On the parallel surfaces of the non-developable surfaces

Abstract

In the differential geometry of curves and surfaces, the curvatures of curves and surfaces are often calculated and results are given. In particular, the results given by using the apparatus of the curve - surface pair are important in terms of what kind of surface the surface indicates. In this study, some relationships between curvatures of the parallel surface pair ( X,Xr ) via structure functions of non - developable ruled surface X ( u,v ) = a ( u ) + vb ( u ) are established such that a ( u ) is striction curve of non - developable surface and b ( u ) is a unit spherical curve in E 3. Especially, it is examined whether the non - developable surface Xr is minimal surface, linear Weingarten surface and Weingarten surface. X and its parallel Xr are expressed on the Helicoid surface sample. It is indicated on the figure with the help of S W P. Moreover, curvatures of curve - surface pairs ( X,a ) and ( Xr,β ) are investigated and some conclusions are obtained.

Keywords

• parallel surfaces
• non-developable ruled surface
• striction line
• Gaussian curvature
• mean curvature
• curvatures of curve-surface pair