Terrain Modeling & Delaunay Triangulation

As I discussed in a previous post, TIN (Triangular Irregular Network) creation is an art. But the art has its roots firmly planted in mathematics.

The first triangulated irregular network program was written by Randolph Franklin at Simon Fraser University in 1973 and was based upon the concept of Delaunay Triangulation. This method of triangulation is how Civil 3D builds a TIN.

The triangulation method was invented by Boris Delaunay in 1934. Based upon his definition, the circumcircle (The circle that passes through all vertices of a polygon, in this case, a triangle) of a triangle formed by three points can’t contain points (or vertices) other than the three that define it. Other points are only permitted at the perimeter, or “border” of the surface.

The usefulness of Delaunay Triangulation is only as good as the points mathematically analyzed. More points = better triangulation (to a degree, of course). For our purposes, that means the more representative the shots taken in the field are to the actual terrain, the better the triangulation. Bad shots = bad triangulation. Too few shots = bad triangulation. The goal of a good topographical survey is to collect as much information about feature changes as possible. In Civil 3D, the points are analyzed by various mathematical algorithms, and the result is Delaunay Triangulation. Breaklines and borders are added manually (or via figure commands) as a means of establishing the path at least two vertices of a triangle must take, even if that leads to a Delaunay violation.

The algorithms that do the math do the hard part. It’s human intervention that adds the accuracy to the network. Notice I’ve not made a distinction between “existing” triangles, or “proposed” triangles. The software doesn’t care, it’s just doing mathematical analysis. Therefore, when modeling terrain, you can use all the tools at your disposal to model both. In modeling existing ground, you are limited by features that physically exist. In modeling proposed ground, you’re limited only by your creative approach and knowledge of both the software, and the analysis tools it uses.

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