Volume of Arbitrary Shapes from Boundary Curvature and Medial Scale

The volume contained within any closed, simple, piece-wise smooth boundary can be determined by integrating over the boundary a function whose only parameters are the principal boundary curvatures and the distance to the medial manifold. The method is an extension of the common concept of mining rights, by which the interior of the earth is parceled out to patches of real estate on the surface. This concept can be extended to any object in m for m 2 independent of topological genus, yielding a one-to-one mapping between the boundary and the interior, which can be used to compute properties such as volume. Introduction The relationship between an object's interior and its boundary is of fundamental concern in geometry, and provides an approach by which properties of the object's interior, such as its volume, can be determined from its surface. Since an m-dimensional object has an m -1 () - dimensional boundary, the boundary may provide a more convenient domain for such calculatio..