Help with a special kind of Torus

[ejhong]

Hi,

Consider a cone with height H and radius of base circle R. Take a point on the circular edge of the cone and make that the center of another circle of radius r whose normal points at the apex of the cylinder. Rotate this circle around the axis of the cone to create a surface. Given an angle theta for the circle of radius R, and an angle phi for the circle of radius r, you get a point on the surface. I believe a point on the surface can correspond to multiple theta/phi solutions (not sure though).

I'd like to learn more about this type of surface. In particular, among other things, I'd like to be able to:

- Find the closest point on the surface to a given point P, and find the solution(s) of theta/phi that correspond to this point.
- Find the point(s) on the surface a given distance from a point P, and find the solutions(s) of theta/phi that correspond to this point.

I'm trying to attach a prismatic joint to the smaller circle connected to a fixed point P, and find a path that causes the largest change of theta + phi as the joint is extended.

Would appreciate pointers - thanks in advance!

 

[StatusX]

It seems like the shape you're describing isn't a torus but is some kind of curved band, with two edges but no inside. To see this, note we get the same shape by rotating only half of the circle (where we split the circle into halves with a plane containing the center of the circle and the axis of the cone).