BEZIER CURVES WITH POWER PARAMETERIZATION

Abstract

The modern development of various equipment is characterized by the widespread use of computer information technologies during its design, manufacture and operation. These tools for industrial production are usually integrated CAD/CAM/CAE/PLM (Computer-Aided Design/Computer-Aided Manufacturing/Computer-Aided Engineering/Product Life-cycle Management) systems. In domestic practice, they correspond to the abbreviation CAD, that is, computer-aided design systems. One of their fundamental bases is geometric modeling. Currently, the most widely used mathematical apparatus for computer modeling is the form of NURBS (Non-Uniform Rational B-Splines), i.e. non-uniform rational B-splines. They are the basis for the further formation of surfaces, and then solid figures. The listed geometric objects ensure the proper design of parts, assembly units, and the development of technologies for their manufacture and operation. Therefore, the improvement of NURBS lines is an actual scientific and applied problem. Bezier curves are a separate type of NURBS lines that have certain advantages and disadvantages, in other words, their own sphere of rational use. This publication proposes a power-law parameterization of these curves that differs from the traditional linear parameterization. The article shows that in this case, not only the certain useful properties of Bezier lines are preserved, but also new ones are obtained, appropriate for the computer-aided design of many industrial products, in particular, in the field of mechanical engineering. This applies to the construction of compound contours of the first and second order of smoothness, that is, by tangent and curvature, for example, the convenience of including straight line segments in them. Mathematical processing of outlined geometric figures requires the solution of relevant scientific problems, which is the subject of further research in the field of computer geometric modeling and automated design of various industrial products. Also important is the task of generalizing the obtained theoretical results, their thorough practical verification, and implementation in real production.