A general method is proposed to derive equations of irregular curves O x y D ( , ) = 0 and of irregular surfaces O x y z G ( , , ) = 0 in implicit form, where the functions O x y D , ) and O x y z G ( , , ) belong to a prescribed differentiability class. The method essentially involves interlineation and interflation of functions. An example is considered.Introduction. The paper [1] considers the approximation methods for multivariable functions with the use of interlination and interflation of functions and some of their applications in modern computer-based technologies. We will analyze the methods of approximation of functions of two and three variables in order to derive equations for compound lines and surfaces in implicit form with the use of functions that belong to a specified differentiability class. Problems of deriving implicit equations of surfaces and curves has wide practical application (for example, in variational methods for solving stationary and nonstationary boundary-value problems with two and three spatial variables). Deriving implicit equations of surfaces with the possibility of choosing surface model parameters allows obtaining mathematical models of surfaces with prescribed properties such as modeling aerodynamic surfaces with allowance for not only geometrical properties (the continuity of derivatives up to some order) but also physical properties such as heat conductivity, strength, etc. [2,3]. Therefore, it is expedient to solve these problems with the use of a mathematical model of the surface by choosing its parameters using some criteria.Such an approach is expedient in setting up mathematical models of aerodynamic surfaces [2,3]. Relevance of the Subject. One of the most difficult stages in setting up implicit mathematical models of curves and surfaces is to ensure the continuity of derivatives of prescribed order. Therefore, of current interest is to construct and study implicit mathematical models of curves and surfaces that meet these requirements.Available Methods to Solve the Problem Posed. The implicit form of curves and surfaces, respectively, is
