P-curves and surfaces: Parametric design with global fullness control

Abstract

A new curve representation (P-curves) that is well-suited for computer-aided geometric design is proposed. While several properties of Bézier and B-spline curves are inherited, new useful features have also been introduced. A P-curve is defined by a control polygon; it forms a singleC∞ continuous segment with endpoint interpolation. The new basis functions have been inspired by the Mean Value generalized barycentric coordinates. P-curves actually represent a family of curves with a continuously changing fullness parameter that determines the proximity between the curve and its control polygon. It is fairly straightforward to increase the degree of design freedom of P-curves, as the new control point will always be inserted on a selected chord retaining both the full control polygon and the shape of the curve.In this paper, we describe the construction of P-curves and prove their basic mathematical properties. Several examples will be shown to compare P-curves with Bézier and B-spline curves. The adjustment of the fullness parameter will also be demonstrated. The new basis functions can also be used to define tensor product P-surfaces with a global control to loosely or tightly approximate the control grid.