How would you call geometric objects that lie on a single surface, e.g. a sphere, plane, torus, etc. I'm looking for an extension of the name coplanar to something like "cosurfacial", but I guess their must be a correct term..
Edit: In the comments, the context was asked for where I would use that term. I should have done that in the first place.
My intent is to design an algorithm that checks for all faces of a solid (in euclidian 3D space) which faces share the same underlying surface. That could be a plane, but also a cylinder, a sphere, etc.. So these faces could be called "co-surfical" (but that doesn't sound very well)
 A: It seems to me that you need to mention which surface the two objects lie on, because simply saying that they both lie on some surface doesn't say very much. Saying it another way ... given any two objects, it is very often (maybe always) possible to find some surface that contains them both, so "co-surfacity" doesn't mean very much without some mention of the surface.
On the other hand, given two curves, or two finite point sets, it is often possible to find several surfaces that contain them both. So, again it seems important to mention which common containing surface you have in mind.
I have heard "co-cylindrical" used, but I don't know of a general term.
Even if there is a term, it's probably an obscure one that is unfamiliar to most people, so better to avoid it.
Given two objects $A$ and $B$ and a surface $S$, I would just say that "$A$ and $B$ are contained within $S$", or $A \cup B \subset S$. Those both seem very clear, and the second one has the virtue of brevity.
A: I'm not sure I completely understand the class of objects you are trying to describe, but they sound like (connected) $2$-manifolds.
One related term to "collinear" which applies to circles is concyclic. My gut tells me that if you will find any such term, you will find it in algebraic geometry - the theory of elliptic curves and other algebraic varieties would seem to call for such a term.
