Merge pull request AllAlgorithms#43 from polettix/patch-triangle-area

abranhe · web-flow · commit 4a30f385c846 · 2020-10-03T13:36:10.000-04:00
Add content for "Area of triangle"
diff --git a/docs/area-of-triangle.md b/docs/area-of-triangle.md
@@ -4,4 +4,123 @@ title: Area of triangle
 sidebar_label: Area of triangle
 ---
 
-[Open a pull request](https://github.com/AllAlgorithms/algorithms/tree/master/docs/area-of-triangle.md) to add the content for this algorithm.
+A *triangle* is a polygon with three sides and three vertices.
+Calculating its area efficiently depends on what is known about it.
+
+## From base and height
+
+The basic formula assumes knowledge of the length of one of the sides
+(called *base*) and the length of the *height* of the triangle with
+respect to that side. This *height* is the segment that originates from
+the vertex that is *not* on the side we know about, and intersects the
+side we know about at a right angle.
+
+Assuming the base is put horizontally and the third vertex above it, the
+following picture results. Quantity *b* is the length of the bottom
+side, and *h* the height with respect to it.
+
+<img
+   src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangle.TrigArea.svg/535px-Triangle.TrigArea.svg.png"
+   alt="Triangle">
+
+Height can sometimes be referred to as *altitude*.
+
+### Formula
+
+In this case, the area is calculated as follows:
+
+<img
+    src="https://latex.codecogs.com/png.latex?T=\frac{b&space;\cdot&space;h}{2}"
+    alt="T=\frac{b \cdot h}{2}" title="S=\frac{b \cdot h}{2}" />
+
+
+### Algorithm
+
+The algorithm can be derived directly from the defining formula above:
+
+```
+triangle_area_basic (b, h) {
+   return b * h / 2;
+}
+```
+
+## In the euclidean plane
+
+In this section, it is assumed that the triangle is known by the
+coordinates of its three vertices A, B, and C on the plane:
+
+<img src="http://latex.codecogs.com/png.latex?\dpi{110}&space;\\\mathbf{A}&space;=&space;(x_A,&space;y_A)&space;\\\mathbf{B}&space;=&space;(x_B,&space;y_B)&space;\\\mathbf{C}&space;=&space;(x_C,&space;y_C)&space;" alt="A, B, and C in the plane" />
+
+### Formula
+
+In this case, the area of the triangle can be calculated as follows:
+
+<img src="http://latex.codecogs.com/png.latex?\dpi{110}&space;T&space;=&space;\frac{|(x_B&space;-&space;x_A)(y_C&space;-&space;y_A)&space;-&space;(x_C&space;-&space;x_A)(y_B&space;-&space;y_A)|}{2}&space;" alt="T = \frac{|(x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A)|}{2} " />
+
+### Algorithm
+
+The following algorithm is a direct translation of the formula:
+
+```
+triangle_area_plane (x_A, y_A, x_B, y_B, x_C, y_C) {
+    return abs((x_B - x_A) * (y_C - y_A) - (x_C - x_A) * (y_B - y_A)) / 2;
+}
+```
+
+## In the euclidean space
+
+In this section, it is assumed that the triangle is known by the
+coordinates of its three vertices A, B, and C in the three-dimensional
+space:
+
+<img src="http://latex.codecogs.com/png.latex?\dpi{110}&space;\\\mathbf{A}&space;=&space;(x_A,&space;y_A,&space;z_A)&space;\\\mathbf{B}&space;=&space;(x_B,&space;y_B,&space;z_B)&space;\\\mathbf{C}&space;=&space;(x_C,&space;y_C,&space;z_C)&space;" title="http://latex.codecogs.com/png.latex?\dpi{110} \\\mathbf{A} = (x_A, y_A, z_A) \\\mathbf{B} = (x_B, y_B, z_B) \\\mathbf{C} = (x_C, y_C, z_C) " />
+
+### Formula
+
+The formula to calculate the area involves calculating the determinant
+of three matrices:
+
+<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c599953dad11fdc117c9ecf5ad8c56de3563e0" alt="triangle area in the plane">
+
+The absolute value of each determinant is the double of the area of the
+triangle obtained by projecting the target triangle onto one of the
+coordinate planes, which allows reusing the formula for the triangle in
+the plane in a previous section.
+
+### Algorithm
+
+The following algorithm leverages the triangle area calculation in the
+plane, described in a previous section and not repeated here:
+
+```
+/* Assume that A, B, and C are 3-dimensional arrays
+   X at index 0, Y at index 1, Z at index 2         */
+triangle_area_space (A, B, C) {
+    x = triangle_area_plane(A[1], A[2], B[1], B[2], C[1], C[2]);
+    y = triangle_area_plane(A[2], A[0], B[2], B[0], C[2], C[0]);
+    z = triangle_area_plane(A[0], A[1], B[0], B[1], C[0], C[1]);
+    return sqrt(x * x + y * y + z * z);
+}
+```
+
+*Note*: the division by two in the formula is already included in the
+result from the calculation of the area in the plane.
+
+## Performance
+
+All algorithms for calculating the area of a triangle in the sections
+above execute in constant time (O(0)).
+
+## Implementations
+
+| | Language | Link |
+|:-: | :-: | :-: |
+| | | |
+
+## Helpful Links
+
+- [Triangle][]
+- [Area of Triangles and Polygons][]
+
+[Triangle]: https://en.wikipedia.org/wiki/Triangle
+[Area of Triangles and Polygons]: http://geomalgorithms.com/a01-_area.html
