Applications of envelopes
Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. More precisely, there are three different, but related, types of envelopes associated to most families of curves. In this paper, we illustrate how one can use these envelopes to approach problems appearing in fields like matrix theory and hyperbolic geometry. As a first example, we use envelopes to fill in the details of a proof by Donoghue of the elliptical range theorem; this use of envelopes provides insight into how certain numerical ranges arise from families of circles. More generally, envelopes can be used to compute boundaries of families of curves. To demonstrate how such arguments work, we first deduce a general relationship between the envelopes and boundaries of families of circles and then use this to compute the boundaries of several families of pseudohyperbolic disks.
Complex Analysis and its Synergies
Bickel, Kelly; Gorkin, Pamela; Tran, Trung; Applications of envelopes. Complex Anal. Synerg. 6 (2020), no. 1, Paper No. 2.