Fundamental difficulties with projective normalization of planar curves

There can not exist continuous projective invariants, and a projective normalisation scheme can not have both the properties of continuity and uniqueness, because uniqueness might be preferred but it is not essential for recognition.

Abstract

In this paper projective normalization and projective invariants of planar curves are discussed. It is shown that there exists continuous affine invariants. It is shown that many curves can be projected arbitrarily close to a circle in a strengthened Hausdorff metric. This does not infer any limitations on projective invariants, but it is clear that projective normalization by maximizing compactness is unsuitable. It is also shown that arbitrarily close to each of a finite number of closed planar curves there is one member of a set of projectively equivalent curves. Thus there can not exist continuous projective invariants, and a projective normalisation scheme can not have both the properties of continuity and uniqueness. Although uniqueness might be preferred it is not essential for recognition. This is illustrated with an example of a projective normalization scheme for non-algebraic, both convex and non-convex, curves.