
Journal for Geometry and Graphics 13 (2009), No. 1, 029040 Copyright Heldermann Verlag 2009 Note on Flecnodes Boris Odehnal Inst. of Discrete Mathematics and Geometry, University of Technology, Wiedner Hauptstr. 810/104, 1230 Vienna, Austria boris@geometrie.tuwien.ac.at The flecnodes F_{i} on a regular and non torsal ruling R_{0} of a ruled surface R are the points where R's asymptotic tangents along R_{0} hyperosculate the ruled surface. The name flecnode characterizes the intersection curve c_{i} of the tangent plane τ_{i} with R at F_{i}. It has a double point (a node) at F_{i} and this node is an inflection point for both linear branches of c_{i} at F_{i}. We show a way to parameterize the smooth oneparameter family of flecnodes of R which in general forms a curve with two branches. For that we derive the equation of the ruled quadric on three given lines in terms of Plücker coordinates of the given lines. Keywords: Ruled surface, flecnode, line geometry, Lie's osculating quadric. MSC: 53A05; 53A25 [ Fulltextpdf (1807 KB)] for subscribers only. 