X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fcode.git;a=blobdiff_plain;f=src%2Fbezier.h;h=3f47555c662d5bd63f5fc13486ccf8fc27c2465c;hp=8e7cedc6d6bc868efe03e8541e57e11332951fd0;hb=326f04a375ce3120f7e8957e3d7cd5f296f513e3;hpb=4119777e3dd950ac5f25ba5925e308882036adf3 diff --git a/src/bezier.h b/src/bezier.h index 8e7cedc..3f47555 100644 --- a/src/bezier.h +++ b/src/bezier.h @@ -1,68 +1,101 @@ #ifndef _BEZIER_H #define _BEZIER_H -#include "real.h" +#include +#include #include "rect.h" +#include "real.h" + + + namespace IPDF { + typedef Real BReal; + typedef TRect BRect; + extern int Factorial(int n); extern int BinomialCoeff(int n, int k); - extern Real Bernstein(int k, int n, const Real & u); + extern BReal Bernstein(int k, int n, const BReal & u); + extern std::pair BezierTurningPoints(const BReal & p0, const BReal & p1, const BReal & p2, const BReal & 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 BReal & a, const BReal & b, const BReal & c, const BReal & min = 0, const BReal & 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... - 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; - - } + extern std::vector SolveCubic(const BReal & a, const BReal & b, const BReal & c, const BReal & d, const BReal & min = 0, const BReal & max = 1, const BReal & delta = 1e-9); /** A _cubic_ bezier. **/ struct Bezier { - Real x0; Real y0; - Real x1; Real y1; - 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) + BReal x0; BReal y0; + BReal x1; BReal y1; + BReal x2; BReal y2; + BReal x3; BReal y3; + + typedef enum {UNKNOWN, LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type; + Type type; + + //Bezier() = default; // Needed so we can fread/fwrite this struct... for now. + Bezier(BReal _x0=0, BReal _y0=0, BReal _x1=0, BReal _y1=0, BReal _x2=0, BReal _y2=0, BReal _x3=0, BReal _y3=0) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x3), y3(_y3), type(UNKNOWN) { + + } + + 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) + BReal a1 = (x1-x0)*BReal(3); + BReal a2 = (x0- x1*BReal(2) +x2)*BReal(3); + BReal a3 = (x3 - x0 + (x1 - x2)*BReal(3)); + + BReal b1 = (y1-y0)*BReal(3); + BReal b2 = (y0- y1*BReal(2) +y2)*BReal(3); + BReal b3 = (y3 - y0 + (y1 - y2)*BReal(3)); + // d vector (d0 = 0 since all w = 1) + BReal d1 = a2*b3 - a3*b2; + BReal d2 = a3*b1 - a1*b3; + BReal d3 = a1*b2 - a2*b1; + + if (Abs(d1+d2+d3) < BReal(1e-6)) + { + type = LINE; + //Debug("LINE %s", Str().c_str()); + return type; + } + + BReal delta1 = -(d1*d1); + BReal delta2 = d1*d2; + BReal delta3 = d1*d3 -(d2*d2); + if (Abs(delta1+delta2+delta3) < BReal(1e-6)) + { + type = QUADRATIC; + + //Debug("QUADRATIC %s", Str().c_str()); + return type; + } + + BReal discriminant = d1*d3*BReal(4) -d2*d2; + if (Abs(discriminant) < BReal(1e-6)) + { + type = CUSP; + //Debug("CUSP %s", Str().c_str()); + } + else if (discriminant > BReal(0)) + { + type = SERPENTINE; + //Debug("SERPENTINE %s", Str().c_str()); + } + else + { + type = LOOP; + //Debug("LOOP %s", Str().c_str()); + } + //Debug("disc %.30f", discriminant); + return type; } - 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) {} std::string Str() const { @@ -75,7 +108,7 @@ namespace IPDF * 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) + Bezier(const Bezier & cpy, const BRect & t = BRect(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; @@ -95,9 +128,14 @@ namespace IPDF y3 += t.y; } - Rect SolveBounds() const; + BRect SolveBounds() const; + + std::pair GetTop() const; + std::pair GetBottom() const; + std::pair GetLeft() const; + std::pair GetRight() const; - Bezier ToAbsolute(const Rect & bounds) const + Bezier ToAbsolute(const BRect & bounds) const { return Bezier(*this, bounds); } @@ -106,12 +144,12 @@ namespace IPDF * (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 + Bezier ToRelative(const BRect & 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}; + // BRect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, BReal(1)/bounds.w, BReal(1)/bounds.h}; Bezier result; if (bounds.w == 0) { @@ -145,54 +183,175 @@ namespace IPDF return result; } - Bezier ReParametrise(const Real& t0, const Real& t1) + // Performs one round of De Casteljau subdivision and returns the [t,1] part. + Bezier DeCasteljauSubdivideLeft(const BReal& t) { - // 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; + BReal one_minus_t = BReal(1) - t; - 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); + // X Coordinates + BReal x01 = x1*t + x0*one_minus_t; + BReal x12 = x2*t + x1*one_minus_t; + BReal x23 = x3*t + x2*one_minus_t; - 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; + BReal x012 = x12*t + x01*one_minus_t; + BReal x123 = x23*t + x12*one_minus_t; - // 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); + BReal x0123 = x123*t + x012*one_minus_t; - 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; + // Y Coordinates + BReal y01 = y1*t + y0*one_minus_t; + BReal y12 = y2*t + y1*one_minus_t; + BReal y23 = y3*t + y2*one_minus_t; + BReal y012 = y12*t + y01*one_minus_t; + BReal y123 = y23*t + y12*one_minus_t; + BReal 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 [t,1] part. + Bezier DeCasteljauSubdivideRight(const BReal& t) + { + BReal one_minus_t = BReal(1) - t; + + // X Coordinates + BReal x01 = x1*t + x0*one_minus_t; + BReal x12 = x2*t + x1*one_minus_t; + BReal x23 = x3*t + x2*one_minus_t; + + BReal x012 = x12*t + x01*one_minus_t; + BReal x123 = x23*t + x12*one_minus_t; + + BReal x0123 = x123*t + x012*one_minus_t; + + // Y Coordinates + BReal y01 = y1*t + y0*one_minus_t; + BReal y12 = y2*t + y1*one_minus_t; + BReal y23 = y3*t + y2*one_minus_t; + + BReal y012 = y12*t + y01*one_minus_t; + BReal y123 = y23*t + y12*one_minus_t; + + BReal y0123 = y123*t + y012*one_minus_t; + + return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3); + } + + Bezier ReParametrise(const BReal& t0, const BReal& 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] + BReal new_t0 = t0 / t1; + //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()); return new_bezier; } + std::vector ClipToRectangle(const BRect & r) + { + // Find points of intersection with the rectangle. + Debug("Clipping Bezier to BRect %s", r.Str().c_str()); + + + // Find its roots. + std::vector x_intersection = SolveXParam(r.x); + //Debug("Found %d intersections on left edge", x_intersection.size()); + + // And for the other side. + + std::vector x_intersection_pt2 = SolveXParam(r.x + r.w); + x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end()); + //Debug("Found %d intersections on right edge (total x: %d)", x_intersection_pt2.size(), x_intersection.size()); + + // Find its roots. + std::vector y_intersection = SolveYParam(r.y); + //Debug("Found %d intersections on top edge", y_intersection.size()); + + std::vector y_intersection_pt2 = SolveYParam(r.y+r.h); + y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end()); + //Debug("Found %d intersections on bottom edge (total y: %d)", y_intersection_pt2.size(), y_intersection.size()); + + // Merge and sort. + x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end()); + x_intersection.push_back(BReal(0)); + x_intersection.push_back(BReal(1)); + std::sort(x_intersection.begin(), x_intersection.end()); + + //Debug("Found %d intersections.\n", x_intersection.size()); + /*for(auto t : x_intersection) + { + BReal 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.size() <= 2) + { + all_beziers.push_back(*this); + return all_beziers; + } + BReal t0 = *(x_intersection.begin()); + for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it) + { + BReal t1 = *it; + if (t1 == t0) continue; + //Debug(" -- t0: %f to t1: %f: %f", Double(t0), Double(t1), Double((t1 + t0)/BReal(2))); + BReal ptx, pty; + Evaluate(ptx, pty, ((t1 + t0) / BReal(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; + } /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/ - void Evaluate(Real & x, Real & y, const Real & u) const + void Evaluate(BReal & x, BReal & y, const BReal & u) const { - Real coeff[4]; + BReal 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]; } + std::vector Evaluate(const std::vector & u) const; + + std::vector SolveXParam(const BReal & x) const; + std::vector SolveYParam(const BReal & x) const; + + // Get points with same X + inline std::vector SolveX(const BReal & x) const + { + return Evaluate(SolveXParam(x)); + } + // Get points with same Y + inline std::vector SolveY(const BReal & 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);} };