Circles and Quadratic Maps Between Spheres

Consider an analytic map from a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of vector lines to germs of circles. Such map is called rounding. Two roundings are equivalent if they take the same lines to the same circles. We prove that any rounding whose differential at 0 has rank at least 2 is equivalent to a fractional quadratic rounding. The latter gives rise to a quadratic map between spheres. Results of P. Yiu on quadratic maps between spheres have some interesting implications concerning roundings.