While it is usually not difficult to compute principal curvatures of a smooth surface of sufficient differentiability, it is a rather difficult task when only a polygonal approximation of the surface is available, because of the inherent ambiguity of such representation. A number of different approaches has been proposed in the past that tackle this problem using various techniques. Most papers tend to focus on a particular method, while an comprehensive comparison of the different approaches is usually missing.
We present results of a large experiment, involving both common and recently proposed curvature estimation techniques, applied to triangle meshes of varying properties. It turns out that none of the approaches provides reliable results under all circumstances. Motivated by this observation, we investigate mesh statistics, which can be computed from vertex positions and mesh connectivity information only, and which can help in deciding which estimator will work best for a particular case. Finally, we propose a meta‐estimator, which makes a choice between existing algorithms based on the value of the mesh statistics, and we demonstrate that such meta‐estimator, despite its simplicity, provides considerably more robust results than any existing approach.
Although hydraulic erosion modeling on a GIS terrain models has been addressed by a body of previous work, it still remains an open problem. In GIS, raster representation and triangular irregular networks (TIN) are the most commonly used surface models, because they are simple and offer implicit topological information. However, these data structures do not allow the simulation of erosion on concave terrain features, such as caves or overhangs. Other methods, more commonly used in the computational fluid dynamics, use volumetric data representation. They are able to model the 3D features, but they usually have high memory requirements and are computationally demanding. We propose a novel solution to the hydraulic erosion modeling problem that uses a triangular mesh data structure. Our framework allows for adaptive changes of the mesh resolution according to the local complexity of the terrain, which leads to lower memory requirements when compared to the volumetric approaches. Our data structure also supports the visualization of the concave 3D features, allowing the simulation and visualization of erosion on terrain elements such as tunnels or caves.
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