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Concept
Euclidean distance
Summary
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and 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 distance between any two points on the real line is the absolute value of the numerical difference of their coordinates, their absolute difference. Thus if and are two points on the real line, then the distance between them is given by:
A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:
In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value.
In the Euclidean plane, let point have Cartesian coordinates and let point have coordinates . Then the distance between and is given by:
This can be seen by applying the Pythagorean theorem to a right triangle with horizontal and vertical sides, having the line segment from to as its hypotenuse. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.
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