Dupin Cyclide and Second-Order Curves. Part 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.