Improvement of free-form surfaces for product styling applications

Abstract

There are a number of ways of describing free-form surfaces within geometric modelling systems. For product styling and related activities, there is a need to ensure that surface quality is good and that patches join together smoothly. Additional parameters can be introduced to allow surfaces to be modified. This raises the question of whether these can be chosen automatically, and this in turn requires measures of what is a ‘fair’ surface. Measures based on surface curvature are discussed and applied to adjust surface patches presented in terms of point meshes.

Introduction

Conventional computer aided design (CAD) systems are good at modelling standard engineering components. The geometry of these often consists of standard primitive shapes such as straight lines, circular arcs, and surfaces of revolution. These have simple definitions which can be processed easily by computer. However, CAD systems have limited benefit when trying to model products which are composed of free-form surfaces. This is because the definition of what is required is less precise. The designer is usually looking to improve the aesthetics of a product, and it is difficult to quantify this for computer processing [1], [2], [3]. To obtain good results, the designer needs to be provided with specialised manipulations of the underlying geometry. Ideally, this should not require of the designer a deep understanding of the modelling process used. While the definitions of standard geometric entities are well known, what makes a good object from an aesthetic point of view is less well understood. Some measures of the fairness of a surface have been proposed [4], [5], [6], [7], [8], [9], particularly for work in surface fitting. These are often based upon analogies with either internal strain energy or reflection of light.

One approach to the creation of free-form components is that of Gregory [10]. Here, an object is initially defined in terms of a number of curves in space. These could define certain aspects of the required style of the final object. The use of curves simplifies the designer's task as he/she now works in terms of one-dimensional objects (embedded into a three-dimensional world space). Additionally, the processes for creating curves and smoothing them are better understood and are, perhaps, more intuitive. The styling curves are assumed to intersect and in so doing they define a number of faces. The basic Gregory technique allows these to be filled by free-form surface patches which join together to make an overall surface which is tangent plane continuous. The faces which appear between the curves most often have four sides, but this is not always the case. Other numbers can occur, particularly in regions where an object becomes reentrant. This is another limitation of the usual CAD approach which is often restricted to four-sided patches.

There are drawbacks to the basic Gregory approach. The surface created is uniquely defined by the styling curves, and so there is no opportunity for a designer to refine the result. The lack of higher order continuity between surface patches can sometimes spoil the overall aesthetics. Recent work has shown how to increase the degree of continuity to second order in some cases [11] and how to introduce additional parameters which can be used to perform small modifications to surface patches without affecting their basic continuity with neighbouring patches [12], [13].

However, the manipulation of these modifying parameters is not something which a designer can undertake easily without a detailed knowledge of the mathematics underlying the modelling approach. This leads to the question of whether they can be chosen automatically through partial interaction with the designer. In turn, this requires a good measure of the fairness of surface patches.

In this paper, we begin by reviewing the Gregory technique as a means for creating free-form surface patches. The method for allowing modifications is described. This introduces a large number of additional variables which need to be selected, and we look at possible measures of surface quality based upon curvature. The application of such a measure is demonstrated working with a surface in terms of a mesh of points lying within it. This allows manipulation of the patch directly without the need to deal with the mathematical representation. If required, such a representation can subsequently be fitted to the revised points. Some examples of the adjustment scheme are given.