Mathematica Eterna
Open Access

Abstract

A Conformal Approach to Bour’s Theorem

ZEHRA BOZKURT, ISMA˙IL GOK, F. NEJAT EKMEKC˙I and YUSUF YAYLI

In this paper, we give relation between Bour’s theorem and conformal map in Euclidean 3−space. We prove that a spiral surface and a helicoidal surface have a conformal relation. So, a helix on the helicoid correspond to a spiral on the spiral surface. Moreover we obtain that a spiral surface and a rotation surface have a conformal relation. So, spirals on the spiral surface correspond to parallel circles on the rotation surface. When the conformal map is an isometry we obtain the Bour’s theorem ,i.e, we obtain an isometric relation between the helisoidal surface and the rotation surface, which was given by Bour in [1]. Thus this paper is a generalization of Bour’s theorem.