Свойства циклид дюпена и их применение. Часть 4: приложения

Abstract

In the first and second parts of the work there were
 considered mainly properties of Dupin cyclide, and given some
 examples of their application: three ways of solving the problem of
 Apollonius using only compass and ruler, using the identified properties
 of cyclide; it is determined that the focal surfaces of Dupin
 cyclid are degenerated in the lines and represent curves of the
 second order – herefrom Dupin cyclide can be defined by conic curve and a sphere whose center lies on the focal curve. Polyconic
 compliance of these focal curves is identified. The formation of the
 surface of the fourth order on the basis of defocusing curves of the
 second order is shown.
 In this issue of the journal the reader is invited to consider the
 practical application of Dupin cyclide’s properties. The proposed
 solution of Fermat’s classical task about the touch of the four
 spheres by the fifth with a ruler and compass, i.e., in the classical
 way. This task is the basis for the problem of dense packing. In the
 following there is an application of Dupin cyclide as a transition
 pipe element, providing smooth coupling of pipes of different diameters
 in places of their connections. Then the author provides
 the examples of Dupin cyclide’s application in the architecture as
 a shell coating. It is shown how to produce membranes from the
 same cyclide’s modules, from different modules of the same cyclide,
 from the modules of different cyclides, from cyclides with the inclusion
 of other surfaces, special cases of cyclides in the educational
 process. The practical application of the last problem found the
 place in descriptive geometry at the final geometrical education of
 architects in the "Construction of surfaces". Here such special cased
 of cyclides as conical and cylindrical surfaces of revolution.