Shape recognition in three dimensions

The subshape recognition problem for three-dimensional shapes under linear transformations is considered. The problem is analysed in a series of cases, some that provide a determinate number of solutions and others that have indeterminately many solutions. Procedures for its solution for general shapes are developed. Difficulties posed by strict adherence to rational transformations are examined. As a corollary, an outline of a procedure for determining the symmetries of a shape is presented. Introduction A central computational problem associated with shape grammar theory is discussed, namely, under which transformations is one shape a subshape of another? Further, the question is examined whether these transformations—if they exist—between shapes that can be rationally described also can be described by rational coefficients. In other words, can the subshape recognition problem for rational shapes always be resolved using exact arithmetic? The importance of the subshape problem can be gauged from the fact that its resolution is a prerequisite for the composition of shapes by the application of spatial rules. A spatial relation (a, ft) between shapes a and ft can be considered as a spatial rule that applies to a shape y if we can find a similarity (in general, a linear transformation) x such that x(a) occurs as a shape in y in which case x(fi) replaces x(a) in y under rule application. The subshape recognition problem in the form considered here has direct application to shape grammars (Stiny, 1980a; 1990), and to the ways in which spatial relations between shapes can be used to specify shape rules (Earl, 1986; Earl and Krishnamurti, 1984; Stiny, 1980b). Although acquaintance with shape grammar concepts would be an advantage, it is not crucial. The properties of shapes and definitions of spatial terms necessary for the arguments in this paper will be developed as needed. The treatment for two-dimensional shapes given in Krishnamurti (1980; 1981) is the basis for implementations of shape generation systems (Chase, 1989; Krishnamurti, 1982; Krishnamurti and Giraud, 1986). A discussion of threedimensional shapes is presented in Earl (1986), and an independent approach both for two-dimensional and for three-dimensional shapes is given in Stiny (forthcoming). This paper completes the details of the three-dimensional case. The following ideas are basic. A shape is a finite set of maximal straight lines of finite, nonzero length, where each line is specified by the coordinates of its endpoints. A shape is rational if its lines have endpoints given by rational coordinates, and real otherwise. The lines in a shape are maximal in the sense that colinear lines are separated by a gap, but 586 R Krishnamurti, C F Earl otherwise lines may touch or intersect. A subshape of a shape is a shape the lines of which each have endpoints that are coincident with a line in the original shape. The verification of the subshape relation for any two shapes s and s' can be approached as follows. Let s denote a shape and the set of maximal lines that describes it. Shape s may be structured by an equivalence relation that partitions the set into equivalence classes of colinear maximal lines. That is, each line / is associated with an unambiguous descriptor, say co(l), such that colinear lines have identical descriptors. An equivalence class of colinear lines can be organised by the order relation, <, extended to n-tuples as follows. Let A and B denote two arbitrary n-tuples of numbers, A = (al9 a2,..., an) and B = (bl9 b2,..., bn). Then A < B, if for some j £ {1,2,..., n}, ak = bk,k < j , and af < bj. Equality between A and B holds whenever the components of A and B are identical. The endpoints of a line segment / can be ordered such that one is designated as the tail(l) and the other as the head(l) where tail(l) < head(l). A line / may be thus described by the triple (co(l), tail(l), head(l)). A set of colinear maximal lines can be arranged as a sequence of lines in (increasing) order of head or tail values. A further structuring of a shape can be carried out. Suppose that the descriptor function co is expressed as a tuple of numbers, then the equivalence classes of colinear lines can be arranged as a sequence in order of their descriptor values. Two shapes can then be checked for equality, denoted s = s, by comparing the sequences in their description term by term. Observe that by arranging the description of a shape in the manner suggested above, shape equality can be tested for in a time linear in the number of lines in shapes s or s'. In a similar fashion, the subshape relation s < s' can be verified. Here the description of s must be wholely contained within the description of s. That is, for any two colinear lines, / = (co(l), tail{l), head{l)) in s, and /' = (co{l), tail{l'\ head(l')) in s', I is contained in /' if and only if: co(l) = co(l'), tail{l) > tail(l'), and head(l) < head{l'). The subshape relation can be decided in a time linear in the number of lines in s or s'. Line geometry in three dimensions Points are described in homogeneous coordinates {x,y, z, w) in order to facilitate the description of points with rational coordinates as 4-tuples of integers. For real shapes we take w = 1. For rational shapes, any three-dimensional point (x,y,z) is expressed as a 4-tuple of relatively prime integers (xw, yw, zw, w) for some integer w chosen such that the integers are relatively prime. A simple way to do this is to set w = w \a, where w is the product of the denominators of the nonzero x, y, and z, and a = gcd{xw', yw', zw', w'), the greatest common divisor. We will assume that all coordinates are given in their (in the case of rational shapes, to their reduced relatively prime) homogeneous coordinate form. Points can be defined by vectors (which are indicated by boldface). A point p with coordinates {x,y,z, w) can be represented as p = {x,y,z,w). The operations of vector arithmetic, scalar, and cross products are defined for any two vectors P = (xp,yP,Zp,Wp) and q = (xq,yq, zq, wq) as: Xp = (Xxp,Xyp,Xzp,wp), (1) p±q = {xpwq ±xqwp, ypwq ±yqwp, zpwq ±zqwp, wpwq), (2) Shape recognition in three dimensions 587