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);
- }
-
- 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);
-
- 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)
- {
- Real real_root = (Real(-1) / (Real(3) * a)) * (b + C + delta0 / C);
-
- roots.push_back(real_root);
+ extern std::vector<Real> SolveQuadratic(const Real & a, const Real & b, const Real & c, const Real & min = 0, const Real & max = 1);
- 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)
- {
- 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-4);
/** A _cubic_ bezier. **/
struct Bezier
Real x2; Real y2;
Real x3; Real y3;
- typedef enum {LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
+ 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)
{
- //TODO: classify the curve
- type = SERPENTINE;
+
}
+ const 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 (d1 == d2 && d2 == d3 && d3 == 0)
+ {
+ 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 (delta1 == delta2 && delta2 == delta3 && delta3 == 0)
+ {
+ type = QUADRATIC;
+
+ //Debug("QUADRATIC %s", Str().c_str());
+ return type;
+ }
+
+ Real discriminant = d1*d3*4 -d2*d2;
+ if (discriminant == 0)
+ {
+ 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());
+ }
+ 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 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(UNKNOWN)
{
x0 *= t.w;
y0 *= t.h;
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
{
git.ucc.asn.au / ipdf / code.git / blobdiff
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