Both constructive solid geometry and boundary representation techniques are also used outside of the engineering domain, most notably in computer graphics work. The most difficult problems in computer graphics seem to involve the realistic representation of natural objects, rather than man-made ones. This means that new techniques developed for graphics systems, such as texture mapping, have not yet influenced the fields of either CAD or robotics, which deal mainly with man-made objects.
A number of fields make extensive use of computerised shape representation, but are not concerned at all with solid shape, because they are building on commonly used two dimensional abstractions. A typical example is computer assisted mapping, where there is a definite correspondance between the two dimensional map and the three dimensional shape of the land, but mapping conventions allow all infomation about the map to be described in two dimensions only. Such systems may include sophisticated two dimensional shape representation (such as extended polygons, in Geovision's ``AMS'' system), but represent three dimensional shape only in that they allow the draughtsperson to draw contour lines.
Another example of a two dimensional abstraction that has lent itself to computerisation is plane geometry. Gelernter's 1963 geometry-theorem proving machine used descriptions of two dimensional shape that specified a range of possible coordinate values for each point in the shape described, thus allowing the system to test its proofs over a range of cases.
This chapter has covered a variety of methods for representation of, and reasoning about, shape and space in two and three dimensions. Several of the methods discussed here have influenced the development of the qualitative shape representation presented in chapter 4, and they will be referred to again in that chapter. This survey has also given some indication of the limits of spatial reasoning capabilities both in robotics and in other fields - limitations arising from lack of accuracy, ambiguity, or difficulty in constructing the representation either from a vision system, or from a human programmer. Chapter 6 discusses ways in which a qualitative spatial representation can overcome some of these limitations.