P-Bézier and P-Bspline curves – new representations with proximity control

Abstract

Proximity curves represent a family of curves that are associated with a given parametric curve, defined by control points and basis functions. Proximity curves continuously sweep from this curve to its control polygon depending on a proximity value, that determines the location of an intermediate curve and its fullness (or tension). The proximity value also determines sensitivity, i.e. how strongly the shape is affected by displacing control points. An important feature of proximity curves relates to the insertion of new control points: while in other schemes a new degree of freedom leads to repositioning the existing control points, in our case the new control points are always placed on some chord of the control polygon.Our first proximity curve scheme – called P-curves – has been published recently (Kovács and Várady, 2017), having C∞ continuity and G1 endpoint interpolation. The basis functions were constructed by means of generalized barycentric coordinates, and a somewhat limited algorithm for control point insertion was proposed.Our current paper takes a different approach: the basis functions are calculated by a much simpler algebra that is capable to reproduce standard formulations like Bézier and B-splines curves, and can maintain Cn end constraints. We introduce the Proximity-Bézier, shortly P-Bézier, and P-Bspline curves, including the construction of basis functions and the most important mathematical properties. A general control point insertion algorithm is also described. Several examples are shown to compare the classical and the new representations. A tensor product generalization of the scheme is also demonstrated.