Construction of a spatial curve composed of rational cubic segments

The paper describes a rational cubic segment by a formula similar to the Bezier formula, but, unlike this formula, the coordinates of the vertices of the characteristic broken line of a rational cubic segment are supplemented with weighting coefficients. The paper considers two problems. The first task is to construct a composite G2-smooth curve passing through points given in three-dimensional space and touching pre-specified directing lines at these points. Smoothness G2 means a continuous change in the curvature vector along the constructed curve (without jumps in magnitude and direction). The second task is to insert a cubic segment into the gap between two fixed spatial cubic curves. To solve the problems, constructive step-by-step graphic-analytical algorithms have been developed. A distinctive feature of the proposed algorithms is the significant use of tools and methods of three-dimensional computer graphics. The paper proposes to solve the first problem sequentially: we add a second segment to the first segment, providing a common tangent and a common curvature vector of the connected segments at the joint point. We add the next segment to the second segment, also ensuring that the tangents and curvature vectors at the joint point coincide. To solve the second problem, it is necessary to find the characteristic broken line of the inserted segment based on the condition of coincidence of the touching planes at the joint points. The segment weights are calculated taking into account the G2 smoothness requirement. The examples of solving problems 1 and 2 presented in the paper are transparent and can be used in the educational process.