Confocal conics and 4-webs of maximal rank

TL;DR

This paper explores the geometric properties of confocal conics and associated 4-webs, demonstrating their maximal rank and characterizing confocal conics through web theory, enriching the understanding of planar geometric structures.

Contribution

It introduces a novel web-theoretic characterization of confocal conics and proves the maximal rank property of specific 4-web configurations involving confocal conics.

Findings

Each of the four constructed webs is of maximal rank.

Confocal conics can be characterized by web theory.

The paper establishes geometric conditions for web maximality.

Abstract

Confocal conics form an orthogonal net. Supplementing this net with one of the following: 1) the net of Cartesian coordinate lines aligned along the principal axes of conics, 2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, 3) the net of tangents to a conic of the confocal family, we get a planar 4-web. We show that each of these 4-webs is of maximal rank and characterize confocal conics from the web theory viewpoint.