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.