A possibility of application of Plücker's conoid for mathematical modeling of contact of two smooth regular surfaces in the first order of tangency

Abstract

The purpose of the paper is to investigate a possibility of application of Plücker's conoid in engineering geometry of surfaces. The research reveals that Plücker's conoid itself is a complex 3D characteristic image. On the premises of Plücker's conoid, the 2D characteristic curves of novel kind are introduced. We refer to these characteristic curves as to the An R ( P)-indicatrix and An k ( P)-indicatrix of the surface P. The characteristic curves An R ( P) and An k ( P) in many cases are similar to the corresponding Dupin's indicatrices Dup R ( P) and Dup k ( P). For two interacting surfaces the developed approach yields introducing of the Plücker's relative indicatrices Pl R ( P/T) and Pl k ( P/T). Finally, the 3D characteristic curves Pl R ( P/T) and Pl k ( P/T) reduced to the planar characteristic curves An R ( P/T) and An k ( P/T), i.e., to the so-called An R ( P/T)- and An k ( P/T)-relative indicatrices. The characteristic curves An R ( P/T) and An k ( P/T) could be employed. (a) for analytical description of local topology of composite surfaces along the linkage curve; (b) for developing of the optimal tool-paths for sculptured surface machining on multiaxis NC machine, as well as for analytical description of conditions of proper sculptured surface machining; (c) for improving accuracy of solution to the problem of contact of two elastic bodies, etc. These characteristic curves reported in the paper in many cases are similar to the indicatrices of conformity Cnf R ( P/T) and Cnf k ( P/T) of two smooth rugular surfaces earlier developed by the author. The obtained results perfectly correlate with the results obtained by other researchers, as well as with the results earlier developed by the author.