Totally break Quadtree Béziers.
[ipdf/code.git] / src / bezier.h
index 722ed57..6dac0d5 100644 (file)
@@ -11,11 +11,44 @@ namespace IPDF
        
        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
        {
@@ -111,7 +144,100 @@ namespace IPDF
                        }
                        return result;
                }
+
+               Bezier ReParametrise(const Real& t0, const Real& t1)
+               {
+                       // This function is very, very ugly, but with luck my derivation is correct (even if it isn't optimal, performance wise)
+                       // (Very) rough working for the derivation is at: http://davidgow.net/stuff/cubic_bezier_reparam.pdf
+                       Bezier new_bezier;
+                       Real tdiff = t1 - t0;
+                       Real tdiff_squared = tdiff*tdiff;
+                       Real tdiff_cubed = tdiff*tdiff_squared;
+
+                       Real t0_squared = t0*t0;
+                       Real t0_cubed = t0*t0_squared;
+                       
+                       // X coordinates
+                       Real Dx0 = x0 / tdiff_cubed;
+                       Real Dx1 = x1 / (tdiff_squared - tdiff_cubed);
+                       Real Dx2 = x2 / (tdiff - Real(2)*tdiff_squared + tdiff_cubed);
+                       Real Dx3 = x3 / (Real(1) - Real(3)*tdiff + Real(3)*tdiff_squared - tdiff_cubed);
+
+                       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;
+                       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;
+                       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;
+                       new_bezier.x0 = Dx0 - Dx1 + Dx2 - Dx3 + new_bezier.x1 - new_bezier.x2 + new_bezier.x3;
+
+                       // Y coordinates
+                       Real Dy0 = y0 / tdiff_cubed;
+                       Real Dy1 = y1 / (tdiff_squared - tdiff_cubed);
+                       Real Dy2 = y2 / (tdiff - Real(2)*tdiff_squared + tdiff_cubed);
+                       Real Dy3 = y3 / (Real(1) - Real(3)*tdiff + Real(3)*tdiff_squared - tdiff_cubed);
+
+                       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;
+                       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;
+                       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;
+                       new_bezier.y0 = Dy0 - Dy1 + Dy2 - Dy3 + new_bezier.y1 - new_bezier.y2 + new_bezier.y3;
+
+
+                       return new_bezier;
+               }
                
+               std::vector<Bezier> ClipToRectangle(const Rect& r)
+               {
+                       // Find points of intersection with the rectangle.
+
+                       // Convert bezier coefficients -> cubic coefficients
+                       Real xa = x0-x1+x2-x3;
+                       Real xb = x1 - Real(2)*x2 + Real(3)*x3;
+                       Real xc = x2 - Real(3)*x3;
+                       Real xd = x3 + r.x;
+
+                       // Find its roots.
+                       std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
+
+                       // And for the other side.
+                       xd = x3 + r.x + r.w;
+
+                       std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
+                       x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
+
+                       // Similarly for y-coordinates.
+                       // Convert bezier coefficients -> cubic coefficients
+                       Real ya = y0-y1+y2-y3;
+                       Real yb = y1 - Real(2)*y2 + Real(3)*y3;
+                       Real yc = y2 - Real(3)*y3;
+                       Real yd = y3 + r.y;
+
+                       // Find its roots.
+                       std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
+
+                       // And for the other side.
+                       yd = y3 + r.y + r.h;
+
+                       std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
+                       y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
+
+                       // Merge and sort.
+                       x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
+
+                       Debug("Found %d intersections.\n", x_intersection.size());
+                       
+                       std::vector<Bezier> all_beziers;
+                       if (x_intersection.empty())
+                       {
+                               all_beziers.push_back(*this);
+                               return all_beziers;
+                       }
+                       Real t0 = *(x_intersection.begin());
+                       for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
+                       {
+                               Real t1 = *it;
+                               all_beziers.push_back(this->ReParametrise(t0, t1));
+                               t0 = t1;
+                       }
+                       return all_beziers;
+               }
 
                /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
                void Evaluate(Real & x, Real & y, const Real & u) const

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