A conformal mapping for a rectilinear congruence
The author studies line congruences in a three-dimensional Euclidean space \(E^3\) with the property (c): the one-to-one mapping \(f: P(u, v)\to M(u, v)\), between their middle surface \(\overline{OP}= P(u, v)\) and their middle envelope \(\overline{OM}= M(u, v)\), is conformal. Apart from other interesting properties of them, it is proved that ``the determination of all congurences that have a given minimal surface \(M(u,v)\) as middle envelope and possess the property (c), is reduced to the integration of a non linear system of second order partial differential equations for an unknown function''. Additionally, in case \(M(u,v)\) is a right helicoid, the general solution of this system is found.
scientific article; zbMATH DE number 5118000