A vertex-first parametric algorithm for polyhedron blending

Abstract

This paper enhances the conventional parametric algorithms for polyhedron blending, by strategically inverting the edges-first approach to vertex-first, so that matching the vertex blending surface (using a triangular or tensor product Bézier surface, or an $S$-patch) with the edge blending surfaces (generated by Hartmann method) becomes essentially easier. Based on a study of cross boundary derivatives (those of $S$-patches are deduced herein), $G^g$-continuity between all the above surfaces and the primary planar faces is achieved by a novel trick as a first step: assigning the vertex, some edge points and some face points to be the proper control points. This still leaves enough free parameters usable for changing the blending configuration. The new algorithm is illustrated with two practical examples involving miscellaneous vertices up to 6-edge convex–concave.

Introduction

Blending is widely used in geometric modeling for manufacturing, strength, aesthetic and usage purposes. A lot of blending is applied to smooth the vertices and edges of a polyhedral model.

Early in the late 1970s, Doo and Sabin [1] (also Catmull and Clark at the same time [2]) tackled this problem. Being the classical papers on surface subdivision [3], [4], both focused on the polyhedron blending. Their solutions were expressed in recursive formulas for subdivision, rather than blending surface equations. Another numerical method is described by a rolling ball: Choi and Ju [5] put forth the constant-radius rolling ball method, whereas Chuang and Hwang [6] presented an approach of varying the radius of rolling ball (VRRB). Farouki and Sverrisson [7] blended the free-form parametric surfaces with prescribed-precision approximation of constant-radius. Later, researchers aimed to find the exact analytical solutions. According to the blending surface equation forms, the relevant publications can be classified into implicit and parametric. In implicit forms, Braid described various blending for the boundaries of models [8]. Hartmann carried out $G^n$-blending for vertices by functional splines [9]. Mou et al. presented $G^2$-blending for convex polyhedra by algebraic splines and provided a free parameter to control the shape of the blending surface [10]. Kosters made use of the quadratic surfaces to blend complex corners [11].

In general, parametric forms of surfaces are more convenient to use. For example, it is trivial to determine points, curves and trimmed patches on a parametric surface. In parametric forms, Szilvasi–Nagy presented a flexible local blending operation for polyhedra [12]. Varady and Rockwood produced vertex blending with elegant setbacks [13]. However, both algorithms blended the edges before the vertex. As a result, smoothly connecting the vertex blending surface with the edge blending surfaces as well as the primary planar faces was quite complicated, since a vertex collects several edges. Great efforts have been made to attain $G^1$-continuity. At last, they often turned to splitting the vertex blending surface.

Obviously, adopting an inversed, vertex-first strategy will greatly facilitate the matching. In particular, Hartmann already provided an efficient way [14] to blend the edges if the vertices have been blended. When $m$, number of edges meeting at a vertex, equals 3, 4 or more, TB (triangular Bézier) surfaces, TPB (tensor product Bézier) surfaces and regular $S$-patches would be the most promising candidates for the respective vertex blending surfaces. Nevertheless, there still exist large gaps between the candidates and the goal surfaces.

To bridge the gaps, after investigating the schematic control points related to the cross boundary derivatives, by designating the vertex, some edge points and some face points to be the proper control points, we were pleased to turn out a required vertex blending surface, keeping $G^g$-continuity with the primary planar faces. The correspondence of the control points in schema to the positional control points (not always one to one but sometimes several to one) is the crux of our approach (look ahead at Fig. 6, Fig. 8, Fig. 10, Fig. 11).

In what follows, Section 2 reviews some useful properties of Bézier surfaces and regular $S$-patches. The latter cross boundary derivatives are derived in Section 3 as a preparation for later sections. In Section 4, we construct the vertex blending surface and discuss the influence of their control points on the blending. In Section 5, the edge blending surfaces are generated as the convex combination of two reparameterized local base patches on the constructed vertex blending surfaces. The influence of the trimmed patches on the edge blending surface is also explained. Section 6 gives two practical examples–a die and a monument–to demonstrate the application. Finally, Section 7 concludes the paper.