X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fcode.git;a=blobdiff_plain;f=src%2Fbezier.h;h=3f47555c662d5bd63f5fc13486ccf8fc27c2465c;hp=17bc5b65b00d8b6b60aad88e3f91c773045ea697;hb=326f04a375ce3120f7e8957e3d7cd5f296f513e3;hpb=67fbce330b046b1f0d63222f04d83410dc1b2faa diff --git a/src/bezier.h b/src/bezier.h index 17bc5b6..3f47555 100644 --- a/src/bezier.h +++ b/src/bezier.h @@ -3,92 +3,100 @@ #include #include - -#include "real.h" #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 + 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); - // delta = 18abcd - 4 b^3 d + b^2 c^2 - 4ac^3 - 27 a^2 d^2 + /** A _cubic_ bezier. **/ + struct Bezier + { + BReal x0; BReal y0; + BReal x1; BReal y1; + BReal x2; BReal y2; + BReal x3; BReal y3; - 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); + typedef enum {UNKNOWN, LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type; + Type type; - 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); - - if (false && discriminant < 0) + //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) { - 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) + + Type GetType() { - 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)))) + 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)) { - Debug("Found root of %fx^3 + %fx^2 + %fx + %f at %f (%f)", a, b, c, d, x, y); - roots.push_back(x); + type = LINE; + //Debug("LINE %s", Str().c_str()); + return type; } - 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, 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; + 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; } + std::string Str() const { std::stringstream s; @@ -100,7 +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), type(cpy.type) + 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; @@ -120,9 +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); } @@ -131,12 +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) { @@ -171,140 +184,135 @@ namespace IPDF } // Performs one round of De Casteljau subdivision and returns the [t,1] part. - Bezier DeCasteljauSubdivideRight(const Real& t) + Bezier DeCasteljauSubdivideLeft(const BReal& t) { - Real one_minus_t = Real(1) - t; + BReal one_minus_t = BReal(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; + BReal x01 = x1*t + x0*one_minus_t; + BReal x12 = x2*t + x1*one_minus_t; + BReal x23 = x3*t + x2*one_minus_t; - Real x012 = x01*t + x12*one_minus_t; - Real x123 = x12*t + x23*one_minus_t; + BReal x012 = x12*t + x01*one_minus_t; + BReal x123 = x23*t + x12*one_minus_t; - Real x0123 = x012*t + x123*one_minus_t; + BReal 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; + BReal y01 = y1*t + y0*one_minus_t; + BReal y12 = y2*t + y1*one_minus_t; + BReal y23 = y3*t + y2*one_minus_t; - Real y012 = y01*t + y12*one_minus_t; - Real y123 = y12*t + y23*one_minus_t; + BReal y012 = y12*t + y01*one_minus_t; + BReal y123 = y23*t + y12*one_minus_t; - Real y0123 = y012*t + y123*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 [0,t] part. - Bezier DeCasteljauSubdivideLeft(const Real& t) + // Performs one round of De Casteljau subdivision and returns the [t,1] part. + Bezier DeCasteljauSubdivideRight(const BReal& t) { - Real one_minus_t = Real(1) - t; + BReal one_minus_t = BReal(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; + BReal x01 = x1*t + x0*one_minus_t; + BReal x12 = x2*t + x1*one_minus_t; + BReal x23 = x3*t + x2*one_minus_t; - Real x012 = x01*t + x12*one_minus_t; - Real x123 = x12*t + x23*one_minus_t; + BReal x012 = x12*t + x01*one_minus_t; + BReal x123 = x23*t + x12*one_minus_t; - Real x0123 = x012*t + x123*one_minus_t; + BReal 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; + BReal y01 = y1*t + y0*one_minus_t; + BReal y12 = y2*t + y1*one_minus_t; + BReal y23 = y3*t + y2*one_minus_t; - Real y012 = y01*t + y12*one_minus_t; - Real y123 = y12*t + y23*one_minus_t; + BReal y012 = y12*t + y01*one_minus_t; + BReal y123 = y23*t + y12*one_minus_t; - Real y0123 = y012*t + y123*one_minus_t; + BReal 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) + Bezier ReParametrise(const BReal& t0, const BReal& 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); + 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()); + //Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str()); return new_bezier; } - std::vector ClipToRectangle(const Rect& r) + std::vector ClipToRectangle(const BRect & r) { // Find points of intersection with the rectangle. - Debug("Clipping Bezier to Rect %s", r.Str().c_str()); + Debug("Clipping Bezier to BRect %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); + std::vector x_intersection = SolveXParam(r.x); + //Debug("Found %d intersections on left edge", x_intersection.size()); // And for the other side. - xd = x0 - 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 yd = y0 - r.y; - Real yc = Real(3)*(y1 - y0); - Real yb = Real(3)*(y2 - y1) - yc; - Real ya = y3 - y0 - yc - yb; + //Debug("Found %d intersections on right edge (total x: %d)", x_intersection_pt2.size(), x_intersection.size()); // 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 = SolveYParam(r.y); + //Debug("Found %d intersections on top edge", y_intersection.size()); - 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()); + //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(Real(0)); - x_intersection.push_back(Real(1)); + 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()); + //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.empty()) + if (x_intersection.size() <= 2) { all_beziers.push_back(*this); return all_beziers; } - Real t0 = *(x_intersection.begin()); + BReal t0 = *(x_intersection.begin()); for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it) { - Real t1 = *it; + BReal t1 = *it; if (t1 == t0) continue; - Debug(" -- t0: %f to t1: %f", t0, t1); - Real ptx, pty; - Evaluate(ptx, pty, ((t1 + t0) / Real(2))); + //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)", ptx, pty); + //Debug("Segment removed (point at %f, %f)", Double(ptx), Double(pty)); } t0 = t1; } @@ -312,14 +320,38 @@ namespace IPDF } /** 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);} };