Fundamentals of TIN triangulation in ArcGIS

ArcGIS supports the Delaunay triangulation method. The Delaunay triangulation ensures that no vertex lies within the interior of any of the circumcircles of the triangles in the network. If the Delaunay criterion is satisfied everywhere on the TIN, the minimum interior angle of all triangles is maximized. The result is that long, thin triangles are avoided as much as possible.

At ArcGIS 10, a constrained Delaunay triangulation has been introduced for TIN generation.

Constrained Delaunay triangulation

A constrained Delaunay triangulation method follows Delaunay rules everywhere except along breaklines.

Using traditional Delaunay triangulation, breaklines are densified by the software with Steiner points to ensure that the resulting TIN remains Delaunay conforming. Therefore, one input breakline segment can result in multiple triangle edges. Using a constrained Delaunay triangulation, no densification occurs, and each breakline segment is added as a single edge.

Setting the constrained property on a TIN does not alter the existing state of the triangulation. It only controls what happens from that point forward. Therefore, if breaklines are added prior to setting the constrained triangulation property, those features will remain unchanged. Any densification points added to enforce those breaklines will not be removed.

Once the constrained property has been set on a TIN, there is no way to unset it. Constrained TINs can't be converted into or saved as Delaunay conforming TINs.

Delaunay or constrained Delaunay?

Delaunay conforming triangulations are recommended over constrained triangulations. This is because the resulting TINs are likely to contain fewer long, skinny triangles, which are undesirable for surface analysis. Additionally, natural neighbor interpolation and Thiessen (Voronoi) polygon generation can only be performed on Delaunay conforming triangulations.

A constrained Delaunay triangulation can be considered when you need to explicitly define certain edges that are guaranteed not be modified (i.e., split into multiple edges) by the triangulator.

Constrained Delaunay triangulations are also useful for minimizing the size of a TIN, since they have fewer nodes and triangles where breaklines are not densified.