Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance.
These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid’s Elements, distances were not represented as numbers but line segments of the same length, which were considered “equal”. The notion of distance is inherent in the compass tool used to draw a circle, whose points all have the same distance from a common center point. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century.
The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.
The Euclidean metric, or Euclidean distance, is the “straight-line” distance between two points in Euclidean space, calculated using the Pythagorean theorem. To find the distance between two points, you square the differences of their corresponding coordinates, sum these squares, and then take the square root of the total sum. This generalizes the familiar distance in a 2D plane to n-dimensional spaces.
Formula
In two dimensions (for points $((x_{1},y_{1})) and ((x_{2},y_{2}))):(d=sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}})$
In n dimensions (for points $((x_{1},x_{2},…,x_{n})) and ((y_{1},y_{2},…,y_{n}))):(d=sqrt{(y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}+…+(y_{n}-x_{n})^{2}})$
Key characteristics
- Symmetry: The distance from point A to point B is the same as from B to A.
- Positivity: The distance is zero only if the two points are identical, and positive for any two distinct points.
- Triangle inequality: The distance between two points is always less than or equal to the distance traveled by going through a third point.
- Pythagorean theorem: The metric is a direct generalization of the Pythagorean theorem, which forms the basis for its calculation.
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[1] https://mathworld.wolfram.com/EuclideanMetric.html
[3] wikipedia/en/Euclidean_distance
[6] https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch7.pdf
[7] https://study.com/academy/lesson/euclidean-distance-calculation-formula-examples.html