Descartes's Lost Theorem
Curvature and polyhedral curvature
Gauss's local integral formula. One way of measuring the Gaussian curvature at a point in a surface is to take a closed path about the point. The path should be made up of segments which are as straight as they can be on the surface (``geodesics''); it can then be described as a geodesic polygon. Gauss tells us (in slightly different words) that the sum of the exterior angles of this polygon is equal to 2 minus the integral of the curvature over the enclosed area or, in other terms, to 2 minus the product: average value of the Gaussian curvature inside the polygon times the area of the polygon. Taking smaller and smaller polygons about the point will zero in on the exact value of the curvature at the point. Examples: Polyhedral curvature. Let us apply this same construction to the surface of a polyhedron. Since the faces are planar, a geodesic polygon will be made up of straight line segments. Suppose the polygon encloses a vertex. Then an elementary calculation of angles shows that the sum of the exterior angles of the polygon is equal to the sum of the face angles at the vertex. So if the polyhedron had Gaussian curvature, the integral of that curvature over any polygon enclosing the vertex would be 2 minus the sum of the face angles at that vertex. The only way to interpret this is to say that
This is in fact how polyhedral curvature is defined. On to next Descartes page. Back to previous Descartes page. Back to first Descartes page. |