Generalized trigonometry

Ordinary trigonometry studies triangles in the Euclidean plane ⁠ $\mathbb {R} ^{2}$ ⁠. There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions^{[broken anchor]}, definitions via differential equations^{[broken anchor]}, and definitions using functional equations. Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space. A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and $n$ -simplices.