LIDS - P - 1921 Curvatures of Surfaces and their Shadows

The relationship between the n-dimensional surfaces of smooth, strictly convex objects and the m-dimensional surfaces of their orthogonal projections, or shadows, is investigated. Our main results concern the relationships between the local properties of the surface at a point and those of its shadow. Specifically, the curvature Hessian of the projection at a boundary point is shown to be simply the projection of the curvature at the point's pre-image. Further, necessary and sufficient conditions are presented for solution of the inverse problem of determining the surface curvature at a point, given the curvatures of a series of projections involving the point. These conditions imply, for example, that knowledge of the projection of a surface point onto two hyperplanes and an additional two-dimensional subspace is necessary and sufficient to determine the local surface curvature there. These local results are then combined with a curvature based object representation on the Gaussian sphere to both construct the shadows of objects and to elucidate the inverse problem of reconstructing object shape from shadows. These results serve to illuminate and extend the work of Van Hove [1] for obtaining the two-dimensional shadow of an object in three-dimensional space.