Darboux Vector and Stress Analysis of Centro-Affine Frame

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 A₃ as affine 3-space associated with IR³.

The first written sources that can be achieved about affine space curve theoryare 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

References

1. Cayley, A., The algebraic structure of the orthogonal group and the other classical groups in a field of characteristic zero or a prime characteristic, J. Reine Angew.Math., 32,(1846).
2. Ces`aro, E., Lezioni di Geometria Intrinseca, Napoli, Italy, 1896.
3. Ferdinand P. Beer and E. Russell Johnson, Jr, Mechanics of Materials, Second Edition, McGraw-Hill, Inc, 1992.
4. Gardner, R. B. and Wilkens, G. R. The fundamental theorems of curves and hypersurfaces in centro-a_ne geometry, Bull. Belg. Math. Soc. Simon Stevin 4, 379(1997).
5. Hu, N., Centro-a_ne Space Curves with Constant Curvatures and Homogeneous Surfaces, J. Geom. 102(2011), 103-104.
6. Salkowski, E., and Schells, W., Allgemeine Theorie der Kurven doppelter Krümmung, Leipzig und Berlin, 1914.
7. Su, B., Some Classes of Curves in The A_ne space, Tohoku Math. Journ. 31,(1929), 283-291.
8. Su, B., A_ne Di_erential Geometry, Science Press, Beijing, China, 1983.