We present an error metric based on the potential energy of water flow to evaluate the quality of lossy terrain simplification algorithms. Typically, terrain compression algorithms seek to minimize RMS (root mean square) and maximum error. These metrics fail to capture whether a reconstructed terrain preserves the drainage network. A quantitative measurement of how accurately a drainage network captures the hydrology is important for determining the effectiveness of a terrain simplification technique. Having a measurement for testing and comparing different models has the potential to be widely used in numerous applications (flood prevention, erosion measurement, pollutant propagation, etc). In this paper, we transfer the drainage network computed on reconstructed geometry onto the original uncompressed terrain and use our error metric to measure the level of error created by the simplification. We also present a novel terrain simplification algorithm based on the compression of hydrology features. This method and other terrain compression schemes are then compared using our new metric.

A terrain M can be represented as a triangulation of the plane along with a height function associated with the vertices (and linearly interpolated within the edges and triangles) of M. We investigate the problem of answering contour queries on M: Given a height ℓ and a triangle f of M that intersects the level set of M at height ℓ, report the list of the edges of the connected component of this level set that intersect f, sorted in clockwise or counterclockwise order. Contour queries are different from level-set queries in that only one contour (connected component of the level set) out of all those that may exist is expected to be reported. We present an I/Oefficient data structure of linear size that answers a contour query in O(log B N + T/B) I/Os, where N is the number of triangles in the terrain and T is the number of edges in the output contour. The data structure can be constructed using O(Sort(N)) I/Os.

We study the complexity and the I/O-efficient computation of flow on triangulated terrains. We present an acyclic graph, the descent graph, that enables us to trace flow paths in triangulations i/o-efficiently. We use the descent graph to obtain i/o-efficient algorithms for computing river networks and watershed-area maps in O(Sort(d + r)) i/o’s, where r is the complexity of the river network and d of the descent graph. Furthermore we describe a data structure based on the subdivision of the terrain induced by the edges of the triangulation and paths of steepest ascent and descent from its vertices. This data structure can be used to report the boundary of the watershed of a query point q or the flow path from q in O(l(s) + Scan(k)) i/o’s, where s is the complexity of the subdivision underlying the data structure, l(s) is the number of i/o’s used for planar point location in this subdivision, and k is the size of the reported output. On α-fat terrains, that is, triangulated terrains where the minimum angle of any triangle is bounded from below by α, we show that the worst-case complexity of the descent graph and of any path of steepest descent is O(n/α 2), where n is the number of triangles in the terrain. The worst-case complexity of the river network and the above-mentioned data structure on such terrains is O(n 2 /α 2). When α is a positive constant this improves the corresponding bounds for arbitrary terrains by a linear factor. We prove that similar bounds cannot be proven for Delaunay triangulations: these can have river networks of complexity Θ(n 3). 1

Consider rain falling at a uniform rate onto a terrain T represented as a triangular irregular network. Over time, water collects in the basins of T, forming lakes that spill into adjacent basins. Our goal is to compute, for each terrain vertex, the time this vertex is flooded (covered by water). We present an I/O-efficient algorithm that solves this problem using O(sort(X) log(X/M) + sort(N)) I/Os, where N is the number of terrain vertices, X is the number of pits of the terrain, sort(N) is the cost of sorting N data items, and M is the size of the computer’s main memory. Our algorithm assumes that the volumes and watersheds of the basins of T have been precomputed using existing methods.