If I'm not mistaken, C2 continuity means that the derivative of the Bezier curve is continuous and differentiable. I understand why C0 and C1 continuity are necessary to produce a smooth curve, but why must C2 continuity be ensured?
moridin22
I think C2 continuity roughly just means that the curve is even smoother than a C1 continuous curve, and it is a nice property we happen to get from Bezier curves. For an example of a function that is C1 but not C2, there's this function (defined as the integral of the Weierstrass function): https://i.stack.imgur.com/aeQ5W.jpg You can kind of tell that it's not terribly smooth, although the visual distinction is definitely harder to make with these higher levels of continuity.
aparikh98
Is there a general trend to how the shape of joined bezier curves compare to higher order curves defined on the (combined) control points?
Wikipedia (https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Degree_elevation) mentions using degree elevation to allow for use of lower order curves in softwares that only support higher order curves, but this seems to be more related to defining new control points that maintain the same points.
julialuo
Alternative way to ensure C2 continuity is here: https://stackoverflow.com/questions/12295773/joining-two-b%C3%A9zier-curves-smoothly-c2-continuous. I believe these two methods are equivalent.
glee-
My understanding of the C2 continuity is slightly different from the previous responses. Having a C1 continuity would mean that the transition between the two bezier curves is smooth. Having a C2 continuity means that the derivative of each bezier curve at the nearby are the same. I would say that this implies that if you were to pick any two continuous points along the piecewise bezier curve, it would be a bezier curve itself.
If I'm not mistaken, C2 continuity means that the derivative of the Bezier curve is continuous and differentiable. I understand why C0 and C1 continuity are necessary to produce a smooth curve, but why must C2 continuity be ensured?
I think C2 continuity roughly just means that the curve is even smoother than a C1 continuous curve, and it is a nice property we happen to get from Bezier curves. For an example of a function that is C1 but not C2, there's this function (defined as the integral of the Weierstrass function): https://i.stack.imgur.com/aeQ5W.jpg You can kind of tell that it's not terribly smooth, although the visual distinction is definitely harder to make with these higher levels of continuity.
Is there a general trend to how the shape of joined bezier curves compare to higher order curves defined on the (combined) control points?
Wikipedia (https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Degree_elevation) mentions using degree elevation to allow for use of lower order curves in softwares that only support higher order curves, but this seems to be more related to defining new control points that maintain the same points.
Alternative way to ensure C2 continuity is here: https://stackoverflow.com/questions/12295773/joining-two-b%C3%A9zier-curves-smoothly-c2-continuous. I believe these two methods are equivalent.
My understanding of the C2 continuity is slightly different from the previous responses. Having a C1 continuity would mean that the transition between the two bezier curves is smooth. Having a C2 continuity means that the derivative of each bezier curve at the nearby are the same. I would say that this implies that if you were to pick any two continuous points along the piecewise bezier curve, it would be a bezier curve itself.