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Applications of envelopes

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Abstract

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.

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Acknowledgements

The authors would like to thank the referee for a careful and detailed report that substantially improved many aspects of this paper. The authors would also like to thank U. Daepp for providing the illustration of Steiner’s theorem in Fig. 2, Elias Wegert for sharing the envelope algorithm, and Phuong Nguyen for her work on pseudohyperbolic disks and in particular, her contributions to this new proof of Theorem 1.

Funding

K. Bickel was supported in part by National Science Foundation DMS Grant #1448846, and P. Gorkin was supported in part by Simons Foundation Grant #243653.

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  1. Department of Mathematics, Bucknell University, 380 Olin Science Building, Lewisburg, PA, 17837, USA

    Kelly Bickel, Pamela Gorkin & Trung Tran

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  1. Kelly Bickel
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  2. Pamela Gorkin
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  3. Trung Tran
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Correspondence to Kelly Bickel.

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Bickel, K., Gorkin, P. & Tran, T. Applications of envelopes. Complex Anal Synerg 6, 2 (2020). https://doi.org/10.1007/s40627-019-0039-z

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