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.

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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 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.

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In Computational geometry, a triangulation is one of two things:

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

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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.

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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.