# Online Encyclopedia

# Triangulation

In trigonometry and elementary geometry, **triangulation** is the process of finding a distance to a point by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other reference points.

Some identities often used (valid only in flat or euclidean geometry):

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

Triangulation is used for many purposes, including surveying, navigation, astrometry, binocular vision and gun direction of weapons.

See: Parallax.

In advanced 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 triangles, hence the name.

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

A **triangulation** T of is a subdivision of 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 intersects only finitely many simplices in T.

A **triangulation** of a discrete set of points is a triangulation of such that the set of points that are vertices of the subdividing simplices coincides with *P*.

In computational geometry, triangulation may be performed for various objects.

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 .

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

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.