Bianchi transformation 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 non-linear 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. The Bianchi transform for the Minding coil is constructed. Using a mathematical package, the Minding coil and its Bianchi transform are constructed.

Keywords

• gaussian curvature
• surface of revolution
• the minding coil
• bianchi transform