Deformation of 3D objects along curve or surface?

Question

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":
http://www.youtube.com/watch?v=5WNNsJRot-Y (RhinoGuide)
http://www.youtube.com/watch?v=yiFFRbyV0DU (digitaltoolbox).

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.

Answer 1

The idea actually goes back to Bézier's thesis, though it's almost always attributed to Sederberg and Parry: Tom Sederberg and Scott Parry. "Free Form Deformation of Solid Geometric Models." SIGGRAPH, Association of Computing Machinery. Volume 20, Number 4, 1986. 151-159. Google their original paper -- easy to find. Many CAD packages are essentially just implementing these old ideas. Intuitively, you embed your target object in a cube of jello, and deform the jello, and this carries your object along with it. Or, for bending deformations, you do essentially what you described -- "attach" your object to a straight line, and bend the line, which bends your object, too. As you say, it's not much more than a change of coordinates.

There's a huge amount of literature on the subject. Look up "free-form deformation". The wikipedia entry is pretty miserable, but the references it gives are quite good.

Answer 2

This is too long for a comment, so I'm posting an answer. There doesn't seem to be a definite answer, nor do I claim that what is suggested below is the most computationally optimal way of accomplishing the task, but it is the first thing that comes to mind:

Let $O$ be a 3D object that we wish to flow along a surface $S$. Let us assume that $S$ is the image under some map $F$ of a plane $\Pi$. For simplicity, we don't care about specific orientation of $O$. Foliate $O$ by copies of $\Pi$ along the direction perpendicular to $\Pi$, say along some line segment $l\subset O$. Then $O = O\bigcap \left[\cup_{p\in l}\Pi_p\right]$ where $\Pi_p$ is the plane passing through $p$. Now define $\widetilde{F}: \bigcup_{p\in l}\Pi_p \rightarrow\mathbb{R}^3$ by $\widetilde{F}(\pi, p) = (F(\pi), p)$, where $\pi$ is a point on $\Pi_p$. Then $\widetilde{O} := \widetilde{F}(O)$ is the object $O$ "flow along" the surface $S$. Notice in particular that $\widetilde{F}(\bigcup_{p\in l}\Pi_p) = \bigcup_{p\in l}S\times\{p\}$.