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.


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.

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© copyright 1999, American Mathematical Society.