Hausdorff metric
The Hausdorff metric , named after the mathematician Felix Hausdorff , measures the distance between non-empty compact subsets , a metric space .
Clearly, two compact subsets have a small Hausdorff distance if for each element of one set there is an element of the other set to which it has a small distance.
definition
As an aid, one defines the distance between a point and a non-empty compact subset with recourse to the metric of the space as
Then one defines the Hausdorff distance between two non-empty compact subsets and as
One can show that there is indeed a metric on the set of all compact subsets of .
Equivalently, the Hausdorff distance can be defined as
- ,
in which
- ,
this is the set of all points with a distance of at most from the set .
Applications
In the theory of iterated function systems , fractals are generated as sequence limit values in the sense of the Hausdorff metric.
See also
literature
- MI Voitsekhovskii: Hausdorff metric . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Individual evidence
- ↑ James Munkres: Topology . Prentice Hall, 1999, ISBN 0-13-181629-2 , pp. 280-281 ( google.com ).