The Application Of Arc Splines And Construction Of Dual Functionals To B Splines

In this paper,first,we improved the method of interpolation and fairing by arc splines. Then we constructed dual functionals with local supports to B-splines.On one hand,we improved the method of fairing planar points by arc splines.Our method is not to reduce as many inflexions as we can,but to preserve the ones which we need. In order to make all of the radii for arc segments are in the same direction,we keep the same concave-convex feature between the two adjacent inflexions.On the other hand,we modified the inevitable errors which were made in arcs computation.Our method not only keeps smoothness at the joint points,but also uses an easier operator.We proposed a new method to construct dual functionals with local supports to B-splines. For the equidistant univariate B-splines of degree n,first,we choose 2n-1 B-splines in the local supports,then select one knot in each local supports of B-spline.So there are the same numbers of spline knots as that of interpolation knots.Second,we deduced dual functionals from B-spline interpolation,which are a linear combination of function values at the selected interpolation knots.And coefficients of the linear functional are elements of a special matrix which can be easily computed.We also prove the symmetry of these elements and reduce the dual functionals.For the non-uniform univariate B-splines,we used the same method to construct dual functionals.But for unimodular Box splines,we only gave a simple example to illustrate the feasibility of our method to multivariate splines.