Add content for distance between points

Author: Flavio Poletti

docs/distance-between-points.md

@@ -4,4 +4,143 @@ title: Distance between points
 sidebar_label: Distance between points
 ---
 
-[Open a pull request](https://github.com/AllAlgorithms/algorithms/tree/master/docs/distance-between-points.md) to add the content for this algorithm.
+The **distance between two points** can have several different
+definitions depending on the specific situation, although without any
+further specification it usually refers to the length of the segment
+connecting the two points (also called [*Euclidean distance*][euclidean]).
+
+## Euclidean Distance
+
+The [Euclidean distance][euclidean] is generally defined as the lenght
+of the segment connecting two points.
+
+<img
+   src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Euclidean_distance_2d.svg/500px-Euclidean_distance_2d.svg.png"
+   alt="Euclidean distance">
+
+### Formula
+
+The [Euclidean distance][euclidean] can be easily calculated when the
+coordinates of the two points are known. For example, in the XY plane,
+the formulas would be the following:
+
+<img
+   src="https://latex.codecogs.com/gif.latex?P%20%3D%20%28P_x%2C%20P_y%29"
+   alt="P = (P_x, P_y)">
+
+<img
+   src="https://latex.codecogs.com/gif.latex?Q%20%3D%20%28Q_x%2C%20Q_y%29"
+   alt="Q = (Q_x, Q_y)">
+
+<img
+   src="https://latex.codecogs.com/gif.latex?d%28P%2C%20Q%29%20%3D%20%5Csqrt%7B%28P_x%20-%20Q_x%29%5E2%20&plus;%20%28P_y%20-%20Q_y%29%5E2%7D"
+   alt="d(P, Q) = \sqrt{(P_x - Q_x)^2 + (P_y - Q_y)^2}">
+
+The formula above can be generalized for vector spaces of any dimension,
+by taking the square root of the sum of the squares of differences in
+each dimension:
+
+<img
+   src="https://latex.codecogs.com/gif.latex?P%20%3D%20%28P_1%2C%20P_2%2C%20...%2C%20P_n%29"
+   alt="P = (P_1, P_2, ..., P_n)">
+
+<img
+   src="https://latex.codecogs.com/gif.latex?Q%20%3D%20%28Q_1%2C%20Q_2%2C%20...%2C%20Q_n%29"
+   alt="Q = (Q_1, Q_2, ..., Q_n)">
+
+<img
+   src="https://latex.codecogs.com/gif.latex?d%28P%2C%20Q%29%20%3D%20%5Csqrt%7B%28P_1%20-%20Q_1%29%5E2%20&plus;%20%28P_2%20-%20Q_2%29%5E2%20&plus;%20...%20&plus;%20%28P_n%20-%20Q_n%29%5E2%7D"
+   alt="d(P, Q) = \sqrt{(P_1 - Q_1)^2 + (P_2 - Q_2)^2 + ... + (P_n - Q_n)^2}">
+
+<img
+   src="https://latex.codecogs.com/gif.latex?d%28P%2C%20Q%29%20%3D%20%5Csqrt%7B%5Csum_%7Bi%3D1%7D%5En%20%28P_i%20-%20Q_i%29%5E2%7D"
+   alt="d(P, Q) = \sqrt{\sum_{i=1}^n (P_i - Q_i)^2}">
+
+### Algorithm
+
+The algorithm for the [Euclidean distance][euclidean] can be derived
+directly from the defining formula above:
+
+```
+/* Assume that P and Q are N-dimensional arrays */
+distance(P, Q) {
+   N = dim(P)
+   square_sum = 0.0                              /* initialize running sum */
+   for dimension i from 1 to N {                 /* iterate over all dimensions */
+      square_sum = square_sum + (P[i] - Q[i])^2  /* sum squares of differences */
+   }
+   distance = sqrt(square_sum);                  /* distance is the square root of the sum of squared differences */
+   return distance;
+}
+```
+
+## Manhattan Distance
+
+Among the many possible definitions of distance, another one commonly
+found in computer science is the [Manhattan distance][manhattan]. This
+is typically a distance used when movements are constrained to happen
+over a grid:
+
+<img
+   src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/500px-Manhattan_distance.svg.png"
+   alt="Manhattan distance">
+
+### Formula
+
+The formula below applies when calculating the [Manhattan
+distance][manhattan] over an N-dimensional grid where adjacent nodes are
+assumed to have distance 1:
+
+<img
+   src="https://latex.codecogs.com/gif.latex?P%20%3D%20%28P_1%2C%20P_2%2C%20...%2C%20P_n%29"
+   alt="P = (P_1, P_2, ..., P_n)">
+
+<img
+   src="https://latex.codecogs.com/gif.latex?Q%20%3D%20%28Q_1%2C%20Q_2%2C%20...%2C%20Q_n%29"
+   alt="Q = (Q_1, Q_2, ..., Q_n)">
+
+<img
+   src="https://latex.codecogs.com/gif.latex?d%28P%2C%20Q%29%20%3D%20%5Csum_%7Bi%3D1%7D%5En%20%7CP_i%20-%20Q_i%7C"
+   alt="d(P, Q) = \sum_{i=1}^n |P_i - Q_i|">
+
+### Algorithm
+
+The algorithm can be implemented as a direct derivation of the formula:
+
+```
+/* Assume that P and Q are N-dimensional arrays */
+manhattan_distance(P, Q) {
+   N = dim(P)
+   distance = 0.0                             /* initialize running sum */
+   for dimension i from 1 to N {              /* iterate over all dimensions */
+      distance = distance + abs(P[i] - Q[i])  /* sum squares of differences */
+   }
+   return distance;
+}
+```
+
+## Performance
+
+Both algorithms for each respective distance definition are linear
+(O(n)) with respect to the number of dimensions with which each point is
+represented.
+
+
+## Implementations
+
+| | Language | Link |
+|:-: | :-: | :-: |
+| | | |
+
+## Helpful Links
+
+- [Euclidean distance][euclidean]
+- [Manhattan distance][manhattan]
+- [Norm][norm] a generalization of the concept of distance
+- [L<sup>p</sup> space][lp-space] a generalization of a class of norms
+
+
+[euclidean]: https://en.wikipedia.org/wiki/Euclidean_distance
+[manhattan]: https://en.wikipedia.org/wiki/Taxicab_geometry
+[norm]: https://en.wikipedia.org/wiki/Norm_(mathematics)
+[lp-space]: https://en.wikipedia.org/wiki/Lp_space