Циклида Дюпена и кривые второго порядка. Часть 1

Abstract

Making smooth shapes of various products is caused
 by the following requirements: aerodynamic, structural, aesthetic,
 etc. That’s why the review of the topic of second-order curves is
 included in many textbooks on descriptive geometry and engineering
 graphics. These curves can be used as a transition from the one
 line to another as the first and second order smoothness. Unfortunately,
 in modern textbooks on engineering graphics the building of Konik
 is not given. Despite the fact that all the second-order curves are banded by a single analytical equation, geometrically they unites
 by the affiliation of the quadric, projective unites by the commonality
 of their construction, in the academic literature for each of
 these curves is offered its own individual plot. Considering the
 patterns associated with Dupin cyclide, you can pay attention to
 the following peculiarity: the center of the sphere that is in contact
 circumferentially with Dupin cyclide, by changing the radius of the
 sphere moves along the second-order curve. The circle of contact
 of the sphere with Dupin cyclide is always located in a plane passing
 through one of the two axes, and each of these planes intersects
 cyclide by two circles. This property formed the basis of the graphical
 constructions that are common to all second-order curves. In
 addition, considered building has a connection with such transformation
 as the dilation or the central similarity. This article considers
 the methods of constructing of second-order curves, which are
 the lines of centers tangent of the spheres, applies a systematic
 approach.