Computational methods for evaluating swept object boundaries Laxmi Parida* and S.P. Mudur National Centre for Software Technology, Gulmohar Cross Road No. 9, Juhu, Bombay 400 049, India * e-mail: laxmi @ saathi.ncst.ernet.in Many systems support the design of two- dimensional (2D) regions and three-di- mensional (3D) volumes by sweeping a generator contour over a spine. Well-es- tablished and widely used rendering sys- tems accept high-level boundary descrip- tions of regions and solids but not in the contour-spine form. Hence the swept ob- ject design specification must be such as to lend itself to efficient evaluation of a compact and high-level boundary descrip- tion. In this paper, we identify a sufficiently general class of swept objects and classify the sweep rules. We also present computa- tional methods for directly evaluating the boundary representation which implicitly simulate the sweep process. The boundary is generated as a set of piecewise curves/ surfaces in a compact form suitable for di- rect input to rendering systems like Post- Script(R) in 2D and Renderman(R) in 3D. Key words: Swept objects - Sweep rules - Boundary evaluation - Volume model- ing - Region modeling - Body modeling - Pen-stroke modelling Correspondence to: L. Parida 1 Introduction Pen-stroke modeling in two dimensions (2D) (Knuth 1979; Ghosh and Mudur 1984) and body modeling in three dimensions (3D) (Magnenat- Thalmann and Thalmann 1988) are typical exam- ples of spine-based modeling. Designing such re- gions/volumes as a spine with a generator contour sweeping over it has the advantages of not only providing a more natural way of modeling but also of effectively parameterizing the essential geometric features of the model. For example, various arms may have the same spine and generator contour with different sweeping rules to indicate differing "fatness" and shapes of the arm. With advances in hardware, there are a number of systems which accept high-level geometric de- scriptions and render them directly on paper or on the screen. For 2D we have page description languages which accept filled regions defined by their boundaries. Lines, conics and parametric cub- ics can be used for describing the boundary. For 3D we have workstations with special rendering processors which also accept solids defined by their boundaries and some even support boolean opera- tions. Once again, polygons, quadric surfaces, bi- cubics and non-uniform rational B-splines (NURBS) may be used to describe the boundary surfaces of the solids. For example, PostScript (Adobe Systems 1985) accepts regions bounded by lines and B6zier curves and also strokes of uniform thickness, but not a pen-stroke specification. Simi- larly, Renderman (Upstill 1990) accepts quadrics, polyhedral solids and NURBS but not swept sur- faces or swept solids. It, therefore, becomes impera- tive to convert the designs into boundary descrip- tions that can be accepted by these systems which are becoming de facto standards. Any modeling system based on the sweep object design technique will have to generate a description of the boundary of the solid modeled which can then be passed on to the rendering processor. This paper primarily addresses the problem of computing a compact high-level boundary descrip- tion of the swept object which implicitly simulates the sweep operation on the spine. 2 Other related work The notion of swept objects has been extensively used in the fields of computer-aided design (Ahn et al. 1992; Woodward 1987) and computer vision (Marr 1982), where it goes by the name of general- The VisualComputer(1994) 10:26(~276 266 9 Springer-Verlag 1994