Generalized constant ratio surfaces in \(\mathbb E^3\)
The constant angle surfaces in the Euclidean 3-space, i.e., oriented surfaces whose normals make a constant angle with some fixed direction, have been intensively studied in the last years. One possible generalization for this topic is to consider surfaces with the property that the angle between the normal and the position vector is constant, these surface are called constant slope surfaces and they were classified by the second author in [``From golden spirals to constant slope surfaces'', J. Math. Phys. 51, No. 7, Article ID 073507 (2010; \url{doi:10.1063/1.3459064})]. A remarkable property of these special surfaces is that the tangential component of the position vector in the tangent plane of the surface is a principal direction. In this work the authors extend the concept of constant slope surfaces by using this property and call them generalized constant ratio surfaces (GCR surfaces) in order to point out the connection with the constant ratio hypersurfaces in the Euclidean \(n\)-space defined by \textit{B.-Y. Chen} in [Soochow J. Math. 27, No. 4, 353--362 (2001; Zbl 1007.53006)]. They obtain interesting results about flat and constant mean curvature GCR surfaces and give some pictures using MATLAB.
scientific article; zbMATH DE number 6307362
scientific article