  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.}
