Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Distance is a numerical description of how far apart objects are In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry By using this formula as distance, Euclidean space becomes a metric space (even a Hilbert space). In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined This article assumes some familiarity with Analytic geometry and the concept of a limit. Older literature refers to this metric as Pythagorean metric. The technique has been rediscovered numerous times throughout history, as it is a logical extension of the Pythagorean theorem.

1 Definition
2 One-dimensional distance
3 Two-dimensional distance
4 Three-dimensional distance
5 See also

Definition

The Euclidean distance between points $P=(p_1,p_2,\dots,p_n)\,$ and $Q=(q_1,q_2,\dots,q_n)\,$ , in Euclidean n-space, is defined as:

$\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + \cdots + (p_n-q_n)^2} = \sqrt{\sum_{i=1}^n (p_i-q_i)^2}.$

One-dimensional distance

For two 1D points, $P=(p_x)\,$ and $Q=(q_x)\,$ , the distance is computed as:

$\sqrt{(p_x-q_x)^2} = | p_x-q_x |$

The absolute value signs are used since distance is normally considered to be an unsigned scalar value.

Two-dimensional distance

For two 2D points, $P=(p_x,p_y)\,$ and $Q=(q_x,q_y)\,$ , the distance is computed as:

$\sqrt{(p_x-q_x)^2 + (p_y-q_y)^2}$

Alternatively, expressed in circular coordinates (also known as polar coordinates), using $P=(r_1, \theta_1)\,$ and $Q=(r_2, \theta_2)\,$ , the distance can be computed as:

$\sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2)}$

Three-dimensional distance

For two 3D points, $P=(p_x,p_y,p_z)\,$ and $Q=(q_x,q_y,q_z)\,$ , the distance is computed as

$\sqrt{(p_x-q_x)^2 + (p_y-q_y)^2+(p_z-q_z)^2}.$

Dictionary

-noun

(geometry) The distance between two points defined as the square root of the sum of the squares of the differences between the corresponding coordinates of the points; for example, in two-dimensional Euclidean geometry, the Euclidean distance between two points a = (a_x, a_y) and b = (b_x, b_y) is defined as:

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