Examples of surfaces of constant mean curvature

Abstract

A surface in E3 is called parallel to the surface M if it consists of the ends of constant length segments, laid on the normals to the surfaces at points of this surface. The tangent planes at the corresponding points will be parallel. For surfaces in E3 the theorem of Bonnet holds: for any surface that has constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature. Using Bonnet's theorem for a surfaces of revolution of constant posi­tive Gaussian curvature, surfaces of constant mean curvature are con­structed. It is proved that they are also surfaces of revolution. A family of plane curvature lines (meridians) is described by means of elliptic inte­grals. The surfaces of constant Gaussian curvature are also described by means of elliptic integrals. Using the mathematical software package, the surfaces under consideration are constructed.

Keywords

• parallel surface
• mean curvature
• gaussian curvature
• bonnet theorem
• elliptic integrals