Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometric space analogous to Euclidean space, but such that Euclid's …

The hyperbolic case occurs when the extended map has no fixed point in L itself but has two fixed points at infinity: we examine this transformation in the half- space model H for …

6 days ago · There are no similar triangles in hyperbolic geometry. The best-known example of a hyperbolic space are spheres in Lorentzian four-space. The Poincaré hyperbolic disk is a …

We want to know what it means for a space (X, d) to be hyperbolic. There are several definitions, all of them being equivalent. We will give various versions of Gromov’s hyperbolic criterion.

In hyperbolic geometry, through a point not on a given line there are at least two lines parallel to the given line. The tenets of hyperbolic geometry, however, admit the other four Euclidean …

In this chapter we gather together basic information about the geometry of two- and three-dimensional hyperbolic spaces and their isometries. This will set the stage for our study of …

4 days ago · Later, physicists discovered practical applications of these ideas to the theory of special relativity. Hyperbolic geometry also inspired the art of M. C. Escher, and has various …

With his “Circle Limit” series of drawings, Escher explored the endless symmetries inherent in hyperbolic space: in “Circle Limit III,” red, green, blue and yellow fish tessellate their world in a …

We use the hyperbolic metric in order to take advantage of the surprising property that hyperbolic space has more room than our familiar euclidean space. Two parallel lines are always the …

The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities).