Generalized Similar Frenet Curves
Abstract
The paper is devoted to differential geometric invariants determining a Frenet curve in En up to a direct
similarity. These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of
the spherical indicatrices. Then, they expressed by focal curvatures of the curve. And then, we give
the relationship between curvatures of evolute curve and shape curvatures. Morever, the geometric
interpretations of these invariants are given.
Keywords
Self-similar curves,Similar curves,Shape curvatures,Focal curvatures
References
- [1] Encheva, R., Georgiev, G., Shapes of space curves. J. Geom. Graph 7 (2003), 145-155.
- [2] Encheva, R., Georgiev, G., Georgi, H., Similar Frenet curves. Results Math. 55 (2009), no. 34, 359-372.
- [3] Greub W., Linear Algebra. Springer Verlag, Heidelberg, 1967.
- [4] Millman, Richard S., Parker George D., Elements of differential geometry. Prentice Hall, 1977.
- [5] Struik, D. J., Lectures on Classical Differential Geometry. Dover, New York, 1988.
- [6] Uribe Vargas, R., On vertices focal curvatures and differential geometry of space curves. Bull. Braz. Math. Soc. 36 (2005), no. 3, 285-307.