THE ASSOCIATED LOCUS OF SOME HYPERSURFACES IN R^(n+1)

Abstract

For a smooth hypersurface of the space Rⁿ⁺¹ project orthogonally the origin of Rⁿ⁺¹ on its tangent hyperplanes and call the set of all projections the associated locus of the given hypersurface. In this paper we are going to find the equation of the associated locus for some given hypersurfaces and to show that it is a smooth hypersurface diffeomorphic with the initial one. We will also show that in one particular case both of them, the hypersurface and its associated locus, are diffeomorphic with the n-dimensional sphere.