inline std::pair<Real,Real> SolveQuadratic(const Real & a, const Real & b, const Real & c)
{
- Real x0((b + Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
- Real x1((b - Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
+ Real x0((-b + Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
+ Real x1((-b - Sqrt(b*b - Real(4)*a*c))/(Real(2)*a));
return std::pair<Real,Real>(x0,x1);
}
+ inline std::vector<Real> SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d)
+ {
+ // This is going to be a big one...
+ // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
+
+ // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2
+ /*
+ Real discriminant = Real(18) * a * b * c * d - Real(4) * (b * b * b) * d
+ + (b * b) * (c * c) - Real(4) * a * (c * c * c)
+ - Real(27) * (a * a) * (d * d);
+ */
+ // discriminant > 0 => 3 distinct, real roots.
+ // discriminant = 0 => a multiple root (1 or 2 real roots)
+ // discriminant < 0 => 1 real root, 2 complex conjugate roots
+
+ ////HACK: We know any roots we care about will be between 0 and 1, so...
+ Real maxi(100);
+ Real prevRes(d);
+ std::vector<Real> roots;
+ for(int i = 0; i <= 100; ++i)
+ {
+ Real x(i);
+ x /= maxi;
+ Real y = a*(x*x*x) + b*(x*x) + c*x + d;
+ if (y == Real(0) || (y < Real(0) && prevRes > Real(0)) || (y > Real(0) && prevRes < Real(0)))
+ {
+ roots.push_back(x);
+ }
+ }
+ return roots;
+
+ }
+
+
/** A _cubic_ bezier. **/
struct Bezier
{
Rect SolveBounds() const;
+ Bezier ToAbsolute(const Rect & bounds) const
+ {
+ return Bezier(*this, bounds);
+ }
+
/** Convert absolute control points to control points relative to bounds
* (This basically does the opposite of the Copy constructor)
* ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
*/
- Bezier CopyInverse(const Rect & bounds) const
+ Bezier ToRelative(const Rect & bounds) const
{
// x' <- (x - x0)/w etc
// special cases when w or h = 0