The Triangulation reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

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 Rn+1 is a subdivision of Rn+1 into (n+1)-dimensional simplices such that:

1. any two simplices in T intersect in a common face or not at all;
2. any bounded set in Rn+1 intersects only finitely many simplices in T.

A triangulation of a discrete set of points P in Rn+1 is a triangulation of Rn+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.

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:

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

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.