Spherical orthotomic curve-germs

Abstract

In this paper, it is shown that for an $n$-dimensional spherical unit speed curve $\gamma :I\to S^n$, a given point $P\in S^n$ and a point $s_0$ of the open interval $I$, the spherical orthotomic curve-germ $or{t}_{\gamma ,P}:(I,s_0)\to S^n$ of $\gamma$ relative to $P$ is $\mathcal{L}$-equivalent to the spherical pedal curve-germ $pe{d}_{\gamma ,P}:(I,s_0)\to S^n$ of $\gamma$ relative to $P$ (resp., the spherical dual curve-germ ${\mathbf{u}}_n:(I,s_0)\to S^n$ of $\gamma$) if and only if $P\ne \pm {\mathbf{u}}_n(s_0)$ (resp., if $P=\pm {\mathbf{u}}_n(s_0)$).

10 pages, 2 figures

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