The arithmetic of shapes

Algorithms for the Boolean operations and relations on shapes and labelled shapes are presented. This paper has two parts. The first part accomplishes two objectives. First, an efficient and uniform representation for shapes is presented, which is based on a linear order on the maximal lines of a shape. Second, simple and efficient algorithms for the Boolean operations on shapes (shape union, difference, and intersection) and the Boolean relations on shapes (subshape and shape equality) are presented. The second part deals with the computational aspects involved in performing the Boolean operations and relations on shapes and labelled shapes. An algorithm for the efficient decomposition of the Boolean operations and relations is presented. Last, the data structures required to implement the shape algorithms are described. The relevant definitions and notations upon which this paper is based are given in Stiny (1980). Each shape is assumed to be initially described by a set of maximal lines; each labelled shape is given by a shape and an associated set of labelled points. Rational shapes A restriction is introduced to limit the class of shapes dealt with in this paper. This restriction—albeit a practical one—is necessitated by the fact that algorithms are defined with respect to some form of computing machine. In a random access machine with limited memory a real number is represented by a finite approximation which is determined by the word size of memory. This makes for inexact arithmetic. Moreover, it is usual for shapes to be drawn on some kind of graphics device such as a visual display unit or a digital plotter. On these devices only a limited number of points can be addressed, and the location of each point is given by an integral multiple or a pair of integral multiples of a unit of measurement. Since correct algorithms require exact arithmetic it is convenient to consider just those shapes which can be so described. Therefore, attention will be restricted to shapes which are, in the mathematical sense, rational. The following definition makes the notion of a 'rational shape' precise. Definition: A point p is rational if and only if each of its coordinates x^p),..., xd(p), d > 2, can be expressed as the ratio of two integers. A labelled point p:A is rational if and only if p is rational. A line /, I = {pi, Pi), is rational if and only if its end points, p1 and p2, are rational. A shape s is rational if and only if each of its maximal lines is rational. A labelled shape o, o = (s, p)9 is rational if and only if s is rational and every labelled point in point set P is rational. The ratio of two integers, rn/rd, may be expressed as the ordered pair, <rn, r&), which, in turn, may be described by its unique primitive form. A pair of integers, (rn, r&) is primitive if and only if the following conditions are satisfied: (a) rn and rd are integers,