A Procedure for Generating Floor Plans - Computer Aided Design

PROPERTIES OF SPATIAL LAYOUTS

1. Polyominoes
2. Dimensioned map.

1. Polyominoes
figure 1 Polyominoes are patterns mate by number of square equal-sized cells. At least one edge of each cell must be shares with another. i.e., for each cell at least one relationship A is required.

The relationship B, shown, will not be accepted as a connector between two cells. The count of possible arrangements of cells in a polyomino depends on the number of individual cells and their identity.

A) If cells are not labeled and all are identical the number of patterns which can be created from n cells is given by: The different possibilities for n=1 to 4 is illustrated in Fig. 2.


B) If the cells are labeled then the order of their placing can vary and produce n!/2 different possibilities. For example for 3 elements there are three different orders( Fig.3).

C) For each arrangement there are n(n-1)1/2 links (or distances). In the three element case these are AB, AC, BC. For the four elements case ABCD we have six AB, AC, AD, BC, BD, CD.

Summing up points A to C we see that even if we deal only with 3 elements we have the following number of possible distance links. For four elements it becomes 360 and so forth.

When building a polyomino by adding one cell each time an elimination procedure can be applied rather than keeping all results for further growth considerations. After each step only one pattern will be kept for further growth. We have to be able to determine which alternative pattern to take at any given point.
A pattern created from the set of required distances will depend on the individual distances and their distribution.
If we take the four elements case. and we define the value of the distance as the length of path connecting the centroids of two cells the following distances exist: (Fig. 4 - On left)


2. Dimensioned Map

The dimensional map is a pattern of which each unit can be of different proportions or size. If each of these units is broken into smaller elements so that all the units are a multiple (which can vary for each one) of the same square element, the dimensioned map, too, can be regarded as a polyomino with the same growth rules. If we take two units A and B Fig. 5), A is a 2 x 3 element and B is a 1 x 2 element we find that we actually deal with a polyomino of 8 cells put in two "groups" and the requirement for their growth is that at least one polyomino of the added group share an edge with (i.e., be contiguous to) a polyomino in the other group (a) More edges can also be shared (b). But, two groups sharing only a corner cannot connect together in an allowed growth pattern(c). It becomes evident from Fig. 5, that in the procedure of constructing a map, a far larger number of alternatives exist, depending largely on the size of elements under consideration.

When the two elements are one 2 x 3 and one 1 x 2 there are 7 different and distinct ways of putting them together in allowed growth patterns (fig. 6).
If the elements are 1 x 2 and 1 x 1 the number of distinct alternatives is reduced to 2.



This distance between two such elements is defined here in the conventional way as the length of path connecting their centroids.

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