Triangulation (advanced geometry)

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For other uses of triangulation in mathematics, see Triangulation (disambiguation).

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\subset\mathbb{R}^{n+1} is a subdivision of the convex hull of the points into simplices such that any two simplices intersect in a common face or not at all and the set of points that are vertices of the subdividing simplices coincides with P. The Delaunay Triangulation is a famous triangulation of a set of points where the circum-hypersphere of each simplex contains none of the points.

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