Generating Realistic Terrains with Higher-Order Delaunay Triangulations Thierry de Kok Marc van Kreveld Maarten Löffler Center for Geometry, Imaging and Virtual Environments Utrecht University

Overview Introduction Results on local minima –NP-hard –Two heuristics Results on valley components –A new heuristic

Motivation Terrain modeling for geomorphological applications TIN as terrain representation Realism necessary Choice of triangulation is important

Few local minima Connected valley components Wrong triangulation can introduce undesirable artifacts

Triangulations

Higher-Order Delaunay Triangulations At most k points in circle Order 0 DT is normal DT If k > 0, order k DT is not unique Introduced by Gudmundsson et al. (2002)

Using HODT to Solve the Problem Well shaped triangles, plus room to optimize other criteria We want to minimize local minima For k > 1, optimal order k DT is no longer easy to compute Heuristics are needed

Local Minima Results Computing optimal HODT for minimizing local minima is NP-hard Two heuristics Experimental results comparing the heuristics and analysing HODT

NP-hardness Minimizing local minima for degenerate pointsets is NP-hard Minimizing local minima for non- degenerate pointsets is NP-hard too, when using order k DT Reduction from maximum non- intersecting set of line segments

Flip Heuristic Start with Delaunay triangulation Flip edges that might potentially remove a local minimum Preserve order k property O (n. k 2 + n. k. log n)

New edge must be “lower” than old edge New triangles must be order k

Hull Heuristic Compute a list of all useful order k edges that remove a local minimum Add as many as possible Make sure they do not interfere O (n. k 2 + n. k. log n)

When adding an edge, compute the hull Retriangulate the hull Do not add any other edges intersecting the hull

Experiments on real Terrains

Quinn Peak Elevation data grid 382 x data point = 30 meter

Random sample 1800 vertices Delaunay triangulation 53 local minima

Hull heuristic applied Order 4 Delaunay triangulation 25 local minima

hull heuristic flip heuristic

Drainage on TIN Complex to model due to material properties Water follows path of steepest descent –Over edge –Over triangle

Definitions Three kinds of edges:

Valley component: maximal set of valley edges s.t. flow from these edges reaches lowest vertex of the component

Drainage quality of terrain Quality defined by: –Number of local minima –Number of valley components not ending a local minimum Improve quality by: –Deleting single edge networks –Extending networks downwards to local minima

Isolated valley edge Try to remove it –No new valley edges should be created –New triangle order k Otherwise try to extend it

Extending component Extend: –Single edge network that cannot be removed (at this order) –Multiple edge networks that do end in a local minimum –Multiple edge networks that do not end in a local minimum

Extend if: –bqrp is convex –br is valley edge –brp and bqr are order k –br is steepest descent direction from b –r < b, r < q, r < p –No interrupted valley components in p or q

Results valley heuristic 25-40% decrease in number of valley components +/- 30 % decrease in number of local minima (far less than flip and hull heuristic)

Results on a terrain

Results compared to flip and hull

Delaunay triangulation

Flip-8

Hull-8

Valley-8

Flip-8 + valley heuristic

Hull-8 + valley heuristic

Conclusions Local Minima Low orders already give good results Hull is often better than flip Hull performed almost optimal

Conclusions Drainage Low order already give good results Significant reduction in number of valley components Drainage quality is improved the most when hullheuristic is combined with valley heuristic

Future Work NP-hardness for small k Other properties of terrains –Local maxima –More hydrological features (watersheds) Different local operators for valleyheuristic