On some ruled surfaces of a line congruence
\textit{N. K. Stephanidis} introduced the invariant third quadratic differential form III of a rectilinear congruence \(S\) in \(E^3\) in [Sitzungsber. Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 198, 267-279 (1989; Zbl 0707.53012)] and the III-main ruled surfaces of \(S\) in [ibid. 199, 53-57 (1990; Zbl 0718.53011)]. Using them, the author defines and studies two new classes of ruled surfaces in \(S\). Let \(\Omega=0\) be the differential equation of the III-main ruled surfaces of \(S\), \({1\over D(u,v)}={\text{III}\over\text{I}}\) and \(\zeta(u,v)={\Omega\over\text{I}}\), where I is the first fundamental form of \(S\). More precisely, she introduces the notions of \({1\over D^*}\)-ruled surfaces and \(\zeta^*(u,v)\)-ruled surfaces of \(S\) which correspond to a given function \({1\over D^*(u,v)}={1\over D(u,v)}\) and \(\zeta^*(u,v)=\zeta(u,v)\), respectively. Interesting properties of them are derived. For instance, it is proved that ``the curves on the middle-envelope of \(S\) which correspond to III-main ruled surfaces on the one hand, and to the \({1\over D^*}\)-ruled surfaces on the other hand, intersect each other harmonically''.
scientific article; zbMATH DE number 1320604