Hausdorff distance

Consider, for example, the question What is the shape of the most chiral right triangle in R2 The answer to this question depends on the particular function that is chosen as the chirality measure According to measures based on geometric chirality products,151 on symmetry coordinates,152 on common volumes,153 and on Hausdorff distances,131 154 the smallest internal angles are 18.8°, 30.0°, 37.5°, and 35.2°, respectively.147 The function x(0) = sin 40 mentioned earlier achieves its maximum [ (0) = 1] at 0 = 22.5°. Because there are as many answers to this question as there are functions that can measure the chirality of a triangle, and because there is, in principle, no limit to the number of such functions, the question remains, in a deep sense, unanswerable. 

Electron density and related fuzzy Hausdorff distance problems " ... 

A detailed review of the basic concepts of fuzzy sets can be found in other chapters of this volume. Here only the specific notations and the fuzzy set concepts most relevant to the molecular shape problem are reviewed, followed by a simple proof for a special fuzzy set generalization of the Hausdorff distance, motivated by the quantum chemical properties of fuzzy electronic densities of molecules. 

For the description of shape differences between fuzzy objects, such as molecular electron density clouds, it is useful to generalize the Hausdorff metric for fuzzy sets. The ordinary Hausdorff distance, a formal dis-... 

For a formal definition of the Hausdorff distance, first we shall review some relevant concepts. We assume that A and B are subsets of a set X, and for points of W a distance function is already defined. For example, if X is the ordinary, three-dimensional Euclidean space and if the points a and b of A are represented by their three Cartesian coordinates [, 2) 3) and respectively, where a and b can be written as... 

After these preparations, the Hausdorff distance h A,B) between two subsets A and B of is defined as... 

The Hausdorff distance h A,B) is a proper metric within any family of compact sets, for example, / (/ , B) is zero if and only if the two sets are the same. 

The Hausdorff distance was applied in chemistry in various chirality studies. The Hausdorff distance was used to measure the deviation of a chiral nuclear arrangement from some arbitrary reference arrangement, as proposed by Rassat. Mislow and co-workers used the Hausdorff distance between the object and its optimally overlapping mirror image to provide a chirality measure of the second type. - Using this Hausdorff distance criterion, Buda and Mislow determined the most chiral constrained and unconstrained simplexes in two and three dimensions, that is, the most chiral triangles and tetrahedra." ... 

For all fuzzy sets, including three-dimensional functions of electron density-like continua provided with suitable membership functions, the differences between the corresponding fuzzy sets can be expressed by a metric based on a generalization of the Hausdorff distance. The basic idea is to take the ordinary Hausdorff distances h a) for the a-cuts of the fuzzy sets for all relevant a values, scale the Hausdorff distance h)a according to the a value, and from the family of the scaled Hausdorff distances, the supremum determines the fuzzy metric distance f A,B) between the fuzzy sets A and B. If, in addition, the relative positions of the fuzzy sets A and B are allowed to change, then the infimum of the f(A, B) values obtained for the various positionings determines a fuzzy metric of the dissimilarities of the intrinsic shapes of the two fuzzy sets. 

For two fuzzy sets A and B, consider their a-cuts G (a) and Ggia), respectively, for each membership function value a. The ordinary Hausdorff distances for each pair of a-cuts are denoted by... 

A function g(A, B), equivalent to the fuzzy Hausdorff distance suggested by Puri and Ralescu, is defined as... 

The set fi(G )(a),Gg(a)) in definition (22) of g A,B) contains values of ordinary Hausdorff distances that are nonnegative. Consequently, the supremum over this set is also nonnegative. 

Take three fuzzy sets A, B, and C and their a-cuts G (a), Gg(a), and G ia), respectively, for each membership function value a. Assume that the a-cuts GJ.a), Gg(a and G(-(a) depend at least piecewise continuously on the a parameter from the unit interval [0,1], where the intervals of continuity have nonzero lengths and where continuity is understood within the metric topology of the underlying space X. For the three pairs formed from these three fuzzy sets, the ordinary Hausdorff distances h(G (a),Gg(a)), h(Gg(a), Gcia)), and h GJ,a ... 

The ordinary Hausdorff distance for each set of a-cuts as fulfills the triangle inequality... 

As can be easily proven by a simple modification of the proof presented here for the unsealed fuzzy Hausdorff metric, this scaled fuzzy Hausdorff distance is also a metric in the space of fuzzy subsets of the underlying set X. A proof is given in subsequent text. 

If the fi A, B) supremum in definition (43) is zero, this implies that each a-scaled ordinary Hausdorff distance ahiG ia XGgia )) of a-cuts in the set ah(G (a),GBia)) is zero for any a > 0,... 

The Hausdorff chirality measure is a chirality measure of the second class [Buda and Mislow, 1992]. Let Q and Q denote two enantiomorphous, nonempty, and bounded sets of points defined in the geometrical space (x,y,z). Let d(q,q ) denotes the distance between two points G Q and qi G Q. Then, the Hausdorff distance h between sets Q and Q is defined as... 

This means that the Hausdorff distance between two sets of points, Q and Q, representing geometric obj ects, can be zero only if these two obj ects are identical, that is, achiral mirror images. 

The value of the Hausdorff distance between a geometric object Q and its mirror image Q depends not only on the shape of these objects but also on their size and their relative... 

Electron density contour surfaces have the mathematical property of compactness, a generalization of the properties of closed and bounded . The Hausdorff distance h(A, B) between two (compact) subsets A and B of X is defined as the lowest upper bound h(A, B) = sup g gg(rf(a, B), d b, A) of distances between points a of A and the set B and distances between points b of B and the set A. In particular, the Hausdorff distance between two superimposed molecular contour surfaces (which are closed sets) is the minimum r value such that any point on either contour surface has at least one point of the other contour surface within a distance r. 

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