X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fcode.git;a=blobdiff_plain;f=src%2Fbezier.h;h=03e789cf64af4b17352254d6f62d1d75c6c6707a;hp=3a8e80dec210da86fdc89fbec69a8025fb36456d;hb=d272af0f7f981cea9d1024b6a730be73dd22276a;hpb=b85533b9da0dc3f6cdf8e329250518d4ac82e434 diff --git a/src/bezier.h b/src/bezier.h index 3a8e80d..03e789c 100644 --- a/src/bezier.h +++ b/src/bezier.h @@ -1,6 +1,9 @@ #ifndef _BEZIER_H #define _BEZIER_H +#include +#include + #include "real.h" #include "rect.h" namespace IPDF @@ -8,22 +11,97 @@ namespace IPDF extern int Factorial(int n); extern int BinomialCoeff(int n, int k); extern Real Bernstein(int k, int n, const Real & u); + extern std::pair BezierTurningPoints(const Real & p0, const Real & p1, const Real & p2, const Real & p3); + + 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); + + Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f (Discriminant: %f)", a,b,c,d, discriminant); + // 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 + + Real delta0 = (b*b) - Real(3) * a * c; + Real delta1 = Real(2) * (b * b * b) - Real(9) * a * b * c + Real(27) * (a * a) * d; + + std::vector roots; + + Real C = pow((delta1 + Sqrt((delta1 * delta1) - 4 * (delta0 * delta0 * delta0)) ) / Real(2), 1/3); - /** A _quadratic_ bezier. **/ + if (false && discriminant < 0) + { + Real real_root = (Real(-1) / (Real(3) * a)) * (b + C + delta0 / C); + + roots.push_back(real_root); + + return roots; + + } + + ////HACK: We know any roots we care about will be between 0 and 1, so... + Real maxi(100); + Real prevRes(d); + for(int i = -1; i <= 100; ++i) + { + Real x(i); + x /= maxi; + Real y = a*(x*x*x) + b*(x*x) + c*x + d; + if ( ((y < Real(0)) && (prevRes > Real(0))) || ((y > Real(0)) && (prevRes < Real(0)))) + { + Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y); + roots.push_back(x); + } + prevRes = y; + } + return roots; + + } + + /** A _cubic_ bezier. **/ struct Bezier { Real x0; Real y0; Real x1; Real y1; Real x2; Real y2; + Real x3; Real y3; + + typedef enum {LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type; + Type type; + 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) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2) {} + 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) + { + //TODO: classify the curve + type = SERPENTINE; + } + std::string Str() const { std::stringstream s; - s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << "}"; + s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}"; return s.str(); } - 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) + + /** + * 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), type(cpy.type) { x0 *= t.w; y0 *= t.h; @@ -31,25 +109,232 @@ namespace IPDF y1 *= t.h; x2 *= t.w; y2 *= t.h; + x3 *= t.w; + y3 *= t.h; x0 += t.x; y0 += t.y; x1 += t.x; y1 += t.y; x2 += t.x; y2 += t.y; + x3 += t.x; + y3 += t.y; + } + + Rect SolveBounds() const; + + std::pair GetTop() const; + std::pair GetBottom() const; + std::pair GetLeft() const; + std::pair GetRight() 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; + } + + // Performs one round of De Casteljau subdivision and returns the [t,1] part. + Bezier DeCasteljauSubdivideRight(const Real& t) + { + Real one_minus_t = Real(1) - t; + + // X Coordinates + Real x01 = x0*t + x1*one_minus_t; + Real x12 = x1*t + x2*one_minus_t; + Real x23 = x2*t + x3*one_minus_t; + + Real x012 = x01*t + x12*one_minus_t; + Real x123 = x12*t + x23*one_minus_t; + + Real x0123 = x012*t + x123*one_minus_t; + + // Y Coordinates + Real y01 = y0*t + y1*one_minus_t; + Real y12 = y1*t + y2*one_minus_t; + Real y23 = y2*t + y3*one_minus_t; - Rect ToRect() {return Rect(x0,y0,x2-x0,y2-y0);} + Real y012 = y01*t + y12*one_minus_t; + Real y123 = y12*t + y23*one_minus_t; + + Real y0123 = y012*t + y123*one_minus_t; + + return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123); + } + // Performs one round of De Casteljau subdivision and returns the [0,t] part. + Bezier DeCasteljauSubdivideLeft(const Real& t) + { + Real one_minus_t = Real(1) - t; + + // X Coordinates + Real x01 = x0*t + x1*one_minus_t; + Real x12 = x1*t + x2*one_minus_t; + Real x23 = x2*t + x3*one_minus_t; + + Real x012 = x01*t + x12*one_minus_t; + Real x123 = x12*t + x23*one_minus_t; + + Real x0123 = x012*t + x123*one_minus_t; + + // Y Coordinates + Real y01 = y0*t + y1*one_minus_t; + Real y12 = y1*t + y2*one_minus_t; + Real y23 = y2*t + y3*one_minus_t; + + Real y012 = y01*t + y12*one_minus_t; + Real y123 = y12*t + y23*one_minus_t; + + Real y0123 = y012*t + y123*one_minus_t; + + return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3); + } + + Bezier ReParametrise(const Real& t0, const Real& t1) + { + Debug("Reparametrise: %f -> %f",t0,t1); + Bezier new_bezier; + // Subdivide to get from [0,t1] + new_bezier = DeCasteljauSubdivideLeft(t1); + // Convert t0 from [0,1] range to [0, t1] + Real new_t0 = t0 / t1; + Debug("New t0 = %f", new_t0); + new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0); + + Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str()); + return new_bezier; + } + + std::vector ClipToRectangle(const Rect& r) + { + // Find points of intersection with the rectangle. + Debug("Clipping Bezier to Rect %s", r.Str().c_str()); + + // Convert bezier coefficients -> cubic coefficients + Real xd = x0 - r.x; + Real xc = Real(3)*(x1 - x0); + Real xb = Real(3)*(x2 - x1) - xc; + Real xa = x3 - x0 - xc - xb; + + // Find its roots. + std::vector x_intersection = SolveCubic(xa, xb, xc, xd); + + // And for the other side. + xd = x0 - 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 yd = y0 - r.y; + Real yc = Real(3)*(y1 - y0); + Real yb = Real(3)*(y2 - y1) - yc; + Real ya = y3 - y0 - yc - yb; + + // Find its roots. + std::vector y_intersection = SolveCubic(ya, yb, yc, yd); + + // And for the other side. + yd = y0 - 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()); + x_intersection.push_back(Real(0)); + x_intersection.push_back(Real(1)); + std::sort(x_intersection.begin(), x_intersection.end()); + + Debug("Found %d intersections.\n", x_intersection.size()); + + std::vector all_beziers; + if (x_intersection.size() <= 2) + { + 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; + if (t1 == t0) continue; + Debug(" -- t0: %f to t1: %f", t0, t1); + Real ptx, pty; + Evaluate(ptx, pty, ((t1 + t0) / Real(2))); + if (true || r.PointIn(ptx, pty)) + { + all_beziers.push_back(this->ReParametrise(t0, t1)); + } + else + { + Debug("Segment removed (point at %f, %f)", ptx, pty); + } + 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) + coeff[i] = Bernstein(i,3,u); + x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3]; + y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3]; + } + + bool operator==(const Bezier & equ) const { - Real coeff[3]; - for (unsigned i = 0; i < 3; ++i) - coeff[i] = Bernstein(i,2,u); - x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2]; - y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2]; + return (x0 == equ.x0 && y0 == equ.y0 + && x1 == equ.x1 && y1 == equ.y1 + && x2 == equ.x2 && y2 == equ.y2 + && x3 == equ.x3 && y3 == equ.y3); } + bool operator!=(const Bezier & equ) const {return !this->operator==(equ);} };