X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fcode.git;a=blobdiff_plain;f=src%2Fbezier.h;h=6dac0d51738d5eae781b767d4ce43afc0c95d128;hp=0e9217f7a7d8690efe8074f54bc18e5b6f909680;hb=101df292750d2fe8ed1de541db197dd792232458;hpb=d69b8de94411bee43edc9e16f33bfa0d5d1d6b3b diff --git a/src/bezier.h b/src/bezier.h index 0e9217f..6dac0d5 100644 --- a/src/bezier.h +++ b/src/bezier.h @@ -8,6 +8,46 @@ namespace IPDF extern int Factorial(int n); extern int BinomialCoeff(int n, int k); extern Real Bernstein(int k, int n, const Real & u); + + inline std::pair 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)); + return std::pair(x0,x1); + } + + inline std::vector 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 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 @@ -17,7 +57,10 @@ namespace IPDF Real x2; Real y2; Real x3; Real y3; Bezier() = default; // Needed so we can fread/fwrite this struct... for now. - 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) {} + 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) + { + + } 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) {} @@ -27,6 +70,11 @@ namespace IPDF s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}"; return s.str(); } + + /** + * Construct absolute control points using relative control points to a bounding rectangle + * ie: If cpy is relative to bounds rectangle, this will be absolute + */ 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) { x0 *= t.w; @@ -47,10 +95,152 @@ namespace IPDF y3 += t.y; } - Rect ToRect() {return Rect(x0,y0,x3-x0,y3-y0);} + 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 ToRelative(const Rect & bounds) const + { + // x' <- (x - x0)/w etc + // special cases when w or h = 0 + // (So can't just use the Copy constructor on the inverse of bounds) + // Rect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, Real(1)/bounds.w, Real(1)/bounds.h}; + Bezier result; + if (bounds.w == 0) + { + result.x0 = 0; + result.x1 = 0; + result.x2 = 0; + result.x3 = 0; + } + else + { + result.x0 = (x0 - bounds.x)/bounds.w; + result.x1 = (x1 - bounds.x)/bounds.w; + result.x2 = (x2 - bounds.x)/bounds.w; + result.x3 = (x3 - bounds.x)/bounds.w; + } + + if (bounds.h == 0) + { + result.y0 = 0; + result.y1 = 0; + result.y2 = 0; + result.y3 = 0; + } + else + { + result.y0 = (y0 - bounds.y)/bounds.h; + result.y1 = (y1 - bounds.y)/bounds.h; + result.y2 = (y2 - bounds.y)/bounds.h; + result.y3 = (y3 - bounds.y)/bounds.h; + } + 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 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 x_intersection = SolveCubic(xa, xb, xc, xd); + + // And for the other side. + xd = x3 + r.x + r.w; + + std::vector 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 y_intersection = SolveCubic(ya, yb, yc, yd); + + // And for the other side. + yd = y3 + r.y + r.h; + + std::vector 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 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) + void Evaluate(Real & x, Real & y, const Real & u) const { Real coeff[4]; for (unsigned i = 0; i < 4; ++i)