Total Curvature

(or Gaussian curvature), one of the measures of the curvature of a surface in the neighborhood of a point on the surface. The total curvature is equal to the product of the principal curvatures. In the case of a plane and any developable ruled surface, it vanishes. The total curvature of a sphere is constant and is equal to the reciprocal of the square of the sphere’s radius. For a surface that has the shape of an automobile tire, that is, for a torus, the total curvature is negative at the points contiguous to the wheel and positive at the points outside the wheel.

Suppose a neighborhood of a given point Pona surface is mapped into a sphere of unit radius by placing in correspondence with each point of the neighborhood the end point of a radius whose direction is the same as that of the normal to the surface at the point under consideration. The ratio of the area of the part of the sphere obtained to the area of the neighborhood on the surface approaches the total curvature as the neighborhood shrinks toward P. In order for this assertion to be true in all cases, the areas on the sphere must be assigned plus or minus signs when they are calculated. The sign depends on the direction in which the boundary on the sphere is traversed when the region on the surface is traversed in a certain direction.

The total curvature remains unchanged when the surface is bent— that is, when it is deformed in such a way that the lengths of the curves on the surface are not altered.