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Pasting Spline Surfaces

Cristin Barghiel, Richard Bartels and David Forsey

Abstract. Details can be added to spline surfaces by applying displacement maps. However, the computational intensity of displacement mapping prevents its use for interactive design. In this paper we explore a form of simulated displacement that can be used for interactive design by providing a preview of true displacements at low computational cost.

xIntroduction

Displacement mapping is a standard and useful tool for realistic rendering [3,8], and it has been suggested for application to surface approximation [6]. Because of its computational cost, however, it has not yet been fully explored as a design tool. In [4] a restricted form of displacement mapping was investigated for use in interactive design. Composite surfaces were constructed by layering a sequence of detail surfaces onto a base surface. Each detail was formulated as a vector displacement fleld and mapped onto a region of the base. The process was restricted in that, in order to achieve continuity between detail and base, it was necessary that the detail share a common parametric domain with the base and possess a knot structure derivable from the base by reflnement, a setting studied recently by Weller and Hagen [9]. These restrictions permitted the computations for displacement mapping to be simplifled to such an extent that interactive design was feasible. Here we extend the techniques of [4] to remove some of those restrictions.

True displacement mapping involves a vector-valued function d(r; s) that is nonzero over a compact domain (r; s) 2 D, a 1-1 transformation T of that domain into the domain (u; v) 2 S of a surface (point-valued) S0(u; v), and the resulting composition S1(u; v) = S0(u; v) + d(T?1(u; v)). (Bold, lower case letters denote vectors; bold upper case letters are transformations; Roman and Greek lower case letters are scalars; Roman upper case letters are points, and calligraphic letters are domains.) The continuity of S1 is controlled by the continuity of S0, T and d. In the surface design setting that we wish to consider, the intent of employing a displacement map is to achieve the pasting" of a detail surface D(r; s) onto a base surface S0(u; v). The function d is

Mathematical Methods in CAGD III 1 M. D?hlen, T. Lyche, and L. L. Schumaker (eds.), pp. 1{3.