  Consider a smooth map from a neighborhood of the origin in a real vector
space to a neighborhood of the origin in a Euclidean space. Suppose that this
map takes all germs of lines passing through the origin to germs of Euclidean
circles, or lines, or a point. We prove that under some simple additional
assumptions this map takes all lines passing though the origin to the same
circles as a Hopf map coming from a representation of a Clifford algebra does.
We also describe a connection between our result and the Hurwitz--Radon theorem
about sums of squares.
