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<Real,Real> BezierTurningPoints(const Real & p0, const Real & p1, const Real & p2, const Real & p3);
- inline std::pair<Real,Real> 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<Real,Real>(x0,x1);
- }
+ extern std::vector<Real> SolveQuadratic(const Real & a, const Real & b, const Real & c, const Real & min = 0, const Real & max = 1);
- inline std::vector<Real> 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<Real> 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<Real> 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-9);
/** A _cubic_ bezier. **/
struct Bezier
Real x1; Real y1;
Real x2; Real y2;
Real x3; Real 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(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), 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)
+ Real a1 = (x1-x0)*3;
+ Real a2 = (x0- x1*2 +x2)*3;
+ Real a3 = (x3 - x0 + (x1 - x2)*3);
+
+ Real b1 = (y1-y0)*3;
+ Real b2 = (y0- y1*2 +y2)*3;
+ Real b3 = (y3 - y0 + (y1 - y2)*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 (fabs(d1+d2+d3) < 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 (fabs(delta1+delta2+delta3) < 1e-6)
+ {
+ type = QUADRATIC;
+
+ //Debug("QUADRATIC %s", Str().c_str());
+ return type;
+ }
+ Real discriminant = d1*d3*4 -d2*d2;
+ if (fabs(discriminant) < 1e-6)
+ {
+ type = CUSP;
+ //Debug("CUSP %s", Str().c_str());
+ }
+ else if (discriminant > 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
{
* 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 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;
Rect SolveBounds() const;
+ std::pair<Real,Real> GetTop() const;
+ std::pair<Real,Real> GetBottom() const;
+ std::pair<Real,Real> GetLeft() const;
+ std::pair<Real,Real> GetRight() const;
+
Bezier ToAbsolute(const Rect & bounds) const
{
return Bezier(*this, bounds);
return result;
}
+ // Performs one round of De Casteljau subdivision and returns the [t,1] part.
+ Bezier DeCasteljauSubdivideLeft(const Real& t)
+ {
+ Real one_minus_t = Real(1) - t;
+
+ // X Coordinates
+ 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 = x12*t + x01*one_minus_t;
+ Real x123 = x23*t + x12*one_minus_t;
+
+ Real x0123 = x123*t + x012*one_minus_t;
+
+ // Y Coordinates
+ 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 = y12*t + y01*one_minus_t;
+ Real y123 = y23*t + y12*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 [t,1] part.
+ Bezier DeCasteljauSubdivideRight(const Real& t)
+ {
+ Real one_minus_t = Real(1) - t;
+
+ // X Coordinates
+ 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 = x12*t + x01*one_minus_t;
+ Real x123 = x23*t + x12*one_minus_t;
+
+ Real x0123 = x123*t + x012*one_minus_t;
+
+ // Y Coordinates
+ 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 = y12*t + y01*one_minus_t;
+ Real y123 = y23*t + y12*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)
{
- // 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
Debug("Reparametrise: %f -> %f",t0,t1);
Bezier new_bezier;
- Real tdiff = t1 - t0;
- Real tdiff_squared = tdiff*tdiff;
- Real tdiff_cubed = tdiff*tdiff_squared;
+ // 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);
- 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;
-
-
- Debug("(%f,%f),(%f,%f),(%f,%f),(%f,%f) -> (%f,%f),(%f,%f),(%f,%f),(%f,%f)", x0, y0, x1, y1, x2, y2, x3, y3, new_bezier.x0, new_bezier.y0, new_bezier.x1, new_bezier.y1, new_bezier.x2, new_bezier.y2, new_bezier.x3, new_bezier.y3);
+ Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
return new_bezier;
}
// 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<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
+ std::vector<Real> x_intersection = SolveXParam(r.x);
// And for the other side.
- xd = x3 - r.x - r.w;
- std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
+ std::vector<Real> 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<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
-
- // And for the other side.
- yd = y3 - r.y - r.h;
+ std::vector<Real> y_intersection = SolveYParam(r.y);
- std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
+ std::vector<Real> 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.
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());
+ for(auto t : x_intersection)
+ {
+ Real ptx, pty;
+ Evaluate(ptx, pty, t);
+ Debug("Root: t = %f, (%f,%f)", t, ptx, pty);
+ }
std::vector<Bezier> all_beziers;
- if (x_intersection.empty())
+ if (x_intersection.size() <= 2)
{
all_beziers.push_back(*this);
return all_beziers;
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);
- all_beziers.push_back(this->ReParametrise(t0, t1));
+ Real ptx, pty;
+ Evaluate(ptx, pty, ((t1 + t0) / Real(2)));
+ if (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;
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<Vec2> Evaluate(const std::vector<Real> & u) const;
+
+ std::vector<Real> SolveXParam(const Real & x) const;
+ std::vector<Real> SolveYParam(const Real & x) const;
+
+ // Get points with same X
+ inline std::vector<Vec2> SolveX(const Real & x) const
+ {
+ return Evaluate(SolveXParam(x));
+ }
+ // Get points with same Y
+ inline std::vector<Vec2> 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);}
};
git.ucc.asn.au / ipdf / code.git / blobdiff
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