Abstract
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.
Highlights
Computer synthesis of simple and complex geometrical objects and their transformations is essential for the wide range of design problems
The Bezier curves represent the classic type of smooth curves are of the greatest interest in this current research
The construction of the Bezier curve by an approach proposed in paragraph 3.2 is implemented by means of the algorithm represented in paragraph 4.1
Summary
Computer synthesis of simple and complex geometrical objects and their transformations is essential for the wide range of design problems. The necessity to construct the maximally accurate computer geometrical models for the contemporary design problems remains to be the primary. Nowadays the Set-theoretic operations for the analytical modelling of a space of the complex function (Rvachev function [1,2]) are one of the most curious approaches to constructing the analytically set objects of complex geometrical shape. CAD is traditionally considered to be the main application field of computer geometry though the construction of smooth curves and surfaces in this sphere is based on the parametrical descriptions of functions which makes impossible to apply them both in the analytical geometry and in the R-Functional modelling.
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