On rectification of circles and an extension of Beltrami's theorem (Q2566528)

One considers a bundle of circles and proves the following theorem 1.1: Consider a simple handle of circles passing through the origin such that the set of tangent lines of the circles contains 54 generic lines, then there exists a local diffeomorphism about the origin mapping all the circles into straight lines if and only if all circles in the bundle pass through one common point distinct from the origin. One classifies the rich families of circles and proves that up to a projective transformation of the space of the image and a Möbius transformation of the space of the inverse image, there exist exactly three diffeomorphism classes of rectifiable rich families of circles.