Affine Translation Surface In 3 Dimensional Euclidean Space

Differential geometry is a subject with long history.Its influence on other natural sciences becomes more profound in recent years.Surface and curve theories are two main contents of differential geometry.Translation surfaces received many focus due to its special property in 3 dimensional Euclidean space and Minkowski space.In 3 dimensional Minkowski space,according to the direction of translation,translation surface can be divided into six types.In 3 dimensional Euclidean space,translation surfaces which formed by two regular curves with different parameters are studied less.In this paper,a new kind of curved translation surface is defined in 3 dimensional Euclidean space,called affine translation surfaces.First of all,the affine translation surface is defined by X?u,v?= r₁?u??r₂?v??where r₁?u?and r₂?v?are two regular curves in 3 dimensional Euclidean space.Then,the properties of affine translation surfaces in 3 dimensional Euclidean space are researched according to the affine transformation and the Frenet frame,which include the Gauss curvature,mean curvature,the first and second fundamental form,and the properties of translation surfaces when the curve r₂?v?are some special curves such as general helix,Bertrand curve,Mannheim curve and rectifying curve.