Functional-Voxel Modelling of Bezie Curves

The problem of this research is the impossibility of applying parametric functions in theoretical-multiple modeling, that significantly narrows the range of problems solved by analytical models and methods of computational geometry. To expand the possibilities of R-functional modeling application in the field of computer-aided design systems, it is proposed to solve the problem of finding an appropriate representation of parametric curves using functional-voxel computer models. The method of functional-voxel modeling is considered as a computer graphic representation of analytical functions’ areas on the computer. The basic principles and examples of combining R-functional and functional-voxel methods with obtaining R-voxel modeling have been presented. In this case, R-functional operations have been implemented on functional-voxel models by means of functional-voxel arithmetic. Based on the described approach to modeling of theoretical-multiple operations for the function area represented by graphical M-images, two approaches to construction a functional-voxel model of the Bezier curve have been proposed. The first one is based on the sequential construction of the curve’s interior by intersection a positive area of half-planes, which enumeration is performed by De Castiljo algorithm. This approach is limited by the convexity of the curve’s reference polygon. This problem’s solution has been considered. The second approach is based on the application of a two-dimensional function for local zeroing (FLOZ), i.e., a nil segment on the positive area of function values. By consecutive unification of such segments it is proposed to construct the required parametrically given curve. Some features related to operation and realization of the proposed approaches have been described and illustrated in detail. The advantages and disadvantages of described approaches have been highlighted. Assumptions about applicability of proposed algorithms for Bezier curve functional-voxel modeling in solving of various geometric modeling problems have been made.