Anti-orthotomics of frontals and their applications

Abstract

Let $f:N^n\to {\mathbb{R}}^{n+1}$ be a frontal with its Gauss mapping $\nu :N\to S^n$ and let $P\in {\mathbb{R}}^{n+1}$ be a point such that $(f(x)-P)\cdot \nu (x)\ne 0$ for any $x\in N$. In this paper, for the mapping $\tilde{f}:N\to {\mathbb{R}}^{n+1}$ defined by $$\tilde{f}(x)=f(x)-\frac{||f(x)-P|{|}^2}{2(f(x)-P)\cdot \nu (x)}\nu (x),$$ the following four are shown. (1) $\tilde{f}$ is a frontal with its Gauss mapping $\tilde{\nu }(x)=\frac{f(x)-P}{||f(x)-P||}$ at $\tilde{f}(x)$. (2) $\tilde{f}$ is the unique anti-orthotomic of $f$ relative to $P$. (3) The property $(\tilde{f}(x)-P)\cdot \tilde{\nu }(x)\ne 0$ holds for any $x\in N$. (4) The equality $||\tilde{f}(x)-P||=||\tilde{f}(x)-f(x)||$ holds for any $x\in N$. Moreover, three applications of the main result are given. As the first application, a generalization of Cahn-Hoffman vector formula is given. The second application is to clarify an optical meaning of anti-orthotomics. The third application gives a criterion to be a front for a given frontal.

17 pages, 5 figures; explanations added for sections 1 and 3, e-mail address of 2nd author changed, results unchanged