X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fcode.git;a=blobdiff_plain;f=src%2Fbezier.h;h=b04126027152c9b6c7b12ff2c0c8ef55e4e667fb;hp=6a134b79b2f6540ad177019129762f8252b66bbd;hb=ea748154f1bc7dbc81cb52611a52865e63109439;hpb=85336af25da0c613460bbeda4ff7553933e13064 diff --git a/src/bezier.h b/src/bezier.h index 6a134b7..b041260 100644 --- a/src/bezier.h +++ b/src/bezier.h @@ -11,49 +11,11 @@ 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); - } + extern std::vector SolveQuadratic(const Real & a, const Real & b, const Real & c, const Real & min = 0, const Real & max = 1); - 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... - Debug("Trying to solve %fx^3 + %fx^2 + %fx + %f", a,b,c,d); - Real maxi(100); - Real prevRes(d); - std::vector roots; - 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; - - } + extern std::vector SolveCubic(const Real & a, const Real & b, const Real & c, const Real & d, const Real & min = 0, const Real & max = 1, const Real & delta = 1e-4); /** A _cubic_ bezier. **/ struct Bezier @@ -63,16 +25,73 @@ namespace IPDF Real x2; Real y2; Real x3; Real y3; - typedef enum {LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type; + typedef enum {UNKNOWN, 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, Real _x3, Real _y3) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x3), y3(_y3) + //Bezier() = default; // Needed so we can fread/fwrite this struct... for now. + Bezier(Real _x0=0, Real _y0=0, Real _x1=0, Real _y1=0, Real _x2=0, Real _y2=0, Real _x3=0, Real _y3=0) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x3), y3(_y3), type(UNKNOWN) { - //TODO: classify the curve - type = SERPENTINE; + } + Type GetType() + { + if (type != Bezier::UNKNOWN) + return type; + // From Loop-Blinn 2005, with w0 == w1 == w2 == w3 = 1 + // Transformed control points: (a0 = x0, b0 = y0) + Real a1 = (x1-x0)*Real(3); + Real a2 = (x0- x1*Real(2) +x2)*Real(3); + Real a3 = (x3 - x0 + (x1 - x2)*Real(3)); + + Real b1 = (y1-y0)*Real(3); + Real b2 = (y0- y1*Real(2) +y2)*Real(3); + Real b3 = (y3 - y0 + (y1 - y2)*Real(3)); + + // d vector (d0 = 0 since all w = 1) + Real d1 = a2*b3 - a3*b2; + Real d2 = a3*b1 - a1*b3; + Real d3 = a1*b2 - a2*b1; + + if (Abs(d1+d2+d3) < Real(1e-6)) + { + type = LINE; + //Debug("LINE %s", Str().c_str()); + return type; + } + + Real delta1 = -(d1*d1); + Real delta2 = d1*d2; + Real delta3 = d1*d3 -(d2*d2); + if (Abs(delta1+delta2+delta3) < Real(1e-6)) + { + type = QUADRATIC; + + //Debug("QUADRATIC %s", Str().c_str()); + return type; + } + + Real discriminant = d1*d3*Real(4) -d2*d2; + if (Abs(discriminant) < Real(1e-6)) + { + type = CUSP; + //Debug("CUSP %s", Str().c_str()); + } + else if (discriminant > Real(0)) + { + type = SERPENTINE; + //Debug("SERPENTINE %s", Str().c_str()); + } + else + { + type = LOOP; + //Debug("LOOP %s", Str().c_str()); + } + //Debug("disc %.30f", discriminant); + return type; + } + + std::string Str() const { std::stringstream s; @@ -106,6 +125,11 @@ namespace IPDF 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); @@ -122,7 +146,7 @@ namespace IPDF // (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) + if (bounds.w == Real(0)) { result.x0 = 0; result.x1 = 0; @@ -137,7 +161,7 @@ namespace IPDF result.x3 = (x3 - bounds.x)/bounds.w; } - if (bounds.h == 0) + if (bounds.h == Real(0)) { result.y0 = 0; result.y1 = 0; @@ -155,69 +179,69 @@ namespace IPDF } // Performs one round of De Casteljau subdivision and returns the [t,1] part. - Bezier DeCasteljauSubdivideRight(const Real& t) + 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 x01 = x1*t + x0*one_minus_t; + Real x12 = x2*t + x1*one_minus_t; + Real x23 = x3*t + x2*one_minus_t; - Real x012 = x01*t + x12*one_minus_t; - Real x123 = x12*t + x23*one_minus_t; + Real x012 = x12*t + x01*one_minus_t; + Real x123 = x23*t + x12*one_minus_t; - Real x0123 = x012*t + x123*one_minus_t; + Real x0123 = x123*t + x012*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 y01 = y1*t + y0*one_minus_t; + Real y12 = y2*t + y1*one_minus_t; + Real y23 = y3*t + y2*one_minus_t; - Real y012 = y01*t + y12*one_minus_t; - Real y123 = y12*t + y23*one_minus_t; + Real y012 = y12*t + y01*one_minus_t; + Real y123 = y23*t + y12*one_minus_t; - Real y0123 = y012*t + y123*one_minus_t; + Real y0123 = y123*t + y012*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) + // 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 x01 = x1*t + x0*one_minus_t; + Real x12 = x2*t + x1*one_minus_t; + Real x23 = x3*t + x2*one_minus_t; - Real x012 = x01*t + x12*one_minus_t; - Real x123 = x12*t + x23*one_minus_t; + Real x012 = x12*t + x01*one_minus_t; + Real x123 = x23*t + x12*one_minus_t; - Real x0123 = x012*t + x123*one_minus_t; + Real x0123 = x123*t + x012*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 y01 = y1*t + y0*one_minus_t; + Real y12 = y2*t + y1*one_minus_t; + Real y23 = y3*t + y2*one_minus_t; - Real y012 = y01*t + y12*one_minus_t; - Real y123 = y12*t + y23*one_minus_t; + Real y012 = y12*t + y01*one_minus_t; + Real y123 = y23*t + y12*one_minus_t; - Real y0123 = y012*t + y123*one_minus_t; + Real y0123 = y123*t + y012*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); + Debug("Reparametrise: %f -> %f",Double(t0),Double(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); + Debug("New t0 = %f", Double(new_t0)); new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0); Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str()); @@ -229,35 +253,19 @@ namespace IPDF // Find points of intersection with the rectangle. Debug("Clipping Bezier to Rect %s", r.Str().c_str()); - // 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); + std::vector x_intersection = SolveXParam(r.x); // And for the other side. - xd = x3 - r.x - r.w; - std::vector x_intersection_pt2 = SolveCubic(xa, xb, xc, xd); + std::vector x_intersection_pt2 = SolveXParam(r.x + r.w); 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 = SolveYParam(r.y); - std::vector y_intersection_pt2 = SolveCubic(ya, yb, yc, yd); + std::vector y_intersection_pt2 = SolveYParam(r.y+r.h); y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end()); // Merge and sort. @@ -267,9 +275,15 @@ namespace IPDF std::sort(x_intersection.begin(), x_intersection.end()); Debug("Found %d intersections.\n", x_intersection.size()); + for(auto t : x_intersection) + { + Real ptx, pty; + Evaluate(ptx, pty, t); + Debug("Root: t = %f, (%f,%f)", Double(t), Double(ptx), Double(pty)); + } std::vector all_beziers; - if (x_intersection.empty()) + if (x_intersection.size() <= 2) { all_beziers.push_back(*this); return all_beziers; @@ -279,13 +293,18 @@ namespace IPDF { Real t1 = *it; if (t1 == t0) continue; - Debug(" -- t0: %f to t1: %f", t0, t1); + Debug(" -- t0: %f to t1: %f: %f", Double(t0), Double(t1), Double((t1 + t0)/Real(2))); Real ptx, pty; Evaluate(ptx, pty, ((t1 + t0) / Real(2))); if (r.PointIn(ptx, pty)) { + Debug("Adding segment: (point at %f, %f)", Double(ptx), Double(pty)); all_beziers.push_back(this->ReParametrise(t0, t1)); } + else + { + Debug("Segment removed (point at %f, %f)", Double(ptx), Double(pty)); + } t0 = t1; } return all_beziers; @@ -300,6 +319,30 @@ namespace IPDF 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]; } + std::vector Evaluate(const std::vector & u) const; + + std::vector SolveXParam(const Real & x) const; + std::vector SolveYParam(const Real & x) const; + + // Get points with same X + inline std::vector SolveX(const Real & x) const + { + return Evaluate(SolveXParam(x)); + } + // Get points with same Y + inline std::vector SolveY(const Real & y) const + { + return Evaluate(SolveYParam(y)); + } + + bool operator==(const Bezier & equ) const + { + 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);} };