“Plücker Conoid”: More Characteristic Curves

Abstract

In this chapter, more characteristic curves are derived on the premises of “Plücker conoid”, constructed at a point of a smooth regular part surface. At the beginning main properties of the surface of “Plücker conoid” are briefly outlined. This includes but not limited to basics, analytical representation, and local properties along with auxiliary formulae. This analysis is followed by analytical description of local geometry of a smooth regular part surface. Ultimately, expressions for two more characteristic curves are derived. These newly introduced characteristic curves are referred to as Plücker curvature indicatrix and $An R(P_1)$-indicatrix of a part surface. The performed analysis makes it possible derivation of equations for two more planar characteristic curves for analytical description of the contact geometry of two smooth regular part surfaces at a point of their contact. One of the newly derived characteristic curves is referred to as “$An_{R}(P_1/P_2)$-relative indicatrix of the first kind” of two contacting part surfaces. Another one in a curve inverse to the characteristic curve $An_{R}(P_1/P_2)$. This second characteristic curve is referred to as “$An_{k}(P_1/P_2)$-relative indicatrix of the second kind”. Main properties of both the characteristic curves are briefly discussed in this section of the monograph.