Spherical orthotomic curve-germs

TL;DR

This paper demonstrates that spherical orthotomic and pedal curve-germs are locally equivalent under certain conditions, revealing a fundamental relationship between these curves on the sphere.

Contribution

It establishes a precise condition under which spherical orthotomic and pedal curve-germs are $ ext{L}$-equivalent, clarifying their local geometric relationship on the sphere.

Findings

Orthotomic and pedal curve-germs are $ ext{L}$-equivalent when $P eq ext{±}{f u}_n(s_0)$.

The equivalence depends on the position of $P$ relative to the dual curve.

The result characterizes local curve-germ relationships on the sphere.

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 $ort_{\gamma, P}: (I, s_0)\to S^n$ of $\gamma$ relative to $P$ is $\mathcal{L}$-equivalent to the spherical pedal curve-germ $ped_{\gamma, P}: (I, s_0)\to S^n$ of $\gamma$ relative to $P$ (resp., the spherical dual curve-germ ${\bf u}_n: (I, s_0)\to S^n$ of $\gamma$) if and only if $P\ne \pm{\bf u}_n(s_0)$ (resp., if $P= \pm{\bf u}_n(s_0)$).