Darboux Vector and Stress Analysis of Equi-Affine Frame

Authors : Yilmaz TUNÇER

Pages : 7-11

View : 47 | Download : 9

Publication Date : 2018-12-29

Article Type : Conference Paper

Abstract :A set of points that corresponds a vector of vector space constructed on a  field is called an affine space associate with that vector space. We denote as  affine 3-space A 3 associated with IR 3 . The first written sources that can be achieved about affine space curve theory  are based on the 1890`s when Ernesto Ces\`{a}ro and Die Schon von Pirondini  lived period. From that years to 2000`s there are a some affine fames used in  curve theory. One of them is equi-affine frame. The grup of affine motions special linear transformation consist of volume  preserving linear transformations denoted by and comprising diffeomorphisms of  that preserve some important invariants such curvaures that in curve theory as well. In this study, we separated the matrix representing affine frame as symmetric  and antismmetric parts by using matrix demonstration of the equi-affine frame  of a curve given in affine 3-space. By making use of antisymmetric part, we  obtained the angular velocity vector which is also known as Darboux vector and  then we expressed it in the form of linear sum of affine Frenet vectors. On the other hand, by making use of symmetric part, we obtained the normal  stresses and shear stress components of the stress on the frame of the curve  in terms of the affine curvature and affine torsion. Thus we had the  opportunity to be able to explane the distinctive geometric features of the  affine curvature and affin torsion. Lastly, we made stress analysis of a curve with constant affine curvature and  affine torsion in affine 3-space as an example.
Keywords : Equi affine Frame, Darboux Vector, Stress Analysis

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