View from inside
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.
scientific article; zbMATH DE number 5966460