Mathematical modeling of contact of two surfaces in the first order of tangency

The topic of the paper is in the area of mathematical modeling of the contact of two smooth regular surfaces in the first order of tangency. The paper is aiming to develop an approach for the mathematical description of the geometry of tangency of two smooth regular surfaces, which share a common point, and in such way to describe geometry of contact of two surfaces not only qualitatively, but quantitatively as well. Various approaches of the first-, second-, and higher-order approximation are discussed. To resolve the problem under consideration, a characteristic curve of new kind is introduced. We refer to this characteristic curve as the so-called indicatrix of conformity of two surfaces, which are tangent to each other at a point. Application of the indicatrix of conformity enables one developing an appropriate quantitative and qualitative description of the geometry of contact of two smooth regular surfaces in the first order of tangency. The results presented in the paper will have many practical applications in science and engineering: 1. r optimal tool-path generating in machining sculptured surfaces on multiaxis CNC milling machine, from which the problem arose; or developing a perfect TCA (tooth contact analysis) computer programs for gear pairs and for gear trains; for improving accuracy of solving the problem of two rigid bodies contact; or fitting two surfaces while designing blend surfaces; en performing elastohydrodynamic film thickness calculations, etc