Intersection of two bazier surfaces gives a bezier curve?

Question

Intersection of two bazier surfaces gives a bezier curve?

Answer 1

Most often, Bezier surfaces and curves are considered of third order, that is, you specify a curve with 4 points, two of which are endpoints, and a surface with 16 points, four of which are the corners of the resulting surface segment. In both cases, the manifold is extensible beyond the border, but it does not look any interesting there.

I have said the above because the cubic Bezier curves are the only ones which I have some experience with. I don't know whether it's your case, but provided it is, your claim is FALSE. At least an intersection of two cubic Bezier surfaces does NOT generally give a cubic Bezier line. We could study whether it would be a line of another order, but I think this holds generally.

For my counterexample, you should know that no cubic Bezier curve can be smoothly closed (i.e., look like a circle). However, such intersection of two cubic Bezier surfaces can easily be reached. There is a cubic Bezier surface that looks like a paraboloid. It is not really a paraboloid, not rotationally symmetric, but it is a perfectly smooth bump. Make two of these, one with the vertex at its top, one aiming down, shift the second a bit so that the intersection makes a closed, smooth curve. Here we go ;-)