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# 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):

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 $\mathbb{R}^{n+1}$ is a subdivision of $\mathbb{R}^{n+1}$ 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 $\mathbb{R}^{n+1}$ intersects only finitely many simplices in T.

A triangulation of a discrete set of points $P\in\mathbb{R}^{n+1}$ is a triangulation of $\mathbb{R}^{n+1}$ 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 $h:K\to X$.

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.

Last updated: 02-05-2005 02:18:24
Last updated: 02-27-2005 19:14:39