On a Carathéodory's conjecture on umbilics: representing ovaloids (Q1378845)

Carathéodory's conjecture states that every ovaloid in Euclidean 3-space has at least two umbilics. Apparently, this can now be considered to be proved in the analytic case, but the \(C^r\) case is still open. In this connection, the authors consider representations of ovaloids. For the so-called Bonnet chart they give the differential equation of the curvature lines. The support function, defined on \(S^2\), is arbitrarily extended to a neighborhood of \(S^2\). An explicit formula gives the points of the ovaloid in terms of such an extension.