Numerical shape codes

We have seen that a simple list of Betti numbers of the shape groups can serve as a numerical shape code for a partitioned molecular surface. Some of the alternative topological shape de.scriptors of molecular surfaces, such as the shape matrices s(a,b) and shape graphs g(a,b), can also serve as 3D topological shape codes 143,109,110,158,199]. In Chapter 6, several examples of shape codes are described and used as numerical shape similarity measures. 

The shape types Tj are usually specified by various algebraic methods, for example, by a shape group or a shape matrix, or by some other algebraic or numerical means. The algebraic invariants or the elements of the matrices are numbers, and these numbers form a shape code. The (P,W)-shape similarity technique provides a nonvisual, algebraic, algorithmic shape description in terms of numerical shape codes, suitable for automatic, computer characterization and comparison of shapes and for the numerical evaluation of 3D shape similarity. 

For either code, c(a,b) or c (a,b), the corresponding number can be assigned to the (a,b) location of the parameter map (a,b). Since most small changes in the values of a and b do not change the shape groups of the actual truncated molecular surfaces, the entire (a,b) map will contain only a finite number of different values for the c(a,b) or c (a,b) code. A list of these code values can be regarded as a vector, providing a numerical shape code for the entire (a,b) map (i.e., for all relevant electron density values a and test curvature values b). 

Subsequently, these topological methods have been adopted and modified to the significantly simpler, three-dimensional molecular shape problem, where the shape of the molecule is the quantum mechanical shape of the electron density cloud [13-19], This has led to the development of the shape group methods, where the ranks of homology groups describing relative convexity domains of the complete set of all isodensity surfaces of the molecule, the so-called Shape Group Betti numbers, provided a detailed, numerical shape code for the quantum chemical electron density [13-19]. 

The values of the Betti numbers at the grid points (a,b), or at the points [log(u), log 6 ] of the logarithmic map, form a matrix, M1" 1 1. In either of the direct or the logarithmic representations, this matrix is a numerical shape code for the fuzzy electronic density cloud of the molecule, representing the actual molecular shape. 

The nonvisual shape similarity measures of molecules as well as molecular fragments, using the numerical shape code method, provide the basis for a shape complementarity measure. A simple transformation of the local shape codes generates a representation that is suitable for a direct evaluation of local shape complementarity. 

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