On constant-ratio surfaces of rotation in Euclidean 4-space

Abstract

The general rotational surfaces of [Formula: see text] were first studied by Moore. The Vranceanu surfaces are special examples of this kind of surfaces. These constant-ratio surfaces are surfaces for which the ratio of the norms of the tangent and normal components of the position vector fields is constant. However, spherical surfaces and conical surfaces are also trivial examples of constant-ratio surfaces. Thus, if the norms of the tangent or normal components of the position vector fields are constant, then the given surface is called [Formula: see text]-constant or [Formula: see text]-constant, respectively. In this paper, we considered three types of rotational surfaces lying in [Formula: see text]-dimensional Euclidean space [Formula: see text]. We have obtained the necessary and sufficient conditions for these surfaces to satisfy the [Formula: see text]-constant, [Formula: see text]-constant or constant-ratio conditions. With the help of these results, we characterized the meridian curves of the surfaces. Further, we also give some examples to support the results obtained.