.y3;
190 Debug("(%f,%f),(%f,%f),(%f,%f),(%f,%f) -> (%f,%f),(%f,%f),(%f,%f),(%f,%f)", x0, y0, x1, y1, x2, y2, x3, y3, new_bezier.x0, new_bezier.y0, new_bezier.x1, new_bezier.y1, new_bezier.x2, new_bezier.y2, new_bezier.x3, new_bezier.y3);
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
200 Real xa = x0-x1+x2-x3;
201 Real xb = x1 - Real(2)*x2 + Real(3)*x3;
202 Real xc = x2 - Real(3)*x3;
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
216 Real ya = y0-y1+y2-y3;
217 Real yb = y1 - Real(2)*y2 + Real(3)*y3;
218 Real yc = y2 - Real(3)*y3;
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 std::sort(x_intersection.begin(), x_intersection.end());
234 Debug("Found %d intersections.\n", x_intersection.size());
236 std::vector<Bezier> all_beziers;
237 if (x_intersection.empty())
239 all_beziers.push_back(*this);
242 Real t0 = *(x_intersection.begin());
243 for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
246 Debug(" -- t0: %f to t1: %f", t0, t1);
247 all_beziers.push_back(this->ReParametrise(t0, t1));
253 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
254 void Evaluate(Real & x, Real & y, const Real & u) const
257 for (unsigned i = 0; i < 4; ++i)
258 coeff[i] = Bernstein(i,3,u);
259 x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
260 y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];