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Journal for Geometry and Graphics 16 (2012), No. 1, 029--040
Copyright Heldermann Verlag 2012

Curvature Lines and Normal Congruences of Triangular Bézier Patches

M. Khalifa Saad
Faculty of Science, Sohag University, 82524 Sohag, Egypt

Gunter Weiss
Institute for Geometry, University of Technology, 01062 Dresden, Germany

This paper aims at giving a self-contained description of the focal surfaces of the normal congruence of a triangular Bézier patch in terms of the control points of the patch. The normal congruence of a surface is an Euclidean concept and it is algebraic, if the original surface is algebraic. For a parameter representation of the normal congruence we need derivatives and normal vectors of the patch. For calculating the pair of so called focal points on each generator of the congruence, one has to investigate the curvature lines of the patch in addition. Thus the results become already of high algebraic order. Therefore the treatment is restricted to quadratic and cubic triangular Bézier patches, and focal points of its normal congruence are calculated only for the normals at very special points of the patch: the corner points and the "midpoint". Therewith one can deduce a control point systems for normal congruence. Finally the calculations are applied to a numerical example. The paper is a first attempt to deal with the focal surfaces of a line congruence explicitly and might deliver fundamentals to treat refraction congruences, which have applications in geometric optics.

Keywords: Triangular Bezier patch, normal congruence, curvature lines, focal points and surfaces.

MSC: 53A05; 68U05

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