Hello,
I am working on a project hat involves 2D cubic Bezier curves and I am looking for an explicit formulation of the conditions for G3 geometric continuity.
I found good explanayions of G2 and the formulas work well but I am struggling to work out what exactly the measure is that needs to be equal on both sides of the connection in order to have G3.
The general definition of Gn continuity that refers to the existence of a reparametrization that gives Cn is a bit too abstract for me and I hope someone can help me find something more workable.
One source I found (page 7 in this PDF) says "Gn continuity is like Cn but only requires the derivatives to have the same direction" but that's something I could not confirm for some examples I have with G2 continuity (which is something I understand).
Another source (figure 3 in this PDF) shows an interesting diagram but I struggle to understand what the vectors shown there represent. Does it mean that the vector difference of the unit vectors of the second derivatives has to be parallel to the first derivatives? That's also something I could not confirm with my examples. And what would be the corresponding condition for G3?
What I have done myself is to calculate the derivative of the (signed) curvature with respect to the arc length, which for myself I called "meta-curvature" m:
m(t) = d k(t)/ds = d k(t)/dt * dt/ds = (d k(t)/dt) / (ds/dt) = k'(t) / ||r'(t)||
Does this make sense, or did I do something inappropriate or unnecessarily complicated?
Assuming that the start point of the curve p0 is (0, 0), and given that we only need m(0), the formula doesn't get overly complicated:
m(0) = 2[ (3 x2 y1 - 3 x1 y2 + x1 y3 - x3 y1) (x1^2 + y1^2) - 6(2 x1^2 + 2 y1^2 - x1 x2 - y1 y2) (x2 y1 - x1 y2) ] / 9(x1^2 + y1^2)^3
I can spot some cross/wedge products and norms of the first derivative - maybe this whole thing could be formulated much more elegantly? My main concern is that is is based on a wrong notion of G3, though.
If someone could point me to an easy-to-understand, easy-to-apply definition of G3 (or confirm that what I did is correct) that would be fantastic.
Thanks in advance.
Tim