Anti-orthotomics of frontals and their applications

Let $f:N^n\to {\mathbb{R}}^{n+1}$ be a frontal with its Gauss mapping $\nu: N\to S^n$andlet$P\in $\mathbb{R}$^{n+1}$beapointsuchthat$(f(x)-P)\cdot \nu(x) \ne 0$forany$x\in N$\mathrm{.}Inthispaper,forthemapping$\widetilde{f}: N\to $\mathbb{R}$^{n+1}$definedby$$\tilde{f}(x)=f(x)-\frac{\mathrm{\mid }\mathrm{\mid }f(x)-P\mathrm{\mid }{\mathrm{\mid }}^2}{2(f(x)-P)\cdot \nu (x)}\nu (x),$$ the following four are shown. (1) $\tilde{f}$ is a frontal with its Gauss mapping $\tilde{\nu }(x)=\frac{f(x)-P}{\mathrm{\mid }\mathrm{\mid }f(x)-P\mathrm{\mid }\mathrm{\mid }}$ at $\tilde{f}(x)$. (2) $\tilde{f}$ is the unique anti-orthotomic of $f$ relative to $P$. (3) The property $(\tilde{f}(x)-P)\cdot \tilde{\nu }(x)\mathrm{\ne }0$ holds for any $x\in N$. (4) The equality $\mathrm{\mid }\mathrm{\mid }\tilde{f}(x)-P\mathrm{\mid }\mathrm{\mid }=\mathrm{\mid }\mathrm{\mid }\tilde{f}(x)-f(x)\mathrm{\mid }\mathrm{\mid }$ 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.