Comparing Point Sets under Projection

We say that two sets of n points in the plane, P and Q, are views of the same object when there exists a set of n points S ae ! 3 and two distinct planes A and B, such that P is the projection of S onto A and Q is the projection of S onto B. In the case of orthographic projection, we provide an O(n 3 ) algorithm for deciding whether P and Q are views of the same object. For central projection we provide an O(n 7 ) algorithm. In both cases, the running times are a quadratic factor better than naive methods. We also show how randomization can be used to improve the expected running time by a factor of O(n= log n) when the largest subsets of P and Q that "match" are some fixed fraction of the total size of the sets. 1 Introduction In this paper we provide polynomial-time algorithms for deciding whether two (unlabelled) sets of n points in the plane are projections of the same (unknown) set of points in ! 3 , under either orthographic or central projection. This is a fundame..