MODELING OF SPATIAL ISOTROPIC BÉZIER CURVES ON THE BASIS OF PYTHAGOREAN-HODOGRAPH CURVES

Abstract

Minimal surfaces are essential for geometric modeling, computer science, and for various branches of engineering, and the determination of the conditions for their construction has been studied in many works and includes a large number of methods. However, this topic has not yet been exhausted when using isotropic curves to construct minimal surfaces. These studies focus on methods for determining zero-length curves. For interactive control of such curves, it is advisable to use curves constructed on the basis of characteristic polygons, therefore, studies are based on Bézier curves.

The paper highlights the experience of previous studies in the field of constructing spatial isotropic curves and plane Pythagorean-Hodograph curves. The modeling of spatial Pythagorean-Hodograph curves is based on the theory of quaternions. The authors of the paper present a different approach, namely, the construction of an isotropic spatial curve based on a plain Bézier Pythagorean-Hodograph curve. The transition to the spatial curve is carried out on the basis of determining the third coordinate from the condition that the length is equal to zero. In this case, a plain curve is built on the basis of real values, and the third coordinate will be purely imaginary.

The authors carry out studies based on fifth-order curves. In this case, it is advisable to choose quadratic functions as basic polynomials. Three options are proposed for setting the initial values ​​of the coordinates for determining the Pythagorean-Hodograph curve. The problem of the boundary points of the curve and the determination of the intermediate points of the Bezier curve based on the given dependences have the most practical importance. To prove the reliability of the proposed provisions, the calculation of the points of the curve is performed and examples of simulated curves are given. Further research involves the use of the constructed curves for modeling minimal surfaces and isotropic portions of Bézier.

Key words: isotropic curve, Bézier curve, Pythagorean-Hodograph curve.