Dallas, August 18-22 Volume 20, Number 4, 1986 Free-Form Deformation of Solid Geometric Models Thomas W. Sederberg Scott R. Parry t Brigham Young University Provo, Utah 84602 ABSTRACT A technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, for example. The deformation can be applied either globally or locally. Local deforma- tions can be imposed with any desired degree of deriva- tive continuity. It is also possible to deform a solid model in such a way that its volume is preserved. The scheme is based on trivariate Bernstein polyno- mials, and provides the designer with an intuitive appre- ciation for its effects. CR Categories and Subject Descriptors: 1.3.5 [Com- puter Graphics]: Computation Geometry and Object Modeling- Curve, surface, solid, and object representa- tions; Hierarchy and geometric transformations. KEYWORDS: Solid geometric modeling, free-form surfaces, deformations. 1. INTRODUCTION The fields of solid modeling and surface modeling have been developin$ rather independently over the past fifteen years [Requicha '82], [Varady '84]. Surface model- ing has dealt primarily with parametric surface patches. These patches are generally referred to as free-form sur- faces, or sculptured surfaces, which suggest that they can be shaped with flexibility akin to clay in a sculptor's hands. For this reason, planes, quadrics and tori are gen- erally not considered to be free-form. Most solid model- ing systems use surfaces that are planar, quadric or Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1986 ACM0-89791-196-2/86/008/0151 $00.75 toroidal. Recently, the capability of defining fillets and blended surfaces has also been introduced [Hoffmann '85t, [Middleditch '85], {Rockwood '86]. The problem of defining a solid geometric model of an object bounded by free-form surfaces has long been identified as an important research problem. Most of the approaches to this problem can be classified into one of three categories: 1. Combining existing free-form surface and solid modeling techniques. This extends the surface domain of a solid modeling system to include free- form parametric surface patches. It is currently the most popular approach and some applications can be found in [Kalay '82], [Jared '84], [Chiyokura '83I, [Varady '841, [Riesenfeld '83], [Sarraga '84l, [Steinberg '84], [Thomas '84], and [Kimura '84]. This method must overcome several difficulties such as ensuring representational validity in using the free-form surfaces in a general manner. These problems are described in [Requicha '82]. 2. Trivariate parametric hyperpatch. The hyper- patch is used as a solid modeling primitive. This method has been used for years by the analysis community and has many applications such as finite element mesh generation [Stanton '771, [Casale '85]. [Farouki '85] discusses adding a fourth parameter of time to create a time-space swath useful for motion definition. 3. Implicit surfaces. There has been limited inves- tigation of modeling directly with volumes bounded by implicit or algebraic surfaces. Calcu- lating curves of surface intersection and deciding whether a point lies inside a volume is easier with this definition, especially when the surfaces are of low degree. However, free-form shape definition lends itself more naturally to parametric equations ~Current address: Milliken & Co. LaGrange, GA. 30241 151S I G G R A P H '86 |11 I III than to implicit equations. Sabin was one of the early investigators of modeling with algebraic sur- faces [Sabin '68]. [Ricci '73], [Burr '81], [Rockwood '86], [Owen '86], [Hoffmann '85] and [Blinn '82] explore modeling implicit surfaces other than qua- drics. [Sederberg '85] discusses modeling with piecewise algebraic surface patches. This paper presents an approach to free-form solid modeling which does not fall cleanly into any of the above three categories, although it developed most directly out of the ideas in [Sederberg '85]. This technique is referred to as free-form deformation or FFD, and can be thought of as a method for sculpturing solid models. It is shown to be of value both as a design method, and as a representation for free-form solids. Indeed, the sculptur- ing metaphor is stronger for solids than for surfaces because a lump of clay or a block of marble is a solid. Several researchers have promoted this sculpturing metaphor for geometric modeling, noting that it is a natural and familiar mode of thought for a designer or stylist. For example, [Parent '77] discusses a "computer graphics sculptor's studio" for defining polygonal objects, and [Brewer '77] describes a planar shaping tool for mani- pulating sculptured surfaces. Other "lump-of-clay" modeling techniques are surveyed in [Cobb '84]. None of these sculpturing techniques are directly applicable to solid geometric modeling. Parent's paper deals with polygonal data, and Brewer's work deals with a class of parametric surface patches. FFD involves a mapping from R s to R s through a trivariate tensor product Bernstein polynomial. An ear- lier use of R s to R s mapping is found in Barr's innovative paper on regular deformations of solids [Burr '84 I. While not a free-form modeling technique, Barr's idea of twist- ing, bending and tapering of solid primitives is a powerful and elegant design tool. Brief mention of deformation is Mso
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