In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space (or even any inner product space) becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric.
Despite the development of numerous types of geometry, Euclidean distance is still by far the most geometrical tool of applied scientists. For example, statisticians find it useful to calculate the proximity between samples.
Application in Co-ordinate Geometry:
In two-dimension:
Euclidean Distance= sqrt [ (x1−x2)^{2}+(y1−y2)^{2} ]
Geometrically, it's the length of the segment joining u and v.
In three-dimension:
In three-dimensional Euclidean space, the distance is
Euclidean Distance= sqrt [ (x1−x2)^{2}+(y1−y2)^{2} + (z1-z2)^{2} ]
In N dimensions:
In general, for an n-dimensional space, the distance is
Euclidean Distance= sqrt [ (a1−b1)^{2}+(a2−b2)^{2} + (a1-b2)^{2} +….+ (an-bn)^{2}]
Using the Euclidean distances, we can create what is known as the Euclidean Distance Matrix. A Euclidean distance matrix is a matrix (two-dimensional array) containing the Euclidean distances, taken pair-wise, between a set of points. This matrix will have a size of N×N where N is the number of objects.This matrix is symmetric in nature – that is Xi,j = Xj,I.
Example:
The Euclidean distance matrix would be:
a | b | c | d | e | f | |
a | 0 | 184 | 222 | 177 | 216 | 231 |
b | 184 | 0 | 45 | 123 | 128 | 200 |
c | 222 | 45 | 0 | 129 | 121 | 203 |
d | 177 | 123 | 129 | 0 | 46 | 83 |
e | 216 | 128 | 121 | 46 | 0 | 83 |
f | 231 | 200 | 203 | 83 | 83 | 0 |
Posted by
Saurabh Agarwal
Finance
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