
Journal for Geometry and Graphics 14 (2010), No. 1, 045058 Copyright Heldermann Verlag 2010 Triply Orthogonal Line Congruences with Common Middle Surface Despina Papadopoulou Dept. of Mathematics, Aristotle University, 54124 Thessaloniki, Greece papdes@math.auth.gr Pelagia Koltsaki Dept. of Mathematics, Aristotle University, 54124 Thessaloniki, Greece kopel@math.auth.gr Let S be a non parabolic line congruence in E^{3}, whose middle surface P(u,v) is different from its middle envelope M(u,v). We prove that there exist two line congruences S', S'' orthogonal to S and to each other with common middle surface P(u,v) iff S is isotropic or the straight lines of S', S'' are directed by the tangent vectors of the spherical image of the Sprincipal ruled surfaces of S, in case S is not isotropic. Then, studying the properties of a triplet S, S', S'', we find a new geometric interpretation for the curvature of S. Keywords: Orthogonal line congruences, middle surface, middle envelope, curvature of a line congruence. MSC: 53A25 [ Fulltextpdf (143 KB)] for subscribers only. 