Distinguishing a plane curve from other curves similar to it

Abstract

If C is a convex or smooth simple closed plane curve, then there is a finite subset E of C such that any plane curve which is similar to C and passes through all the points of E must coincide with C. Generalizations to compacta in euclidean spaces are given; on the other hand, there are simple closed plane curves for which no such finite subsets exist. Circles are the only plane continua which separate the plane for which three points suffice, and even for a convex polygon the number of points required may be arbitrarily large.