Boundary Curvature and Medial Scale Stetten 1 Volume of Arbitrary Shapes

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 R 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 calculations than the interior. One may determine a 3D object's interior volume (or a 2D object's area) by establishing a one-to-one mapping between its surface (or boundary contour) and its interior -the mining rights, as it were -and then integrating these rights over the entire surface. Vectors normal to the surface separate the mining rights between neighboring portions of the surface, just as stakes driven into the ground separate the claims of neighboring miners. At the very center of the earth the mining rights transfer to the other side of the planet. The center of the earth is just a special case of the locus of points for any shape called the Blum medial manifold. Brief Review of Blum Medial Manifold For closed contours in R, the Blum medial manifold is that locus of the centers of all circles completely enclosed by the boundary contour that touch the contour in more than one location [1]. Thus for the rectangle in Fig. 1, the centers of all such medial circles form the branching medial manifold shown as thick dotted lines.