This is a general question regarding the problem of deforming or forcing some 3-dimensional mathematical shape to flow along some 3D curve or surface.
What kind of mathematics would one need to describe deforming any object around, say, a cylinder of radius $r$? Is there a general function for applying such a process to any shape, B-spline surface or polyhedra (not taking into account physical parameters such as elasticity)?
I know that the CAD software Rhino 3D has built-in functions "flow along curve" and "flow along surface":
I am looking for some hints on how to achieve such operations were the target 3D curve is, say, a circle, without the use of CAD software. This could be tutorials online, math examples, DOIs for good papers on the subject, etc., etc.
I have tried myself once with flowing a function around a circle in 2D. For me it seemed like it was just a change of coordinates: You just express the coordinates in spherical coordinates, the x-cordinates will the flow around radially and the y-coordinate is the length of the arc. But I am looking for the general mapping solution.