e-MATH
The Differential Geometry of the Sphericon


 
 


3. Using Gauss' Theorem to measure surface curvature

Here is Gauss' Theorem: If a geodesic polygon with n sides encloses an area A, then (in radian measure) the sum of its interior angles minus (n-2)pi is equal to the total curvature of A.

Examples:

The two-dimensional geographers living on their surface can measure curvature safely by enclosing a curved region in a geodesic polygon and adding up the interior angles. Suppose, for example, their surface had a region with the geometry of a cone. To be specific, we make a cone by taking a pie-slice (good to be generous here) with vertex angle beta and gluing the edges together.

Our geometers living on the cone have to avoid the cone point: their insides would be stretched by a factor of 2pi/beta. Imagine that they draw a geodesic triangle around the cone point.

By using Gauss' Theorem, the surface geographers have made this determination without ever touching the cone point itself.


On to Sphericon page 4.

Back to Sphericon page 2.



© copyright 1999, American Mathematical Society.