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 –Triangulation for terrains –Realistic terrains –Higher order Delaunay triangulations Minimizing local minima –NP-hardness –Two heuristics: algorithms and experiments Other realistic aspects

Polyhedral terrains, or TINs Points with (x,y) and elevation as input TIN as terrain representation Choice of triangulation is important

Realistic terrains Due to erosion, realistic terrains –have few local minima –have valley lines that continue local minimum, interrupted valley line after an edge flip

Terrain modeling in GIS Terrain modeling is extensively studied in geomorphology and GIS Need to avoid artifacts like local minima Need correct shape for run-off models, hydrological models, avalanche models, local minimum in a TIN

Delaunay triangulation Maximizes minimum angle Empty circle property

Delaunay triangulation Does not take elevation into account May give local minima May give interrupted valleys

Triangulate to minimize local minima?

Connect everything to global minimum bad triangle shape & interpolation

Higher order Delaunay triangulations Compromise between good shape & interpolation, and flexibility to satisfy other constraints k -th order: allow k points in circle 1 st order 0 th order 4 th order

Higher order Delaunay triangulations Introduced by Gudmundsson, Hammar and van Kreveld (ESA 2000) Minimize local minima for 1 st order: O(n log n) time Minimize local minima for k th order: O(k 2 )-approximation algorithm in O(nk 3 + nk log n) time (hull heuristic)

This paper, results NP-hardness of minimizing local minima NP-hardness for k th order, k = (n ) New flip heuristic: O(nk 2 + nk log n) time Faster hull heuristic: O(nk 2 + nk log n) time Implementation and experiments on real terrains Heuristic to avoid interrupted valleys: valley heuristic

Flip Heuristic Start with Delaunay triangulation Flip edges that remove, or may help remove a local minimum Only flip if 2 circles have k points inside O(nk 2 + nk log n) time flip

Hull Heuristic Start with Delaunay triangulation Compute all useful order k Delaunay edges that remove a local minimum useful order 4 Delaunay edge

Hull Heuristic Add them incrementally, unless –it intersects a previously inserted edge Retriangulate the polygon that appears

Hull Heuristic Add them incrementally, unless –it intersects a previously inserted edge Retriangulate the polygon that appears

Experiments on terrains

Experiments Do higher order Delaunay triangulations help to reduce local minima? How does this depend on the order? Which heuristic is better: flip or hull? Do they create any artifacts? 5 terrains orders 0-10 flip and hull heuristic

Quinn Peak Elevation grid of 382 x 468 Random sample of 1800 vertices Delaunay triangulation 53 local minima

Hull heuristic applied Order 4 Delaunay triangulation 25 local minima

Hull heuristic Flip heuristic

Another realistic aspect Valleys continue normal edgeridge edgevalley edge Valley edges can end in vertices that are not local minima

Valley Heuristic Remove isolated valley edges by flipping them out Extend valley edge components further down O(nk log n) time

Experiments Terrains with valley edges and local minima shown Delaunay, Flip-8, Hull-8, Valley-8, Hull-8 + Valley-8

Delaunay triangulation

Flip-8

Hull-8

Valley-8

Hull-8 + valley heuristic

Conclusions Hull and Flip reduce local minima by % for order 8; Hull is often better Valley reduces the number of valley edge components by 20-40% for order 8 Flip gives artifacts Hull + Valley seems best

Future Work NP-hardness for small k ? Other properties of terrains –Spatial angles –Local maxima –Other hydrological features (watersheds) Improvements valley heuristic