View from inside
type: Item
instance of: scholarly article
title: View from inside
zbMATH Open document ID: 1271.57064
author: Takashi Nishimura
author: Yu Sakemi
published in: Hokkaido Mathematical Journal
publication date: 2011-11-02T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hokmj/1319595861
review text: Let \(N\) and \({\mathbf r}:N\to S^{n+1}\) be a compact connected \(n\)-manifold and an immersion into \(S^{n+1}\subset \mathbb{R}^{n+2}\). In this paper, the authors define the spherical perspective projection \(\pi_{P}|{\mathbf r}(N):{\mathbf r}(N)\to E_P\). Here \(P\in S^{n+1}\), \(E_P\) is the equator with respect to the pole \(P\) and the map \(\pi_P:S^{n+1}\setminus\{\pm P\}\to E_P\) is induced from the intersection between \(E_P\) and the great circle connecting \(X\in S^{n+1}\setminus\{\pm P\}\) and \(P\). The set of critical values of \(\pi_P\circ {\mathbf r}\) is called the silhouette of \({\mathbf r}(N)\) relative to \(\pi_P\). By using the projective dual of \({\mathbf r}\), the authors characterize when the silhouette of \({\mathbf r}(N)\) is empty. If \(n\geq 2\), the authors show that ``the silhouette of \({\mathbf r}(N)\) is empty'' and ``\(\pi_P\circ {\mathbf r}\) is a diffeomorphism'' are equivalent conditions. The authors also introduce the notion of hedgehogs and show that ``the silhouette of \({\mathbf r}(S^n)\) is empty'' and ``the image of projective dual of \({\mathbf r}\) in \(\mathbb{R}^{n+1}\times\{1\}\) is a hedgehog'' are equivalent conditions. From these equivalent conditions, the authors give an example of two plane curves \(C_1\) and \(C_2\) such that \(C_1\) and \(C_2\) are quite similar but their dual curves are different i.e., the dual curve of \(C_1\) is a hedgehog but that of \(C_2\) is not a hedgehog.
reviewed by: Minoru Yamamoto
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 58K30
Mathematics Subject Classification ID: 68T45
zbMATH DE Number: 5966460
zbMATH Keywords: perspective projection
zbMATH Keywords: perspective point
zbMATH Keywords: projective dual
zbMATH Keywords: dual hypersurface
zbMATH Keywords: hedgehog
MaRDI profile type: Publication
OpenAlex ID: W2167575112
DOI: 10.14492/HOKMJ/1319595861
description: scientific article; zbMATH DE number 5966460