8 extern int Factorial(int n);
9 extern int BinomialCoeff(int n, int k);
10 extern Real Bernstein(int k, int n, const Real & u);
12 inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
14 Real x0((-b + Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
15 Real x1((-b - Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
16 return std::pair<Real,Real>(x0,x1);
19 inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
21 // This is going to be a big one...
22 // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
24 // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
26 Real discriminant = Real(18) * a * b * c * d - Real(4) * (b * b * b) * d
27 + (b * b) * (c * c) - Real(4) * a * (c * c * c)
28 - Real(27) * (a * a) * (d * d);
30 // discriminant > 0 => 3 distinct, real roots.
31 // discriminant = 0 => a multiple root (1 or 2 real roots)
32 // discriminant < 0 => 1 real root, 2 complex conjugate roots
34 ////HACK: We know any roots we care about will be between 0 and 1, so...
37 std::vector<Real> roots;
38 for(int i = 0; i <= 100; ++i)
42 Real y = a*(x*x*x) + b*(x*x) + c*x + d;
43 if (y == Real(0) || (y < Real(0) && prevRes > Real(0)) || (y > Real(0) && prevRes < Real(0)))
53 /** A _cubic_ bezier. **/
60 Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
61 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)
66 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) {}
68 std::string Str() const
71 s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
76 * Construct absolute control points using relative control points to a bounding rectangle
77 * ie: If cpy is relative to bounds rectangle, this will be absolute
79 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)
99 Rect SolveBounds() const;
101 Bezier ToAbsolute(const Rect & bounds) const
103 return Bezier(*this, bounds);
106 /** Convert absolute control points to control points relative to bounds
107 * (This basically does the opposite of the Copy constructor)
108 * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
110 Bezier ToRelative(const Rect & bounds) const
112 // x' <- (x - x0)/w etc
113 // special cases when w or h = 0
114 // (So can't just use the Copy constructor on the inverse of bounds)
115 // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h};
126 result.x0 = (x0 - bounds.x)/bounds.w;
127 result.x1 = (x1 - bounds.x)/bounds.w;
128 result.x2 = (x2 - bounds.x)/bounds.w;
129 result.x3 = (x3 - bounds.x)/bounds.w;
141 result.y0 = (y0 - bounds.y)/bounds.h;
142 result.y1 = (y1 - bounds.y)/bounds.h;
143 result.y2 = (y2 - bounds.y)/bounds.h;
144 result.y3 = (y3 - bounds.y)/bounds.h;
150 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
151 void Evaluate(Real & x, Real & y, const Real & u) const
154 for (unsigned i = 0; i < 4; ++i)
155 coeff[i] = Bernstein(i,3,u);
156 x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
157 y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];