Applications of Envelopes

TL;DR

This paper explores the use of envelopes of circle families to analyze matrix theory and hyperbolic geometry, revealing new relationships and boundary characterizations.

Contribution

It introduces novel connections between envelopes of circles and matrix numerical ranges, and applies these to hyperbolic geometry and pseudohyperbolic disks.

Findings

Relationship between envelopes and boundaries of circle families

Characterization of numerical ranges of 2x2 matrices

Boundaries of pseudohyperbolic disks

Abstract

Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. Here we use envelopes of families of circles to study objects from matrix theory and hyperbolic geometry. First we explore relationships between numerical ranges of $2\times 2$ matrices and families of circles to study the elliptical range theorem. Then we deduce a relationship between envelopes and the boundaries of families of intersecting circles and use it to find the boundaries of various families of pseudohyperbolic disks.