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.

Boundary Triangles

The perimeter of the TIN resulting from the triangulation process corresponds to or approximates the convex hull of the data points. This may result in some long thin triangles or "slivers" on the perimeter of the triangulated region. There are several ways to deal with the long thin triangles.

Selecting Boundary Triangles

The thin triangles can be selected and deleted using the normal selection procedures. There is also an option for selecting thin triangles when the Select Triangles tool is selected. If the CTRL key is held down, it is possible to drag out a line with the mouse. All triangles intersecting the line will be selected.

Another technique can be used to select long thin triangles on the perimeter of the TIN. By selecting the Select Boundary Triangles command from the TIN menu, the thin triangles on the perimeter of the TIN are automatically selected.

The Select Boundary Triangles command checks triangles on the outer boundary first. If the length ratio of the triangle is less than the critical length ratio, the triangle is selected and the triangles adjacent to the triangle are then checked. The process continues inward until none of the adjacent triangles violate the minimum length ratio.

Thin Triangle Aspect Ratio

The critical length ratio for selecting thin triangles can be set by selecting Length Ratio from the TIN menu. The length ratio is defined as the longest side of the triangle divided by the sum of the two shorter sides.

A maximum edge length may also be specified in the TIN Options dialog.

Related Topics

• TIN Options
• Creating TINs from Feature Objects