P-spline curves

In this paper, we introduce a new parametric spline curve, named as P-spline curves. Given a control point set associated with a set of knots and parameters, we first define three sets of points using this set of knots and parameters. One set of points lie on a line segment spaced by knots, and two other sets of points lie on two sides of the line segment symmetrically according to the parameters and knots. Then, we compute the mean value coordinates of points on the middle line segment with respect to the two involved quadrilaterals, whose vertices are selected from the three defined point sets. Last, we use these coordinates and blending functions to construct basis functions, which are used to define P-spline curves together with given control point set. There are several desirable features for the P-spline curves. The continuous orders of the resulting curves are determined by the basis functions, and we can adjust the distance between the curves and control points by changing the parameters. Moreover, the construction of P-spline curves is simple, and the relations between the P-spline curves and knots/parameters are intuitive. More importantly, the influences of parameters and control points are local because the four vertices of each quadrilateral only depend on one parameter and three knots. Some numerical examples are used to show that P-spline curves are more local than NURBS (non-uniform rational B-spline) curves, P-curves and P-Bspline curves.