# Triangulation

In geometry, in the most general meaning, **triangulation** is a subdivision of a geometric object into simplices. In particular, in the plane it is a subdivision into triangless, hence the name.

Different branches of geometry use slightly differing definitions of the term.

A **triangulation** T of **R**^{n+1} is a subdivision of **R**^{n+1} into (n+1)-dimensional simplices such that:

- any two simplices in T intersect in a common face or not at all;
- any bounded set in
**R**^{n+1}intersects only finitely many simplices in T.

**triangulation**of a discrete set of points

**P**in

**R**

^{n+1}is a triangulation of

**R**

^{n+1}such that the set of points that are vertices of the subdividing simplices coincides with

**P**.

The following definitions are used in Computational geometry.

A **triangulation** of a polygon **P** is its partition into triangles. In the strict sense, these triangles may have vertices only in the vertices of **P**. In non-strict sense, it is allowed to add more points to serve as vertices of triangles.

Also, a **triangulation** of a set of points **P** is sometimes taken to be the triangulation of the convex hull of **P**.

See also: Delaunay triangulation

Topology generalizes this notion in a natural way as follows. A **triangulation** of a topological space *X* is a simplicial complex *K*, homeomorphic to *X*, together with a homeomorphism *h*:*K*->*X*.

Triangulation is useful in determining the properties of a topological space.

**Triangulation**is the process of finding a distance by calculating the length of one side of a triangle, given a deterministic combination of angles and sides of the triangle. It uses mathematical identities from trigonometry.

Some identities often used:

- The sum of the angles of a triangle is &pi (180 degrees).
- The law of sines
- The law of cosines
- The Pythagorean theorem

See: Parallax.

In the social sciences, **triangulation** is often used to indicate that more than one method is used in a study with a view to double (or triple) checking results. This is also called "cross examination". The idea is that we can be more confident with a result if different methods lead to the same result.