Metric . In mathematics , algebra , geometry and more specifically, Euclidean geometry , coordinate geometry , topology is a magnitude and law which determines the lower level of difference or distance between two objects in a space or geometry given, usually considered points .
By default, it refers to the Euclidean metric of Cartesian geometry and on which the rest of its results have been elaborated. But in cases where there are other types of geometries, the formulation and meaning of distance varies to adapt to that situation. For example, although the Euclidean distance based on the idea that the smallest distance between two points is the length of the line segment that joins them, in the case of spherical geometry the smallest distance would be the length of the arc of maximum circumference u orthodromic connecting the points. In other spaces, it can be other types of curves.
As you can see, the metric concept itself adjusts to each situation, even creating abstract cases of geometries and metric spaces . Although in any case, they must satisfy a series of properties that determine the scope of the “distance” concept.
Summary
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- 1 Definitions
- 2 Examples
- 1 Euclidean metrics
- 2 Binary metrics
- 3 Spherical metric
- 3 Importance
- 4 See also
- 5 Sources
Definitions
Any binary function d (x, y) is called metric or distance , where , where A is any non-empty set and satisfies each of the following properties:
- Non-negativity: d (x, y) ≥0 .
- Symmetry: d (x, y) = d (y, x) .
- Reflexivity: d (x, x) = 0 .
- Triangular inequality: d (x, y) + d (y, z) ≥d (x, z) .
- Reflexivity implication: If d (x, y) = 0 , then x = y .
Examples
There are several formulations of the distance that is applied in different cases. Here are some of them.
Euclidean metric
Main article “Euclidean metrics” .
Let a Cartesian space of N dimensions be the distance between two of its points A , B is defined by the length of the straight line segment that joins them or what is the same:
that for the cases of the three-dimensional space and the plane are represented respectively by:
The demonstration of the five properties inherent in metrics are evident.
Binary metric
Main article “Binary metrics” .
It is defined by binary distance to the function d (x, y) defined according to:
This metric is often used in logic , topology, and artificial intelligence .
Spherical metric
Main article “Spherical metrics” .
The spherical metric is a form of distance that assumes that points A and B are arranged on the surface of a solid sphere centered in O of radius “r” so the distance in question between both points would be the length of the orthodromic or arc of concentric circumference at O and radius r containing at each of its extremes A and B :
Numerically speaking it would be:
where the alphas represent the lengths of their respective points and the betas are the latitudes, exactly the same as on our planet.
In the case, for example, of sea distances, since at sea level the extra in altitude that is added by elevations to distance calculations is evaded, this form of measurement is very useful, because as we know the Earth has a shape spherical and it would therefore be misleading to believe that one travels in a straight line when one is actually sailing on an arc of circumference.
Importance
Metrics are a defining element of geometries, topological spaces and other mathematical entities. But they are also of vital importance in other areas since the distance assigns to the difference between objects a numerical value that allows quantifying and comparing this relationship in the form “how different is object A from B”.
This feature is vital for example in word processing, where it is often necessary to note the syntactic distance between words for their grouping into word families, such as the verb forms of a verb, or also in spelling correction because when it is detected an error is usually made by contrasting the word with others in a language dictionary and if the minimum separation range exceeds a certain value it is declared as an error and possible suggestions for substitutes are proposed. In the ways of calculating the distances between words, the distances of Levenshtein , Oliver or those of phonetic pronunciation can be cited , such as those of Donald Knuth or Lawrence Philips .
It is of great importance in Artificial Intelligence in the areas of pattern recognition , data mining , clustering , information retrieval , non-binary logic , etc.
This extends to the Chemistry softwares where the structural differences between molecules are reviewed, which translates into more or less desirable differences in properties and behavior, as the case may be.
