Shape Group Analysis of Surfaces and Related Techniques

Shape Group Analysis of Surfaces and Related Techniques 

The key notion in shape group analysis is evaluation of topological invariants not only for the original surface, but also for a family of surfaces derived by using geometrical properties of the initial surface. Conceptually, this is the same approach discussed in the preceding section for the derivation of a family of knot polynomials from a given molecular space curve. 

Consider, for instance, a molecular isodensity surface G(a, K) defined for a nuclear configuration K and a value a of the electron density function p(r, K)  

Except for some in special cases, the canonical curvatures are finite properties that can be computed everywhere on the surface. Suppose we now group all the points on G a,K) that satisfy a curvature criterion, say those for which the two curvatures /7j(r) and h2 i) are negative. Such a domain on the surface can be indicated as Dc(a, K), where C indicates the criterion followed for classification. With the criterion of two negative curvatures, Dda, K) [e.g., Diia, K)] corresponds to the regions on the surface that are locally convex. [Note that Dcia, K) can be empty or composed by several disjoint pieces.] Finally, if we now remove (i.e., cut away) this region, we derive a truncated surface from the original one  

A number of topological invariants can be used to describe the truncated surfaces Gci , The Betti numbers Bp, which are the ranks or number 

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