#ifndef _BEZIER_H
#define _BEZIER_H
+#include <vector>
+#include <algorithm>
+
#include "real.h"
#include "rect.h"
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<Real,Real> BezierTurningPoints(const Real & p0, const Real & p1, const Real & p2, const Real & p3);
+
+ extern std::vector<Real> SolveQuadratic(const Real & a, const Real & b, const Real & c, const Real & min = 0, const Real & max = 1);
+
+ 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
{
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), x3(cpy.x3), y3(cpy.y3)
+
+ /**
+ * 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;
y3 += t.y;
}
- Rect ToRect() {return Rect(x0,y0,x3-x0,y3-y0);}
+ 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);
+ }
+
+ /** 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 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)
+ {
+ 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<Bezier> ClipToRectangle(const Rect& r)
+ {
+ // Find points of intersection with the rectangle.
+ Debug("Clipping Bezier to Rect %s", r.Str().c_str());
+
+
+ // Find its roots.
+ std::vector<Real> x_intersection = SolveXParam(r.x);
+
+ // And for the other side.
+
+ 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());
+
+ // Find its roots.
+ std::vector<Real> y_intersection = SolveYParam(r.y);
+
+ 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.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 (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)
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);}
};
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