Rocky Mountain Journal of Mathematics

The geometry of cyclic hyperbolic polygons

Jason DeBlois

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Abstract

We will call a hyperbolic polygon \textit {cyclic}, \textit {horocyclic}, or \textit {equidistant} if its vertices lie on a metric circle, a horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Such convex $n$-gons are parametrized by the subspaces of $(\mathbb {R}^+)^n$ that contain their side length collections, and area and circumcircle or ``collar'' radius determine symmetric, smooth functions on these spaces. We give formulas for and bounds on the derivatives of these functions and make some observations on their behavior. Notably, the monotonicity properties of area and circumcircle radius exhibit qualitative differences on the collection of centered vs non-centered cyclic polygons, where a cyclic polygon is \textit {centered} if it contains the center of its circumcircle in its interior.

Article information

Source
Rocky Mountain J. Math. Volume 46, Number 3 (2016), 801-862.

Dates
First available in Project Euclid: 7 September 2016

Permanent link to this document
http://projecteuclid.org/euclid.rmjm/1473275763

Digital Object Identifier
doi:10.1216/RMJ-2016-46-3-801

Zentralblatt MATH identifier
1350.51003

Subjects
Primary: 51M09: Elementary problems in hyperbolic and elliptic geometries

Keywords
Cyclic polygon convex hyperbolic

Citation

DeBlois, Jason. The geometry of cyclic hyperbolic polygons. Rocky Mountain J. Math. 46 (2016), no. 3, 801--862. doi:10.1216/RMJ-2016-46-3-801. http://projecteuclid.org/euclid.rmjm/1473275763.


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