$3$-webs generated by confocal conics and circles

TL;DR

This paper studies specific geometric configurations involving confocal conics and Apollonian circles, demonstrating they generate trivial 3-webs and providing explicit formulas for these webs.

Contribution

It introduces explicit formulas for 3-webs generated by confocal conics and Apollonian circles with shared foci, revealing their trivial web structure.

Findings

Families of confocal conics and Apollonian circles generate trivial 3-webs.

Exact formulas describing these webs are derived.

The webs' trivial structure is established.

Abstract

We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.