Darboux Vector and Stress Analysis of Equi-Affine Frame

Year 2018, Volume: 10, 7 - 11, 29.12.2018

Yilmaz Tunçer

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 A₃ associated with IR³.

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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