Optimization of polyhedral terrains

A digital terrain model is a representation of a real-world terrain in a computer. Terrain models play an important role in geographic information systems, where they are used for numerous purposes. For example, a terrain model can be used to simulate rainfall and to predict terrain areas that are prone to flooding. Or it can help to locate the best spot where to place a fire lookout tower in a wilderness area, such that the terrain area visible from the tower is as large as possible. One of the main ways to represent a terrain is by a triangulation: some points are sampled from the real terrain, and they are connected by triangles that cover the whole terrain area. This results in a subdivision into triangles. It is well-known that when triangulations are used for terrain modeling, there are several geometric aspects of the triangulation that have an important effect. In particular, it is an established fact that long and skinny triangles should be avoided. Moreover, for terrain modeling one cannot ignore that every point has an elevation, thus there are other requirements the triangulation must fulfill in order to provide a reasonable model for a terrain. For example, terrains formed by natural processes have few depressions. Depending on the triangulation chosen, the number of resulting depressions can be excessively high. If the terrain is meant to be used for hydrologic simulations (such as rainfall simulation), these spurious pits will cause serious disruptions in the water flow, rendering the model useless. In this case, depressions are an example of artifacts of the triangulation that should be avoided. This thesis presents new automated methods to improve terrain models, by finding triangulations with well-shaped triangles and---at the same time---as few artifacts as possible. The methods proposed can produce a more accurate and reliable representation of terrains.