Shape from Chebyshev Nets

Abstract

We consider a special type of wiremesh covering arbitrarily curved (but smooth) surfaces that conserves length in two distinct directions at every point of the surface. Such “Chebyshev nets” can be considered as deformations of planar Cartesian nets (chess boards) that conserve edge lengths but sacrifice orthogonality of the parameter curves. A unique Chebyshev net can be constructed when two intersecting parameter curves are arbitrarily specified at a point of the surface. Since any Chebyshev net can be applied to the plane, such nets induce mappings between any arbitrary pair of surfaces. Such mappings have many desirable properties (much freedom, yet conservation of length in two directions). Because Chebyshev nets conserve edge lengths they yield very strong constraints on the projection. As a result one may compute the shape of the surface from a single view if the assumption that one looks at the projection of a Chebyshev net holds true. The structure of the solution is a curious one and warrants attention. Human observers apparently are able to use such an inference witness the efficaciousness of fishnet stockings and body-suits in optically revealing the shape of the body. We argue that Chebyshev nets are useful in a variety of common tasks.