GEODESICS, LINE OF CURVATURES AND ASYMPTOTIC CURVES VERSUS RELAXED ELASTIC LINES ON AN ORIENTED SURFACE

BAYRAM E., Kasap E.

JOURNAL OF SCIENCE AND ARTS, no.1, pp.37-40, 2017 (ESCI) identifier

  • Publication Type: Article / Article
  • Publication Date: 2017
  • Journal Name: JOURNAL OF SCIENCE AND ARTS
  • Journal Indexes: Emerging Sources Citation Index (ESCI)
  • Page Numbers: pp.37-40
  • Keywords: Relaxed elastic line, geodesic, line of curvature, asymptotic curve, INTRINSIC EQUATIONS, 3-SPACE
  • Ondokuz Mayıs University Affiliated: Yes

Abstract

Let alpha(s) be an arc on a connected oriented surface S in E-3, parameterized by arc length s, with curvature k and length l. The total square curvature K of alpha is defined by K=integral(1)(0)kappa(2)The arc alpha is called a relaxed elastic line if it is an extremal for the variational problem of minimizing the value of K within the family of all arcs of length l on S having the same initial point and initial direction as alpha In this study, we show that a geodesic is a relaxed elastic line if and only if it is planar and an asymptotic curve cannot be a relaxed elastic line. Also, we obtain a criterion for a line of curvature to be a relaxed elastic line.