On the cutting of garments -- variation about a theme of Tchebychev (Q648501)

Parametrizations of surfaces in \({\mathbb R}^3\) for which the coordinate curves are parametrized by arc length were introduced by Tchebychev in an article published 1878 under the same title. They are named Tchebychev nets now. The author calls a surface `dressable' if it can be covered by a Tchebychev net. Locally, every surface is dressable. Globally, only partial results in this direction are known. Here, the following is proved: If from the sphere two arcs of great circles of suitable length intersecting orthogonally at their midpoints are removed, the remaining open set is dressable. A similar result is proved for the Poincaré disk by the same method. Several remarks on the history of the notion of Tchebychev nets and computer images might attract also the interest of non-specialists.