Darboux Vector and Stress Analysis of Centro-Affine Frame

Authors : Yilmaz TUNÇER

Pages : 12-17

View : 39 | Download : 19

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 associated with that vector space. We denote A 3 as   affine 3-space 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 centro-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 centro-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 affine torsion. Lastly, we made stress analysis of a curve with constant affine curvature and   affine torsion in affine 3-space as an example.
Keywords : Centro affine Frame, Darboux Vector, Stress Analysis

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