Feature Column from the AMS

sphericon 3

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) is equal to the total curvature of A.

Examples:

For a planar polygon the curvature is zero everywhere so the total curvature is zero. Gauss' Theorem reduces to the fact that for such a polygon the sum of the angles is (n-2). Special case: sum of angles in a triangle is .
The general definition of total curvature uses an integral, but when the curvature has the constant value K, the total curvature is just K times the area of A.
A sphere of radius R has constant curvature 1/R². The 0 and 90-degree meridians and the equator form a geodesic triangle (n=3) with three right angles. The area of this triangle is one-eighth of the area of the sphere, or (1/8)x(4R²) = R²/2. The total curvature is then R²/2 x 1/R² = /2. Which is equal to the sum of the interior angles minus .

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 2/ beta . Imagine that they draw a geodesic triangle around the cone point.

Suppose for simplicity that one of the vertices of the polygon is on the line there the two sides of the pie-slice are stuck together. Since the cone does not stretch when unrolled into the pie-slice, the triangle unrolls into a 5-sided polygonal figure.
The sum of the interior angles of this figure is exactly the sum of the angles of the triangle plus (two of the angles of the figure coalesce into one of the angles of the triangle when the cone is reassembled).
Since the sum of the interior angles in a planar 5-sided polygon is (5-2) = 3, the sum of the interior angles in the original triangle is 3-.
Gauss' Theorem tells us that the enclosed curvature in the geodesic triangle is the sum of its angles minus , yielding 2 - as the total curvature inside the triangle.
This argument is independent of the size of the triangle. In particular, it holds no matter how small the triangle is. This means that the cone has curvature 2- concentrated at 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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