#include <vector>
#include <algorithm>
-
-#include "real.h"
#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 std::pair<Real,Real> BezierTurningPoints(const Real & p0, const Real & p1, const Real & p2, const Real & p3);
+ 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
+ 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);
- // 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<Real> 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;
* 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;
y3 += t.y;
}
- Rect SolveBounds() const;
+ BRect 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;
+ 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 Rect & bounds) 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)
{
}
// 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<Bezier> ClipToRectangle(const Rect& r)
+ std::vector<Bezier> 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<Real> x_intersection = SolveCubic(xa, xb, xc, xd);
+ std::vector<BReal> 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<Real> x_intersection_pt2 = SolveCubic(xa, xb, xc, xd);
+ std::vector<BReal> 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<Real> y_intersection = SolveCubic(ya, yb, yc, yd);
+ std::vector<BReal> y_intersection = SolveYParam(r.y);
+ //Debug("Found %d intersections on top edge", y_intersection.size());
- // And for the other side.
- yd = y0 - r.y - r.h;
-
- std::vector<Real> y_intersection_pt2 = SolveCubic(ya, yb, yc, yd);
+ std::vector<BReal> 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<Bezier> all_beziers;
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)));
- if (true || r.PointIn(ptx, pty))
+ //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;
}
}
/** 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
{
git.ucc.asn.au / ipdf / code.git / blobdiff
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