We call quadrilaterals of this type hyperbolic quadrilaterals as their sides are geodesics in the

hyperbolic geometry of the unit disk.

As shown in the previous sections, even the simplest instances of such models (namely, those based on elementary hyperbolic surfaces) already allow trajectories of considerable complexity, due to the interplay between the effective force induced by the

hyperbolic geometry and that induced by the scalar potential.

Ungar, Analytic

Hyperbolic Geometry, World Scientific Publishing Co.

Now, we give basic notions for

hyperbolic geometry in [H.sup.3].

The Einstein relativistic velocity model is another model of

hyperbolic geometry. Many of the theorems of Euclidean geometry are relatively similar form in the Einstein relativistic velocity model, Aubel's theorem for gyrotriangle is an example in this respect.

And the researchers have developed an in-depth theory that uses

hyperbolic geometry to describe a negatively curved shape of complex networks such as the Internet.

Before

hyperbolic geometry was discovered, it was thought to be completely obvious that Euclidean geometry correctly described physical space, and attempts were even made, by Kant and others, to show that this was necessarily true.

A complete presentation of

hyperbolic geometry by means of analytic methods is given in Nuut [20].

In areas where the surface has floppy,

hyperbolic geometry, the algorithm will identify many mesh points; where the surface has more tightly curved geometry, the algorithm will identify fewer points.

As the geodesics in the figure is convex towards the lateral direction, it is similar to the geodesics in

hyperbolic geometry. This result suggests the possibility that photographic space is hyperbolic.

Chapters cover

hyperbolic geometry (illustrated with Escher's models of the hyperbolic plane), complex numbers (so essential in quantum mechanics), Riemann surfaces, quaternions, n-dimensional manifolds, fibre bundles, Fourier analysis, G6del's theorem, Minkowski space, Lagrangians, Hamiltonians, and other terrifying topics.

It might have its own interest in the theory of complex

hyperbolic geometry.