Journal for Geometry and Graphics 13 (2009), No. 1, 029--040
Copyright Heldermann Verlag 2009
Note on Flecnodes
Boris Odehnal
Inst. of Discrete Mathematics and Geometry, University of Technology, Wiedner Hauptstr. 8-10/104, 1230 Vienna, Austria
boris@geometrie.tuwien.ac.at
The flecnodes Fi on a regular and non torsal ruling R0 of a ruled surface R are the points where R's asymptotic tangents along R0 hyperosculate the ruled surface. The name flecnode characterizes the intersection curve ci of the tangent plane τi with R at Fi. It has a double point (a node) at Fi and this node is an inflection point for both linear branches of ci at Fi. We show a way to parameterize the smooth one-parameter 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
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