Journal for Geometry and Graphics 14 (2010), No. 1, 045--058
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 E3, 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 S-principal 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
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