Back to Search
Start Over
Descartes Circle Theorem, Steiner Porism, and Spherical Designs
- Publication Year :
- 2018
-
Abstract
- A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be constructed starting with any initial circle, tangent to the parent circles. What do the circles in these 1-parameter family of Steiner chains of length k have in common? We prove that the first k-1 moments of their curvatures remain constant within a 1-parameter family. For k=3, this follows from the Descartes Circle Theorem. We extend our result to Steiner chains in the spherical and hyperbolic geometries and present a related more general theorem involving spherical designs.
- Subjects :
- Mathematics - Metric Geometry
Mathematics - Differential Geometry
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1811.08030
- Document Type :
- Working Paper