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Journal for Geometry and Graphics 11 (2007), No. 1, 001--014
Copyright Heldermann Verlag 2007

Inflection and Torsion Line Congruences

Rashad A. Abdel-Baky
Dept. of Mathematics, University of Assiut, Egypt

Based on E. Study's dual line coordinates, the instantaneous invariants of the relative motion between two dual unit spheres are used for deriving expressions for the velocity and the acceleration of point trajectories (dual curve). The expressions of the curvature and torsion of this curve are related to these invariants. From which the well known inflection, torsion curves and Ball's points of spherical kinematics are calculated in dual space. Then by using E. Study's map two line congruences are introduced and their spatial equivalents are examined in detail. In two ways the invariants of the relative motion are used for deriving a new proof of Disteli's formulae and concise explicit expressions of the inflection line congruence are directly obtained. The obtained explicit equations degenerate into a quadratic form, which can easily give a clear insight into the geometric properties of the inflection line congruence.

Keywords: E. Study's map, axodes, line congruence, Disteli's formulae.

MSC: 53A17; 53A04, 53A05

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