Über die Leitflächen konstanten Mittenabstandes von Geradenkongruenzen des einfach isotropen Raumes

Abstract

In the simply isotropic space J₃ ⁽¹⁾ let S be a congruence of straight lines with the unit vector giving the direction of its lines$\vec e$(u,v) and with middle surface$\vec p$(u,v). A surface of reference$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {p} _t (u,v): = \vec p(u,v) + t(u,v) \vec e(u,v)$ has constant distance from$\vec p$(u,v) if t= const. In this paper we first investigate the mapping, which is established by the lines of S between two arbitrary surfaces of the above type. Next we give a new characterization of the focal points of some hyperbolic lines of the congruence S. Finally, we study some associated ruled surfaces of S, which are connected to the surfaces$\vec p_t$(u,v), t ∈ ℝ.