COMPLETELY SMOOTH APPROXIMATION OF AN ARBITRARY SET OF BEZIER CURVES. PART 2: SOLUTION METHOD AND EXAMPLES

Problems of smooth joining of various types of curves are an integral part of the geometric kernels of modern CAD systems. We consider the problem of approximating an initial arbitrary set of Bezier curves without restrictions on their number, degrees and points of tangency with each other by a set of Bezier curves for which the conditions of completely smooth conjugation are satisfied at the conjugation points (that is, the order of continuity at the conjugation points corresponds to the maximum degree of the original Bezier curves). The construction of an approximating set of curves is reduced to an optimization problem under constraints in the form of equalities, which is solved by the Lagrange multiplier method. The transition is made from a set of fully conjugate Bezier curves to a single Bezier curve of the same degree. In the work, Bezier curves are presented as a special case of a B-spline, which allows you to obtain all derivatives without additional calculations when calculating a point on the curve Additional restrictions can be imposed on the approximating curve in the form of invariance of the starting and ending points and the values of the derivatives at these points. The proposed methods are applicable to 2D and 3D curves.The results of the work have been implemented into the CAD module of the “Sarus” PLM.