Research On Quadratic B-Spline Interpolation Curve With Tangent Constraints

For its convexity,good local property and continuity,B-spline curve has been used and studied widely,but how to control the shape of B-spline interpolation curves efficiently has not been solved perfectly,especially even-degree B-spline curves.Due to the fact that quadratic curves are easy for rendering,subdivision,and intersection computation,most studies about even-degree B-splines focus on quadric B-spline.When data points are given,the shape of quadratic B-spline interpolation curves is determined by parameterization,joint of adjacent curve segments selection and the construction of interpolation curve.Parameterization allots a value to every data point,different parameterization brings different knots vector.A interpolation curve is consist of several curve segments,if the joints of these adjacent segments is put on integer knots,the interpolating points are joint points,otherwise,a interpolating point will belong to a single segment,the typical example is quadratic spline curve,the joint points are put on half knots.Because of having no inflection point,the quadratic B-spline curve will be undesired if it is constructed in the improper way,especially when the data points are uneven or the curvature is big between adjacent points.Therefore,good parameterization,reasonable selection of knots and constructing the interpolation curve in proper way are the key factors to get a satisfied result curve.What's more,putting apart some degree of freedom is one efficient way to control the shape of interpolated curve.Our research focus on the case that knots vector is coincident with parameter values.A new parameterization method and a new structure procedure of interpolation curve are proposed in this thesis.Taking the affinities between the shape of quadratic B-spline curves and the tangents of interpolation points into account,tangent constraints are used in new parameterization.The tangents of data points can be given in applications or estimated by any method.Knots vector and control points are determined recursively with tangent constraints during the interpolation procedure,the geometric characteristics of quadratic B-spline curve are fully utilized and every curve segment has desired shape.Besides adjusting the shape of result curve intuitively by changing tangent vectors,new parameterization method also insure us a small interval between adjacent knots when there is a small curvature between adjacent data points,that is,if the parameterization is looked as a movement of a particle,when the road has a small curvature,the particle will move quickly.When the shape of curve segment proposed by tangents is undesired or there is an inflection point in curve segment,we will depart the segment into 2-4 segments, the result curve is called Complex Quadratic B-spline Curve.All angles of tangent vectors are included in new curve,and degree of freedom is put apart during the procedure of interpolation,at the same time,the general value of freedom degree is given.New curve is convenient to use,and possess good local property.The combination of new parameterization method and new construction method ensure us that the result curve possess given tangent vector at every data point,good robust property.In order to prove the above advantages more perfectly,many experiments are shown in this thesis,contrasts are made between new method and existing methods which is based on tangent constraints,too.