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 (Rfunctions)
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