## Final Report Summary - MFHQTERRAINS (Mathematical Foundations of High-Quality Terrain Models)

This project focused on different mathematical and algorithmic properties of triangulated terrain models. A terrain model is a representation of a terrain surface. Terrain models play an important role in many areas of science, most notably in Geographic Information Science (GIS), where they have plenty of uses, for example in visualization, path planning, and terrain analysis. For instance, two concrete problems that can be solved using terrain models are computing where to locate a fire lookout tower in order to cover as much terrain as possible, or simulating rainfall in order to predict what terrain areas are prone to flooding.

One of the major ways to model a terrain is using a triangulation: a subdivision of the plane into triangles, where the vertices of the triangles are (sampled) points with an elevation. This is called a triangulated terrain. See the figure attached for an illustration.

It is well-known that when triangulations are used for terrain modeling, there are many aspects of the points and the triangles that can have an important effect on the quality of the terrain model. For example, it is an established fact that long and skinny triangles should be avoided. This is in part because a long triangle implies that the height of unsampled points is estimated based on points that are far from each other, something that is likely to produce an incorrect result.

In this project we studied, on the one hand, mathematical aspects of point sets and triangulations of points, and on the other hand, different computational problems on terrains. Our results were obtained using tools from combinatorial geometry, computational geometry, and general algorithmic tools. The results provide new knowledge about basic structures underlying terrain models, as well as new methods to model and analyze terrain data. Altogether, these represent important advances that will allow creating higher quality terrain models.

In the following we highlight two of the most outstanding results obtained in the project.

1) Higher order Delaunay triangulations in practice. The most important properties of a triangulation that represents a terrain have to do with the shape of its triangles. For instance, it is good to avoid very small or very large angles, and often to have a small normal angle between adjacent triangles. The Delaunay triangulation is a well-known triangulation that has certain guarantees about the shape of its triangles. Because of this, most triangulated terrain models use the Delaunay triangulation. However, it is not always the best triangulation to model a terrain. In fact, it has long been recognized that other important criteria like minimizing the angle between normal vectors of adjacent triangles are not addressed by the Delaunay triangulation. As part of the project we have studied a generalization of Delaunay triangulations: Higher-order Delaunay triangulations. These are a family of triangulations that have well-shaped triangles, similar to those of the Delaunay triangulation, but give extra flexibility to optimize additional criteria.

Part of the project focused on performing an experimental study of the applicability of Higher-order Delaunay triangulations to improve on aspects that are not addressed by the Delaunay triangulation. For example, our experiments show that Higher-order Delaunay triangulations allow improving significantly properties related to angles between normal, without compromising the shape of the triangles too much. Our experiments also show that these triangulations can be computed efficiently. Our results demonstrate that Higher-order Delaunay triangulations are a feasible alternative to Delaunay triangulations, which can be computed in a simple way and can produce higher-quality terrain models.

2) Water flow on imprecise terrains. An important issue for producing high-quality terrain models is being able to deal with the inherent imprecision that exists in any computer model (since data is always acquired with error-prone devices). Unfortunately, this fact is often overlooked when designing algorithms for terrain modeling. Usually, when water computations need to be carried out on a terrain, the algorithms assume that the terrain elevation data is exact. However, if the terrain model had a lot of imprecision, the results obtained with such methods are not reliable because they were based on a false assumption of knowing the elevation data exactly.

As part of the project, the researcher and collaborators have studied water flow computations under an explicit imprecision model, in which the elevation of the sampled terrain points is not exact, but consists of an interval. The actual elevation is known to be somewhere within the interval, but it is not known where. Within this context, two natural models of “imprecise terrains” relevant for hydrologic simulations were presented and discussed. In the first one, water is allowed to flow freely across the surface of the terrain triangles. It is shown that in this model water computations are computationally intractable (i.e. it is unlikely that fast and efficient algorithms exist under this model). On the other hand, in the second model, where water is assumed to flow only along the edges of the terrain, the computation of essential hydrologic information in the presence of height imprecision can be done efficiently with simple and practical algorithms. Even though the second model is more limited, the simplification incurred by the model is conceptually reasonable, and is much more powerful than what is allowed by most existing methods. Therefore, the results will be of use for a wide range of terrain analysis tasks that involve flow computations. In addition, all results on the network model are not restricted to triangulated terrains, but can also be applied to the more widespread matrix-based (raster) terrains, increasing even further the applicability of the research.

In addition to the two examples mentioned above, the project has also produced several other results about different properties of point sets, planar graphs, visibility problems on polygons and terrains. These results concern the most basic elements behind the representation of terrain models. Overall, the project has contributed to a deeper understanding of these fundamental structures.

One of the major ways to model a terrain is using a triangulation: a subdivision of the plane into triangles, where the vertices of the triangles are (sampled) points with an elevation. This is called a triangulated terrain. See the figure attached for an illustration.

It is well-known that when triangulations are used for terrain modeling, there are many aspects of the points and the triangles that can have an important effect on the quality of the terrain model. For example, it is an established fact that long and skinny triangles should be avoided. This is in part because a long triangle implies that the height of unsampled points is estimated based on points that are far from each other, something that is likely to produce an incorrect result.

In this project we studied, on the one hand, mathematical aspects of point sets and triangulations of points, and on the other hand, different computational problems on terrains. Our results were obtained using tools from combinatorial geometry, computational geometry, and general algorithmic tools. The results provide new knowledge about basic structures underlying terrain models, as well as new methods to model and analyze terrain data. Altogether, these represent important advances that will allow creating higher quality terrain models.

In the following we highlight two of the most outstanding results obtained in the project.

1) Higher order Delaunay triangulations in practice. The most important properties of a triangulation that represents a terrain have to do with the shape of its triangles. For instance, it is good to avoid very small or very large angles, and often to have a small normal angle between adjacent triangles. The Delaunay triangulation is a well-known triangulation that has certain guarantees about the shape of its triangles. Because of this, most triangulated terrain models use the Delaunay triangulation. However, it is not always the best triangulation to model a terrain. In fact, it has long been recognized that other important criteria like minimizing the angle between normal vectors of adjacent triangles are not addressed by the Delaunay triangulation. As part of the project we have studied a generalization of Delaunay triangulations: Higher-order Delaunay triangulations. These are a family of triangulations that have well-shaped triangles, similar to those of the Delaunay triangulation, but give extra flexibility to optimize additional criteria.

Part of the project focused on performing an experimental study of the applicability of Higher-order Delaunay triangulations to improve on aspects that are not addressed by the Delaunay triangulation. For example, our experiments show that Higher-order Delaunay triangulations allow improving significantly properties related to angles between normal, without compromising the shape of the triangles too much. Our experiments also show that these triangulations can be computed efficiently. Our results demonstrate that Higher-order Delaunay triangulations are a feasible alternative to Delaunay triangulations, which can be computed in a simple way and can produce higher-quality terrain models.

2) Water flow on imprecise terrains. An important issue for producing high-quality terrain models is being able to deal with the inherent imprecision that exists in any computer model (since data is always acquired with error-prone devices). Unfortunately, this fact is often overlooked when designing algorithms for terrain modeling. Usually, when water computations need to be carried out on a terrain, the algorithms assume that the terrain elevation data is exact. However, if the terrain model had a lot of imprecision, the results obtained with such methods are not reliable because they were based on a false assumption of knowing the elevation data exactly.

As part of the project, the researcher and collaborators have studied water flow computations under an explicit imprecision model, in which the elevation of the sampled terrain points is not exact, but consists of an interval. The actual elevation is known to be somewhere within the interval, but it is not known where. Within this context, two natural models of “imprecise terrains” relevant for hydrologic simulations were presented and discussed. In the first one, water is allowed to flow freely across the surface of the terrain triangles. It is shown that in this model water computations are computationally intractable (i.e. it is unlikely that fast and efficient algorithms exist under this model). On the other hand, in the second model, where water is assumed to flow only along the edges of the terrain, the computation of essential hydrologic information in the presence of height imprecision can be done efficiently with simple and practical algorithms. Even though the second model is more limited, the simplification incurred by the model is conceptually reasonable, and is much more powerful than what is allowed by most existing methods. Therefore, the results will be of use for a wide range of terrain analysis tasks that involve flow computations. In addition, all results on the network model are not restricted to triangulated terrains, but can also be applied to the more widespread matrix-based (raster) terrains, increasing even further the applicability of the research.

In addition to the two examples mentioned above, the project has also produced several other results about different properties of point sets, planar graphs, visibility problems on polygons and terrains. These results concern the most basic elements behind the representation of terrain models. Overall, the project has contributed to a deeper understanding of these fundamental structures.