Constructing a G2-Smooth Compound Curve Based on Cubic Bezier Segments

Abstract

The theory and practice of forming composite G2-smooth (two-continuously differentiable) curves, used in technical design since the mid-60s of the 20th century, is still not reflected in any way in the curriculum of technical universities or in Russian textbooks in engineering and computer graphics. Meanwhile, such curves are used in modeling a wide variety of geometric objects and physical processes. The article deals with the problem of constructing a composite G2-smooth curve passing through given points and touching at these points pre-specified straight lines. To solve the problem, cubic Bezier segments are used. The main problem in constructing a smooth compound curve is to ensure the continuity of curvature at the joints of the segments. The article shows that for parametrized cubic curves, this problem is reduced to solving a quadratic equation. A software module has been compiled that allows one to construct a plane G2-smooth curve passing through predetermined points and tangent at these points with predetermined straight lines. The shape of the curve (“completeness” of its segments) is adjusted by the user in the dialog mode of the program module. Solved the problem of constructing a cubic curve smoothly connecting unconnected Bezier segments. An algorithm for constructing a Bezier segment with given tangents and given curvature at its boundary points is proposed. Some properties of the cubic Bezier segment are considered. In particular, it was shown that for the case of parallel tangents, the curvature at the end of a segment is determined by the position of only one control point (Theorem 1). Cases are considered when the curvature at the ends of the Bezier segment is equal to zero (Theorem 2). An approximation of a three-point physical spline is performed using Bezier segments. The approximation error was less than 2%, which is comparable to the error in processing the experimental data. A method is proposed for modeling a spatial G2-smooth curve passing through points set in advance in space and touching at these points arbitrarily oriented lines in space. The article is of an educational nature and is intended for an in-depth study of the basics of computational geometry and computer graphics.