Shape-from-Image via Cross-Sections

Using structural geometry, Whiteley showed that a line drawing is a correct projection of a spherical polyhedron if and only if it has a cross-section compatible with it. We here enlarge the class of drawings to which this test applies, including those of polyhedral disks. Our proof is constructive, showing how to derive all spatial interpretations, it relies on elementary synthetic geometric arguments, and, as a by-product, it yields a simpler and shorter proof of Whiteley’s result. Moreover, important properties of line drawings are visually derived as corollaries: realizability is independent of the adopted projection, it is an invariant projective property, and for trihedral drawings it can be checked with a pencil and an unmarked ruler alone.