Jordan curve

Note the analogy between G, S and G, T. By the same argument, G has also some edges lying on the boundary of H. Thus we can be sure that any closed Jordan curve in the plane separating G and G (which are both connected) must traverse the external region of H and, therefore, intersect exactly two edges which lie on the boundary of H. 

Figure 6.3 Some of the interior filling animals of three Jordan curves, J], J2, and J3, enclosing three planar domains, D], D2, and D3, of different shapes. For cell numbers 1 and 2 the the resolution is not sufficient to distinguish these curves. For cell number n=3 and for any higher cell number, the third curve, J3, has interior filling animals different from those of curves J and J2, but curves J 1 and J2 are distinguishable only for cell number n=8 and beyond. Accordingly, the greatest degree of similarity is found between curves J] and J2 (and the respective domains D] and D2), in agreement with expectation based on visual inspection.

The smaller the squares of the grid, the better the resolution of the representation of D by the animals. By approximately filling up the interior D of J by animals at various levels of resolution, a shape characterization of the continuous Jordan curve J can be obtained by the shape characterization of animals. The animals contain a finite number of square cells, consequently, their shape characterization can be accomplished using the methods of discrete mathematics. As a result, one obtains an approximate, discrete characterization of the shape of the Jordan curve (i.e., the shape of a continuum). The level of resolution can be represented indirectly, by the number of cells of the animals. In particular, one can show [240,243] that the number of cells required to distinguish between two Jordan curves provides a numerical measure of their similarity. 

If one chooses a small enough size s for the length of the side of the square cells, then any finite animal can fit within the given planar domain D. Whether an animal A fits within the interior D of a given Jordan curve J depends on the relative size of J and the cells of the animal. For a given Jordan curve J and a given cell size s there exists a countable family F(J,s) of animals which fit within domain D. Clearly, if the size s is too large, then this family is empty. With reference to J and s, the members A (J,s) of this family F(J,s) are called the inscribed animals of D [240,243]. 

The family F(J,n) of all interior filling animals Aj(J,n) of the given Jordan curve J,... 

In Figure 6.3, three Jordan curves, J], J2, and J3, are shown, with some of their interior filling animals. At both levels n=l and n=2 there is only one interior filling animal, common to all three curves. Hence, at these levels of resolution the shapes of Ji, J2, and J3 appear the same. At level n=3, however. 

The above example illustrates the motivation for the choice of a similarity index io(Ji.J2) of two Jordan curves Jj and J2, defined as the smallest nc value at and above which all interior filling animals of Jordan curves J j and J2 are different, that is,... 

If the shapes of the two domains enclosed by the Jordan curves J and J2 are identical (i.e., if they can be obtained from one another by scaling), then no finite nc value exists and the similarity index io(Ji,J2) = °°- For curves J and J2 of different shapes, the more similar their shapes, the greater the cell number n of the largest common interior filling animals. Consequently, the similarity index io(J >J2) is a large number if the two Jordan curves J and J2 are very similar, and io(Jl>J2) is a small number for highly dissimilar curves. 

The smallest cell number n at which there exist different animals is three, hence the smallest possible value for io(Ji,J2) >s also 3 that justifies the inclusion of the number 2 in the denominator. As a consequence of this definition, the degree of dissimilarity d(Ji,J2) may take values from the [0,1] interval, where greater values indicate greater dissimilarity. For two Jordan curves Jj and J2 of identical shapes d(J],J2) = 0. 

If two Jordan curves Jj and J2 have identical shapes, then their degree of similarity s(J],J2)=l, otherwise s(Ji,J2) is a smaller positive number. 

By analogy with the perimeters of animals and Jordan curves, the surface G(P) of a polycube P is the point set union of all those faces of the cubes C of P that are on precisely one cube. The surfaces of polycubes are used to approximate MIDCO surfaces G(a), and to characterize the shapes of the formal molecular bodies B(a) enclosed by them. 

Since the smallest chiral lattice animals have four cells [54], the minimum possible value for chirality index is n (J)=4. The degree of chirality X(J) of a Jordan curve J is defined as... 

G. If (G , G ) should not contain any edge on a perimeter of G, we could add as many vertices to S as possible such that the added vertices would have the same color as the vertices of S, and the enlarged subset, say S, would still satisfy that (S U N(S )) and G - (S U N(S )> are connected. Now we find that the Jordan curve J can only be a triangle as shown in the below diagram. 

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