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Analytical models are the most accurate method of geometric information representation. Parameterized smooth curves cannot be used in the field of analytical geometry, which explains the necessity for finding of analytical representation of such curves. The article considered the construction of a smooth curve presented in an analytical form and some approaches to finding an analytical model for a parametric Bezier curve. А presentation of a function in the form of its functional areas was chosen as prototype of the analytical model. The selected representation formed on the basis of the De Casteljau's method of constructing the Bezier curve and set-theoretic modeling. The Rvachev functions (R-functions) are used as the mathematical apparatus of set-theoretic operations on function areas. The functional-voxel method makes it possible to simplify the computation of R-functional procedures. An algorithm for constructing the functional area of the Bezier curve is developed on the basis of the presented combined R-voxel approach. The obtained results allow for the conclusions about the adequacy of this approach and its development protentional to construct more complicated structures.

The paper considers the claim that the function-voxel model allows arithmetic operations over the space of values of two different functions given by a single domain. At the same time, there are three possible approaches to solving the problem, leading to a similar result: functional approach -when the analytical representation of functions is involved in the calculations; functional voxel approach -when voxel data representing local geometric characteristics are used in the construction of local functions for further use in the calculation; voxel approach -when exclusively voxel data is used for sequential recalculation of local geometric characteristics of the model. The basic arithmetic operations on functional voxel models are considered, including such procedures as: addition, subtraction, modulo, exponentiation, taking root expressions, multiplication and division of functional voxel models. It is shown that the obtained applicability of arithmetic operations to functional voxel models leads to obtaining new complex functional voxel models.