Extension And Approximate Merging Of B-spline

Changing shape of curves and surfaces and approximate merging two adjacent curves and surfaces are both hot topics in CAGD.The paper mainly focuses on extension and approximate merging of the cubic non-uniform B-spline curves. The conclusion is described as follows:The cubic non-uniform B-spline curve is extended and a class of non-uniform polynomial blending function of degree 4 is created. We get a new piecewise polynomial curve with several local shape parameters. By changing the value of the shape parameters the different polynomial curves about the same control polygon are generated and the curves can approximate the cubic non-uniform B-spline curves from their both sides. The generated curves can be G continuous and have the same geometric constraction and properties with the cubic non-uniform B-spline curves. The changing of the shape parameters will only influence the two adjacent curve segment.The author takes advantage of the above basis functions with local shape parameters to propose an algorithm about reverse calculating the control points of cubic B-spline curves. The algorithm does not need border condition afforded by user. So it simplifies the process of reverse calculating.For cubic uniform B-spline curves, an algorithm of the curves with given tangent polygons is described. The all control points can be generated simply by the vertices of the given tangent polygons. Because of the importing of shape parameter λ and tangent control parameter λ₁ , we can change the location in which the curve is tangent to the give polygon and adjust the approaching degree of the curve to its tangent polygon at the same time.By the means of minimizing L₂ norm distance function and the matrix representation of subdivided cubic non-uniform B-spline curves, the author proposes a new approach of approximate merging two adjacent cubic non-uniform B-spline curves to a single one. We get the explicit formulas of control points of the merged curve which are expressed by matrix form. In this paper, the author gives some examples to illustrate the effect of the method. We can easily see the merged curve approximates the original curves well.