Exploration of describing the vector-parametric bi-spline, defined by the cubic spline with control points incident with surface of appropriate smoothness

Abstract

Explorations carried on within the framework of geometrical simulations are aimed to develop already existing techniques describing spline surface, since under certain circumstances it appears to be hard to construct even outlines applying available methodologies. The proposed technique is based on the principle that control points belong to the curve under consideration.

Basing on the preceding researches the article proposes a technique of bispline configuration as a vectorial-parametrical surface with control points incidental to the relevant curve applying the third degree splines meeting the evenness criteria within the first and the second degrees. To achieve this purpose the vectorial-parametrical spline r = r(u) is extended (or, so to say, pulled out) in the direction identified with v, i. e. in a direction, other than u thus enabling to develop the appropriate surface “portions”. Further, to obtain bispline with appropriate evenness adherence of appropriate surface portions is required preserving the appropriate evenness along the adhering line, obtaining thus the equity of appropriate derivatives (both the first and the second). However, to maintain total evenness of the second degree, i. e. to preserve the continuity of the second quadric form within the entire form it is yet necessary to meet the compound derivatives equity criterion. The test examples of the bicubic splines are included into the work.

The benefit of this research is to develop new, more convenient method that gives the developer more flexible and convenience in its operation that was described above.