Curve Interpolation By Blended Cubic Rational Bezier Curves

The sparse points interpolation is important research contents in Computer Graphics and Computer Aided Geometric Design. Fair, smooth, G" continuous and locally adjusted curves are required in many engineering applications and aesthetic design environments. In the circular blending method we can not avoid unacceptable cases such as loops, self-intersections and so on. We have done a lot of curve interpolation by quadratic rational Bezier curves.The cubic rational Bezier curves can better handle the turning points. Meanwhile, it is smoother than the rational quadratic Bezier representation with the increase of the degrees. What’s more, it can make full use of the geometric properties of given sparse points. As a result,we consider the curve interpolation by blended cubic rational Bezier curves.In this paper, we mainly propose two algorithms for the sparse points interpolation by cubic rational Bezier curves. In the first algorithm we interpolate four successive points locally. Firstly, we estimate the curvature, then we compute the weights by degree elevation, at last, we will reversely compute the control points, do parameter transformation to make the interval parameter change from zero to one and blend the overlapping parts of the blendees. In the second algorithm we interpolate three successive points locally. Firstly, we will estimate the tangent vectors, then we define rules to find the control points, at last, we will reversely compute the weights, do parameter transformation and blend.In this paper, the concrete calculation formulae and their derivations are given for the two proposed algorithms. The algorithms are compatible with the NURBS curves. Several examples are presented to validate the effectiveness of the proposed algorithms. At last, we compare with the traditional algorithms and point out the advantages and disadvantages of the proposed algorithms.