Generalized constant ratio surfaces in \mathbb{E}^3

It iswell-known that the positionvector function is themost basic geometric object for a surface immersed in the three dimensional Euclidean space $\mathbb{E}^3 $ . In 2001, B.-Y. Chen defined constant ratio hypersurfaces in Euclidean n-spaces. Independently, in 2010, by using another approach in dimension 3, the second author classified constant slope surfaces. In this paper, we extend this concept in order to study surfaces with the property that the tangential component of the position vector is a principal direction on the surface.