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, like path planning, visualization, and terrain analysis. 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. There are many ways to triangulate the points, and choosing the right one is essential to obtain terrain models that represent the real terrain faithfully. In particular, the shape of the triangles is very important: long and skinny triangles should be avoided. However, that is not enough. The current approach to triangulating terrains does not take the elevation of the points into account. This can create artifacts---like spurious pits, interrupted valley lines and artificial dams---, a serious obstacle for performing terrain analysis tasks, especially for hydrology or erosion simulations. This research will study combinatorial and geometric properties of triangulations, in order to design new automated methods to find triangulations with well-shaped triangles and---at the same time---as few artifacts as possible.
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