Analytic Representation of Three-dimensional Space

Abstract

In this chapter we shall discuss an exciting recent development in the representation of objects in three-dimensional space. We take a totally different approach to the definition of a scene: instead of approximating surfaces with a polygonal mesh, we define them as combinations of primitive objects. Each primitive object is mathematically defined in terms of an analytic function: we have already introduced this idea in the analytic representation of surfaces in chapter 6. This approach allows a very simple definition of many scenes, but the ease of definition has to be paid for with a large increase in processing overheads, although not necessarily in program complexity. To illustrate these ideas we look at two implementations. The first, the quad-tree (Sidhu and Boute, 1972; Tanimoto, 1977; Hunter and Steiglitz, 1979; Woodwark, 1984), will be used to draw simple molecular models composed of a grouping of spheres of arbitrary radius and position. A program, using listings 1.1, 1.3, 3.3,7.1 and 8.1 linked to the draw-a_picture function of listing 17.1 is used to illustrate it. Apart from the #included file ‘matrix3.c’, also required are transform (listing 7.2), find Q, look 3 (listing 8.1), insource (listing 15.1), findlogicalcolour and colourtable (listing 15.10), and cshade (listing 15.9). Secondly there is the oct-tree method (Clark, 1976; Meagher, 1982): we do not give a program but outline the method and also describe the construction of a binary tree defining a scene as the union, intersection and complement of various primitives. (Such a description can also be used with ray-tracing and the quad-tree method.)