Chinese Journal of Computers   Full Text
Title	Quadratic B-Spline Interpolation Curves with Tangent Constraints on Data Points
Authors	PAN Ri-Jing
Address	(College of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007)
Year	2007
Issue	No.12(2132—2141)
Abstract &   Background	Abstract In this paper a new interpolation method for quadratic B-spline curves is proposed so as to fully utilize the degrees of freedom provided by parameterization and knot vector to control the shapes of the interpolation curves intuitively by the tangent constraints on data points. Without solving any equation systems, the interpolation procedure of the method is a recursive one in which the parameter values at data points, the knots and the control points are determined recursively according to the data points and the tangent constraints on data points. With the method the connection points of adjacent curve segments are not necessarily coincident with data points, so that the shapes of quadratic B-spline interpolation curves are more natural. Furthermore, under the restriction of the tangent constraints, there are still some degrees of freedom in constructing interpolation curves by the method: the shapes of the interpolation curves can be further adjusted by the selection of knots. Besides, the quadratic B-spline interpolation curves constructed by the method possess rather good local properties for the relieving disturbances on data points. Some examples are given to compare the method proposed in the paper with several other interpolation methods. The experimental results show that this method is effective. keywords B-spline curve； interpolation； parameterization； knot vector； tangent constraint background This paper explores the interpolation problem of B-spline curves that is extensively applied in the fields of geometric modeling, computer aided geometric design and reverse engineering etc. Though B-spline interpolation problem has been widely investigated, it remains to be further explored that how to fully utilize the degrees of freedom provided by knot vector and parameterization on data points to control the shapes of the interpolation curves effectively. In this paper, a new interpolation method is proposed to construct quadratic B-spline interpolation curves that satisfy the tangent constraints on data points. The method fully utilizes the degrees of freedom provided by knot vector and parameterization on data points to control the shapes of interpolation curves intuitively. This research is a part of the project—Applications of transformation matrices of B-spline bases in curve and surface modeling—Supported by Nature Science Foundation of Fujian province under granted No.A0610007 and the project—On several key techniques for the fitting of B-spline curves and surfaces—supported by Science Foundation of Education department of Fujian Province under grant No.JA05207. The former project probes into the representations and applications of transformation matrices of B-spline bases in curve and surface modeling, including the interpolation and approximation, data reduction and multiresolution analysis for B-spline curves and surfaces. The latter project probes into several key problems in the fitting of B-spline curves and surfaces, including parameterization, knot vector, object function, and error estimation etc. The interpolation problem of B-spline curves is a common basic aspect for two projects. The research groups of the projects have obtained some results on the representation of transformation matrices of B-spline bases and on the fitting of B-spline curves and surfaces, including a method for constructing quadratic B-spline interpolation curves by dynamic parameterization. Based on these results, this paper provides a further result on B-spline interpolation curves.