WMS:Triangulation

A TIN can be constructed by triangulating a set of vertices. WMS connects the vertices with a series of edges to form a network of triangles. The resulting triangulation satisfies the Delauney criterion The Delauney criterion ensures that no vertex lies within the interior of any of the circumcircles of the triangles in the network (figure shown below).

As the triangulation process proceeds, adjacent triangles are compared to see if they satisfy the Delauney criterion. If necessary, the adjacent edge of the two triangles is swapped (the diagonal of the quadrilateral defined by the two triangles is changed to the other two vertices) in order to satisfy the Delauney criterion. This edge swapping process forms the basis of the triangulation algorithm.

When a new point is inserted into a TIN, the point is incorporated into the TIN and the edges of the triangles adjacent to the new point are swapped as necessary in order to satisfy the Delauney criterion. If the Delauney criterion is satisfied everywhere on the TIN, the minimum interior angle of all of the triangles is maximized. The result is that long thin triangles are avoided as much as possible.

It is important to note that the triangulation described above is used as a preliminary step to creating a TIN conditioned for basin delineation and is not sufficient in most cases for actually doing the drainage delineation. Even if you begin with TIN data you will want to create another TIN using feature objects.

Triangulating

Vertices can be triangulated using the currently selected triangulation algorithm by selecting the Triangles | Triangulate command from the TIN menu. It is important to recognize that the Delauney triangulation is not necessarily the best for performing drainage delineation because it does not insure that important linear features such as streams and ridges will be honored in the TIN as triangle edges. For this reason you should always use a TIN triangulated in this fashion as a "background" elevation source for creating a new TIN from a "conceptual" model of feature objects.

Triangulation Optimization

The Triangles | Optimize Triangulation command of the TIN menu will optimize triangulation according to the following criterion:

• If Angle optimization is selected in the TIN Options dialog, the edges of triangles will be swapped to form edges that match the Delaunay criterion.

• If Area Optimization is selected in the TIN Options dialog, the edges of neighboring triangles will be swapped if the area of one triangle is more than the bias times the area of the smaller triangle.

The criteria is specified in the TIN Options dialog.

Related Topics

• Triangulating
• TIN Options
• Creating TINs from Feature Objects
• Triangulation Optimization