Peter Alfeld Department of Mathematics College of Science University of Utah
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A Triangulation T is a collection of N triangles satisfying the following requirements:

  1. The interiors of the triangles are pairwise disjoint.
  2. Each edge of a triangle in T is either a common edge of two triangles in T or else it is on the boundary of the union D of all the triangles.
  3. D is homeomorphic to a square

The first requirement says that triangles don't overlap. The second requirement rules out combinations of triangles where one has a vertex in the interior of an edge of another triangle, and the strange sounding last requirement rules out holes, pinchpoints (where just 2 triangles meet in a single point) and disjoint sets of triangles.

The reason for the popularity of triangulations is that given any set of points, one can construct a triangulation that has those points as the vertices of the triangulation. Triangulations are a natural generalization of the concept of partitioning an interval into subintervals.

The Figure nearby illustrates a triangulation. The vertices are indicated by circles, and the triangles by grey shading. The domain D of interest therefore is the grey polygon. The triangulation consists of 16 triangles that share 14 vertices, 10 of which lie on the boundary of D .

To facilitate discussions we have to introduce some notation and language:


Triangulations form a huge subject in Mathematics. You may be interested in looking at these two very different graduate level books in Computer Science and Mathematics, respectively:

To whet your appetite, here is a description of just one aspect of triangulations. It is possible to build any triangulation T by starting with a single triangle and then adding one triangle at a time, joining it to the growing triangulation either on one or on two boundary edges, such that at every stage one continues to have a triangulation. This may seem obvious but in fact a similar statement does not hold for triangulations in three dimension (where one uses tetrahedra instead of triangles). Click here to see a counterexample.