Some properties of ruled surfaces under homothety in $E^3$

Abstract

It is clearly that, If $ f : E^3 \rightarrow E^3$ is a homothety and $\varphi$ is a surface in three dimensional Euclidean space $E^3$ , $f (\varphi) = \overline{\varphi}$ is also a surface in $E^3$ . In this paper, the surface $\varphi$ has been taken a ruled surface, specially. It was shown that image surface $f (\varphi) = \overline{\varphi}$ has been a ruled surface, too. Furthermore, It was investigated whether some properties related with ruled surfaces on $\varphi$ have been invariant under the homothety $f$or not.