Geodesic
Curvature Lines and Normal Congruences of Triangular B�zier Patches
Journal for geometry and graphics, Tome 16 (2012) no. 1, pp. 29-4.

Voir la notice de l'article provenant de la source Heldermann Verlag

This paper aims at giving a self-contained description of the focal surfaces of the normal congruence of a triangular B�zier 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 B�zier 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.
Classification : 53A05, 68U05
Mots-clés : Triangular Bezier patch, normal congruence, curvature lines, focal points and surfaces 
@article{JGG_2012_16_1_JGG_2012_16_1_a3,
     author = {M. K. Saad and G. Weiss },
     title = {Curvature {Lines} and {Normal} {Congruences} of {Triangular} {B�zier} {Patches}},
     journal = {Journal for geometry and graphics},
     pages = {29--4},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2012},
     url = {http://geodesic.mathdoc.fr/item/JGG_2012_16_1_JGG_2012_16_1_a3/}
}
TY  - JOUR
AU  - M. K. Saad
AU  - G. Weiss 
TI  - Curvature Lines and Normal Congruences of Triangular B�zier Patches
JO  - Journal for geometry and graphics
PY  - 2012
SP  - 29
EP  - 4
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JGG_2012_16_1_JGG_2012_16_1_a3/
ID  - JGG_2012_16_1_JGG_2012_16_1_a3
ER  - 
%0 Journal Article
%A M. K. Saad
%A G. Weiss 
%T Curvature Lines and Normal Congruences of Triangular B�zier Patches
%J Journal for geometry and graphics
%D 2012
%P 29-4
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JGG_2012_16_1_JGG_2012_16_1_a3/
%F JGG_2012_16_1_JGG_2012_16_1_a3
M. K. Saad; G. Weiss . Curvature Lines and Normal Congruences of Triangular B�zier Patches. Journal for geometry and graphics, Tome 16 (2012) no. 1, pp. 29-4. http://geodesic.mathdoc.fr/item/JGG_2012_16_1_JGG_2012_16_1_a3/