The purpose of this document is to list some outstanding questions in the subject of hyperbolic geometry (real, complex, quaternionic, and the Cayley plane), with the primary focus on the latter three. This list was inspired by the problem sessions held at the conference on Complex Hyperbolic Geometry in Luminy (CIRM) in July of 2003. The questions are roughly grouped by similarity. The name(s)...

There are two types of parallel lines in Hyperbolic Geometry. There are those who diverge from each other in both directions (type 1) and those that diverge in one direction but come arbitrarily close to each other in the other direction (type 2). The first type have a minimum positive distance at points on a common perpendicular. The second type have no minimum distance: the distance tends to ...

(1) Each pair of points can be joined by one and only one straight line segment. (2) Any straight line segment can be indefinitely extended in either direction. (3) There is exactly one circle of any given radius with any given center. (4) All right angles are congruent to one another. (5) If a straight line falling on two straight lines makes the interior angles on the same side less than two ...

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kähler structure. We start with the desingularisations of the quadric cone in C: the smoothing is a natural S-bundle over H, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S-bundle over H with symplectic...

These pages serve two purposes. First, they are notes to accompany the talk Hyperbolic and projective geometry in constraint programming for CAD by Walter Whiteley at the János Bolyai Conference on Hyperbolic Geometry, 8–12 July 2002, in Budapest, Hungary. Second, they sketch results that will be included in a forthcoming paper that will present the equivalence of the first-order rigidity theor...

It is shown how one can draw graphs on surfaces of negative Euler characteristic by using hyperbolic geometry and hyperbolic circle packing representations. The same approach applies to drawings of hyperbolic tessellations.

This article provides a simple pictorial introduction to universal hyperbolic geometry. We explain how to understand the subject using only elementary projective geometry, augmented by a distinguished circle. This provides a completely algebraic framework for hyperbolic geometry, valid over the rational numbers (and indeed any field not of characteristic two), and gives us many new and beautifu...

In order to obtain a global principle for modeling closed surfaces of arbitrary genus, rst hyperbolic geometry and then discrete groups of motions in planar geometries of constant curvature are studied. The representation of a closed surface as an orbifold leads to a natural parametrization of the surfaces as a subset of one of the classical geome-tries S 2 , E 2 and H 2. This well known connec...

Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.