Points on computable curves of computable lengths

Abstract

A computable plane curve is defined as the image of a computable real function from a closed interval to the real plane. As it is showed by Ko [7] that the length of a computable curve is not necessarily computable, even if the length is finite. Therefore, the set of the computable curves of computable lengths is different from the set of the computable curves of finite lengths. In this paper we show further that the points covered by these two sets of curves are different as well. More precisely, we construct a computable curve K of a finite length and a point z on the curve K such that the point z does not belong to any computable curve of computable length. This gives also a positive answer to an open question of Gu, Lutz and Mayordomo in [4]. © 2009 Springer Berlin Heidelberg.