Pseudo-minimality and ruled surfaces

This paper is a follow-up to the author's series of works about shape modeling for an orthotropic elastic material that takes an equilibrium form inside the area with the specified boundaries. V. M. Gryanik and V. I. Loman, based on thin shell equilibrium equations, solved about 30 years ago a similar problem for an isotropic mesh attached to rigid parabolic edges. With a view to extend modeling to orthotropic materials (and other boundary contours), the author in his publications of 2016–2017 proposed an approach to the problem based on the application of surfaces with a constant ratio of principal curvatures. These surfaces are called pseudo-minimal surfaces. A partial differential equation that defines (in the local sense) a class of pseudo-minimal surfaces is very complex for analysis. However, for some classes of surfaces, the analysis is greatly simplified, notably, the analysis can be performed without this inconvenient PDE, but with the method of moving frames. The author is referring to a class of ruled surfaces. This class is interesting not only due to the aforesaid but also due to an evident interest manifested by architects and builders. However, one should discuss not the pseudo-minimal ruled surfaces (they exist but are obviously trivial) but an invariant (principal curvatures ratio), which is not an identical constant on a given surface but its contour lines coincide with the lines of some invariant family. Roughly speaking, there are surfaces whose pseudo-minimal condition is satisfied identically, and surfaces that are pseudo-minimal “in a limited sense -lengthways the lines of a certain family, internally connected with the surface. The article finds that the role of such a family can be obviously played by "equidistant” lines for the striction line of a skew ruled surface, and rays are the carriers of such a ruled surface, they form a regulus with constant Euclidean invariants.