#ifndef _BEZIER_H
#define _BEZIER_H
-#include "real.h"
+#include <vector>
+#include <algorithm>
#include "rect.h"
+#include "real.h"
+
+
+
namespace IPDF
{
+ typedef Real BReal;
+ typedef TRect<BReal> 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<BReal,BReal> BezierTurningPoints(const BReal & p0, const BReal & p1, const BReal & p2, const BReal & 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<BReal> SolveQuadratic(const BReal & a, const BReal & b, const BReal & c, const BReal & min = 0, const BReal & 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...
- Real maxi(100);
- Real prevRes(d);
- std::vector<Real> 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<BReal> 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
{
* 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;
y3 += t.y;
}
- Rect SolveBounds() const;
+ BRect SolveBounds() const;
- Bezier ToAbsolute(const Rect & bounds) const
+ std::pair<BReal,BReal> GetTop() const;
+ std::pair<BReal,BReal> GetBottom() const;
+ std::pair<BReal,BReal> GetLeft() const;
+ std::pair<BReal,BReal> GetRight() const;
+
+ Bezier ToAbsolute(const BRect & bounds) const
{
return Bezier(*this, 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
+ 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)
+ if (bounds.w == BReal(0))
{
result.x0 = 0;
result.x1 = 0;
result.x3 = (x3 - bounds.x)/bounds.w;
}
- if (bounds.h == 0)
+ if (bounds.h == BReal(0))
{
result.y0 = 0;
result.y1 = 0;
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;
+
+ BReal x012 = x12*t + x01*one_minus_t;
+ BReal x123 = x23*t + x12*one_minus_t;
+
+ BReal x0123 = x123*t + x012*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;
+ // 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;
- // 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 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;
- 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;
+ // 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<Bezier> ClipToRectangle(const Rect& r)
+ std::vector<Bezier> ClipToRectangle(const BRect & r)
{
// Find points of intersection with the rectangle.
+ Debug("Clipping Bezier to BRect %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;
+ bool isVerticalLine = (x0 == x1 && x1 == x2 && x2 == x3);
+ bool isHorizontalLine = (y0 == y1 && y1 == y2 && y2 == y3);
// Find its roots.
- std::vector<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
+
+ std::vector<BReal> intersection;
- // And for the other side.
- xd = x3 + r.x + r.w;
+ if (!isVerticalLine)
+ {
+ std::vector<BReal> x_intersection = SolveXParam(r.x);
+ intersection.insert(intersection.end(), x_intersection.begin(), x_intersection.end());
- std::vector<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
- x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
+ // And for the other side.
- // 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;
+ std::vector<BReal> x_intersection_pt2 = SolveXParam(r.x + r.w);
+ intersection.insert(intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
+ }
// Find its roots.
- std::vector<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
-
- // And for the other side.
- yd = y3 + r.y + r.h;
+ if (!isHorizontalLine)
+ {
+ std::vector<BReal> y_intersection = SolveYParam(r.y);
+ intersection.insert(intersection.end(), y_intersection.begin(), y_intersection.end());
- std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
- y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
+ std::vector<BReal> y_intersection_pt2 = SolveYParam(r.y+r.h);
+ intersection.insert(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());
-
- Debug("Found %d intersections.\n", x_intersection.size());
+ intersection.push_back(BReal(0));
+ intersection.push_back(BReal(1));
+ std::sort(intersection.begin(), intersection.end());
std::vector<Bezier> all_beziers;
- if (x_intersection.empty())
+ if (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)
+ BReal t0 = *(intersection.begin());
+ for (auto it = intersection.begin()+1; it != intersection.end(); ++it)
{
- Real t1 = *it;
- all_beziers.push_back(this->ReParametrise(t0, t1));
+ 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<Vec2> Evaluate(const std::vector<BReal> & u) const;
+
+ std::vector<BReal> SolveXParam(const BReal & x) const;
+ std::vector<BReal> SolveYParam(const BReal & x) const;
+
+ // Get points with same X
+ inline std::vector<Vec2> SolveX(const BReal & x) const
+ {
+ return Evaluate(SolveXParam(x));
+ }
+ // Get points with same Y
+ inline std::vector<Vec2> 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);}
};
git.ucc.asn.au / ipdf / code.git / blobdiff
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