| Helical surface is a kind of common curved surface.Due to its geometric properties,it is widely used in mechanical transmission,such as screw rotary pump of petroleum production machinery,various types of screw pump,single screw hydraulic machinery,etc.Affine differential geometry is an important branch of differential geometry,which mainly studies the invariable properties of nondegenerate hypersurfaces in affine space under mono-imitative transformation.Helical surface is an important kind of surface in differential geometry,aiming at the important characteristics and the wide application of helical surface in three-dimensional Euclidean space,this thesis deeply discusses a kind of helical surface in affine differential geometry.The parabolic helical surface discussed in this thesis is a curved surface generated by a plane curve rotating about a fixed axis while moving uniformly along the direction of the axis.And the helical surface can be divided into elliptic,hyperbolic and parabolic forms due to the different rotation axes.This article in three dimensional affine space used the Blaschke measurement to study the parabolic helical surface.Through the research,this thesis has a conclusion that the Gauss curvature and the mean curvature of the helical surface have nothing to do with the rotation factor v and the helicoid are classified when the Gauss curvature is zero or the mean curvature is zero.When h=0.the spiral motion degenerate into rotary motion,the helical surface turn into a rotation surface.In this case,a further classification of the flat and small parabolic surface of revolution have been found and drawn the corresponding image. |
