The construction of shapes

An algorithm for shape rule application is presented. A shape rule a -> |3 applies to a labelled shape 7 whenever there is a transformation r that makes ot a subshape of 7. In this case, a new labelled shape can be obtained by replacing the occurrence of 7(a) in 7 with r(|3). The algorithm required for this process is developed in this paper. This algorithm determines all possible distinct transformations under which a given shape rule applies to a given labelled shape and the corresponding labelled shapes resulting from such applications. The definitions and notations given for labelled shapes and shape grammars by Stiny (1980) are used. Euclidean transformations The transformations of translation, rotation, reflection, scale, and finite compositions of these are referred to as the euclidean transformations, which are hereinafter referred to simply as transformations, denoted by r. A transformation 7 can be expressed, in two dimensions, as a mapping r: R -> R, where R is the set of reals. Furthermore, 7 can be composed as the ordered pair of mappings, r = (TX, ry), where rx and Ty each take the form r z : R 2 > R , z {ax -by, ay + bx), where a = cos0 and b = sin0. (3) Reflection [about (a) the X-axis, and (b) the F-axis]: (a) (x, y) -• (x, -y\ (b) (x, y) -* (-x, y). (4) Scale [or change of size through a scale factor, c > 0] : (x, y) -t (ex, cy). A transformation consisting of a finite sequence of transformations is a composition. The sequence (r l 9 ..., 7/, ..., r„), where each 77, 1 < / < n, is one of the above listed transformations, denotes the composed transformation r1(r2(...(7/(...(rw( ))...))...)). Therefore, in general, a transformation r can be expressed as a mapping of the form T: (X, y) -• (axx-\-bxy-\-cx, ayx + byy + cy), which is clearly linear. The transformation r of a labelled shape 0,0= (s, P), is the labelled shape denoted by 7(a), 7(a) = (T(S), T(P)), which is obtained by changing the spatial disposition and/or size of a. More precisely, 7(a) is defined as follows. Let p,