COMPOSITE GEOMETRICALLY SMOOTH CUBIC HERMITE CURVES IN SPACE AND ON THE PLANE

Abstract

The article considers constructive algorithms for creating compound curves using parametric cubic equations. The possibilities of modern computer technology and systems of computer mathematics make it possible to implement algorithms of graph analytics using fractional rational parametric equations of the fifth, seventh and higher orders. The article considers a constructive algorithm for creating a compound cubic curve passing through given points with direction vectors specified in these points. Direction vectors can be used for monitoring the shape of a constructed curve. At the junction points of the cubic curve segments, second order geometric flatness is ensured, due to the equicontinuity of the camber and curvature. The required curve is made sequentially by attaching a new segment to the previous segment. Each segment must comply with the same group of geometric conditions: frequency at boundary points, contact with direction vectors, fixed curvature at one or both boundary points. The article proposes a constructive algorithm for constructing a three-dimensional or flat cubic Hermite segment that satisfies this group of conditions. The problem of constructing a compound geometrically smooth curve boils down to sequential construction of cubic segments using the proposed algorithm. The method proposed in the article for constructing a smooth compound curve allows: to shape a spatial curve with a given curvature at the starting point; to construct a plane curve with given radii of curvature at the nodes (in particular, with zero curvature at the nodal points); insert a cubic segment into a break in a curved line; find a smooth (without a break in a curve) conjugation of curvilinear and rectilinear sections of the constructed line. A distinctive feature of the algorithm is the significant use of reference tools and design capabilities of three-dimensional computer graphics. All necessary actions are performed using a desktop calculator and any software product that supports three-dimensional graphics (Compass-3D, AutoCAD, etc.). Five transparent examples of constructing flat and spatial composite cubic curves of the second order of smoothness are considered.