Pseudo-minimal surfaces of revolution

Abstract

This paper is a follow-up to the first author’s series of works about shape modeling of orthotropic elastic material that takes the equilibrium form inside the area with the specified boundaries. M.S. Bukhtyak, in a number of his publications of 20162020, proposed an approach to the model building based on the application of surfaces with a constant ratio of principal curvatures. These surfaces are named pseudo-minimal surfaces. The theorem of existence has been demonstrated and the finitely-element model has been built. The condition distinguishing the class of pseudo-minimal surfaces, as applied to ruled surfaces, is either satisfied identically (trivial subclasses) or is satisfied along a family of lines. The corresponding classes of ruled surfaces have been comprehensively characterized geometrically. A partial differential equation that defines (in the local sense) the class of pseudo-minimal surfaces is very complex for analysis, which makes it relevant to consider approximate solutions. The current paper considers the pseudo-minimal surfaces of revolution. Generation of the approximate solutions is complicated by the tendency of the formal Taylor polynomial to diverge. However, the approximate solutions (of course, not ideal) have been generated. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.