On Rectification of Circles and an Extension of Beltrami's Theorem

The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of \r^3, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in \r^3 passing through one point. It turns out that to be rectifiable all circles need to pass through some other common point. The second main result is a complete description of geometries in \r^3 in which all the geodesics are circles. This is a consequence of an extension of Beltrami's theorem by replacing straight lines with circles.}