I am looking for some nifty techniques or methods to help me gather elements of a set which are compatible with a base pattern. The set contains N instances of the base pattern but each instance differs by some multiplicative factor or rotation from the base pattern itself. I want to identify all N instances (N is unknown).

Let me make the problem a bit more concrete with an example since I am probably not using the correct vocabulary for this audience.

Imagine I have a 3D volume in which I draw a series of arcs. The arcs all have a unique radius of curvature R_i. The arc in the above statement is what I call the "base pattern" with say R=1. Then R_i is the multiplicative factor. The orientation of the arc in the 3D volume can vary - thus in the above statement I say a rotation of the base pattern is allowed.

Now the tricky bit (at least for me).

Within this volume, I have measurement planes - parallel planes within the volume where the points at which the arcs cross the plane are known. The measurement planes do not cross - so they are all strictly parallel and spaced along one direction in the 3D volume. All the information I have is the location of the points of crossing of the arcs and the measurement planes (a point in 3D space - (x,y,z)) and the fact that none of the arcs are parallel to the measurement planes. I want to find the set of measurement points which are compatible with an arc and continue that until all arcs are found.

There is no noise in the system - just N arcs with 1 point per measurement plane.

Any idea of how to tackle this problem? Any machine learning techniques that can help?

If this question can be formulated in a way which is more in the language of statistics, I am also appreciative of that information as it will allow me to search the literature myself.

Thank you!!!