Canal Surfaces With Generalized 1-type Gauss Map In Euclidean 3-space

In the 1970s,Chen B Y first proposed the concept of submanifolds of finite-type in Euclidean space and pseudo Euclidean space.Later,the concept of Gauss map of finite-type was extended and the Gauss map of submanifolds of finite-type was defined.In this paper,we applied the concept of Gauss map of finite-type to the canal surface which is generated by a single-parameter spherical family moving along a spatial curve in Euclidean 3-space.The main work is to study the properties of canal surface with generalized 1-type Gauss map.The concept of generalized 1-type Gauss map was proposed by several real examples,that is,the Gauss map G satisfies ΔG=fG+gC,here Δ is the Laplace operator,f and g are nonzero smooth functions,C is a nonzero constant vector.In particular,the 1-type Gauss map and the pointwise 1-type Gauss map were given.Second,according to the parametric equation of the surface of revolution and the parametric equation of the canal surface in Euclidean 3-space,these two surfaces are classified according to their Gauss map.Explicitly,the properties of surface of revolution with generalized 1-type Gauss map,and the characters of canal surface with generalized 1-type Gauss map are discussed.Finally,the canal surface with generalized 1-type Gauss map,the canal surface with pointwise 1-type Gauss map and the canal surface with 1-type Gauss map are studied and compared.At the same time,several examples of special surfaces with generalized 1-type Gauss map are visualized.