Add content for "Area of polygon"

polettix · polettix · commit 765bcb37cecf · 2020-10-04T01:24:54.000+02:00
diff --git a/docs/area-of-polygon.md b/docs/area-of-polygon.md
@@ -4,4 +4,66 @@ title: Area of polygon
 sidebar_label: Area of polygon
 ---
 
-[Open a pull request](https://github.com/AllAlgorithms/algorithms/tree/master/docs/area-of-polygon.md) to add the content for this algorithm.
+A polygon is a closed sequence of segments in the plane. It is often
+convenient to represent it as a sequence of vertices *P<sub>1</sub>,
+P<sub>2</sub>, ..., P<sub>n</sub>*, with the convention that any pair of
+adjacent vertices in the sequence define a segment in the polygon, and
+that the first and last vertices are the same. Each point *P<sub>i</sub>*
+is assumed to have coordinates in the plane *(x<sub>i</sub>,
+y<sub>i</sub>)*.
+
+## Simple polygons
+
+A [simple polygon][] is a non-self-intersecting one, i.e. no pair of
+segments intersects each other.
+
+### Formula
+
+The *signed* area of the polygon is given by the following formula:
+
+<img
+src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d09d3414b2e985a26205bc932010c62d3ffd668d"
+alt="area of a polygon" />
+
+The area is *signed* because going in reverse order means having a result
+with the opposite sign. To get the area it's necessary to take the
+absolute value.
+
+### Algorithm
+
+The following algorithm translates the formula in pseudocode:
+
+```
+/*
+    Vs is an array of vertices. Each vertex is supposed to be
+    an array itself, with X values in index 0 and Y values in
+    index 1. It is also assumed that the last element in Vs is
+    the same as the first one.
+*/
+polygon_area_simple (Vs) {
+    N = Vs.size; # number of elements in Vs
+    sum = 0;
+    for i in 0 .. (N - 1):
+        sum = sum + Vs[i][0] * Vs[i+1][1] - Vs[i+1][0] * Vs[i][1];
+    return abs(sum) / 2;
+}
+```
+
+## Performance
+
+The algorithm for [simple polygons][simple polygon] is linear with respect
+to the number of vertices (O(N)).
+
+## Implementation
+
+| | Language | Link |
+|:-: | :-: | :-: |
+| | | |
+
+## Helpful links
+
+- [Polygon][]
+
+
+[simple polygon]: https://en.wikipedia.org/wiki/Simple_polygon
+[Polygon]: https://en.wikipedia.org/wiki/Polygon
