The Bianchi Transform of the Minding Coil

Abstract

The work is devoted to the study of the Bianchi transform for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Mining top, the Minding coil, the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuens surface and the Dinis surface. The study of surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudospherical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is established. The sin -Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface.