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Interior filling animals

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. 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 family F(J,n) of all interior filling animals Aj(J,n) of the given Jordan curve J,... [Pg.151]

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. [Pg.151]

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,... [Pg.152]

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. [Pg.152]

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. [Pg.151]


See other pages where Interior filling animals is mentioned: [Pg.15]    [Pg.151]    [Pg.152]    [Pg.156]    [Pg.158]    [Pg.15]    [Pg.151]    [Pg.152]    [Pg.156]    [Pg.158]    [Pg.90]    [Pg.598]    [Pg.1193]    [Pg.144]    [Pg.173]    [Pg.577]    [Pg.127]   
See also in sourсe #XX -- [ Pg.15 , Pg.149 , Pg.151 ]




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