arXiv subjects

Aaron Fish

Publications and source records attributed to Aaron Fish.

2 recordsLinked to original sources

arXiv2023

A Poincar\'{e} ball model for taxicab hyperbolic geometry

Taxicab space is a modification of Euclidean space that uses an alternative notion of distance. Similarly, the Poincar\'{e} ball is a model of hyperbolic geometry that consists of a subset of Euclidean space with an alternative notion of distance. In this paper, we merge these two variations to create a taxicab version of the Poincar\'{e} ball. We determine the isometry group for this new space and show that this space is hyperbolic in the sense of Gromov.

math.MG↗

arXiv2018

Apollonian sets in taxicab geometry

Fix two points $p$ and $q$ in the plane and a positive number $k \neq 1$. A result credited to Apollonius of Perga states that the set of points $x$ that satisfy $d(x, p)/d(x, q) = k$ forms a circle. In this paper we study the analogous set in taxicab geometry. We find that while Apollonian sets are not taxicab circles, more complicated Apollonian sets can be characterized in terms of simpler ones.

math.MG↗