View from inside

Abstract

In this paper, we define a perspective projection of a given immersed n-dimensional hypersurface as a C∞ map via a C∞ immersion from the given n-manifold to Sn+1, and characterize when and only when such a perspective projection is non-singular. In order to obtain such characterizations, we consider an immersion from an n-dimensional manifold to Sn+1. We first obtain equivalent conditions for a given point P of Sn+1 to be outside the union of tangent great hyperspheres of a given immersed n-dimensional manifold r(N) in Sn+1 (Theorem 2.4). It turns out that if such a point P exists then the given manifold N must be diffeomorphic to Sn and in the case that n ≥ 2 the given immersion r: N → Sn+1 must be an embedding. Then, we obtain characterizations of a perspective projection of a given immersed n-dimensional manifold to be non-singular. Next, we obtain one more equivalent condition in terms of hedgehogs when the given N isSn and the given immersion is an embedding (Theorem 3.3). We also explain why we consider these equivalent conditions for an embedding r: Sn → Sn+1 instead of an embedding ¥widetilde{r}: Sn → ¥mathbb{R}n+1 in terms of hedgehogs.