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Register a new userQuasi-circles through prescribed points
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Authors
John M. Mackay
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We show that in an L-annularly linearly connected, N-doubling, complete metric space, any n points lie on a K-quasi-circle, where K depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split over any virtually cyclic subgroup, then any geodesic line in G lies in a quasi-isometrically embedded copy of the hyperbolic plane.
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Viera v{C}erv{n}anova (Department of Mathematics
, Computern Science
, Faculty of Education
2020
Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. The configuration and its components, which are two orthogonal frames and two orthogonal families of chains, are in some way connected to classical geometric configurations such as the arbelos or the Pappus chain, or the Apollonian packing from the 20th century. In this paper, we build the fabric and list some of the obvious properties that result from this construction. Next, we focus on the curvature inside the individual components: we show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures of the circles of each chain are arranged in a quadratic bi-sequence. Because solving geometric sangaku problems was a gateway to our discovery of the fabric, we conclude this paper with two sangaku problems and their solutions using our results on curvatures.
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We find all analytic surfaces in space $mathbb{R}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of
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