Hokkaido Mathematical Journal

View from inside

Takashi NISHIMURA and Yu SAKEMI

Full-text: Open access

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: NSn+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: SnSn+1 instead of an embedding ¥widetilde{r}: Sn → ¥mathbb{R}n+1 in terms of hedgehogs.

Article information

Source
Hokkaido Math. J., Volume 40, Number 3 (2011), 361-373.

Dates
First available in Project Euclid: 26 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1319595861

Digital Object Identifier
doi:10.14492/hokmj/1319595861

Mathematical Reviews number (MathSciNet)
MR2883496

Zentralblatt MATH identifier
1271.57064

Subjects
Primary: 57R40: Embeddings 57R42: Immersions 58K30: Global theory 68T45: Machine vision and scene understanding

Keywords
perspective projection perspective point projective dual dual hypersurface hedgehog

Citation

NISHIMURA, Takashi; SAKEMI, Yu. View from inside. Hokkaido Math. J. 40 (2011), no. 3, 361--373. doi:10.14492/hokmj/1319595861. https://projecteuclid.org/euclid.hokmj/1319595861


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