From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: area of triangles in spherical/hyperbolic 2-D geometry
Date: 11 May 1999 16:07:01 GMT
Newsgroups: sci.math

David Bernier  <bernier@my-dejanews.com> wrote:

>I seem to remember that, for a spherical triangle
>T, one has excess(T) = area(T)  where
>   excess(T) :=  (sum of interior angles of T) - pi.

>I have heard that the sum of the interior angles of
>a hyperbolic triangle is always less than pi.
>If T is a hyperbolic triangle and we define
>defect(T) := pi - (sum of interior angles of T),
>is there a simple relationship between area(T) and
>defect(T)?

Is "equality" a simple relationship? :-)

On any surface one may define the (Gaussian) curvature at a point by the 
limiting ratio between excess and area of a suitably decreasing family of 
triangles around that point. You've picked up on the examples of 
constant-positive and constant-negative curvature (of course the plane is an 
example of constant-zero curvature).

>What would be a good introductory text on
>spherical/hyperbolic geometries (including the
>Poincare model of the hyperbolic plane) that
>doesn't deal too much with tensors and Riemannian
>metrics, yet gives a good intuitive understanding
>of these geometries?

Barrett O'Neil has a nice undergraduate textbook on curves and surfaces.
(It's pretty old now but I'm out of touch with current textbooks in this
area.)

dave