Generating Realistic Terrains with Higher-Order Delaunay

Triangulations

Thierry de Kok

Marc van Kreveld

Maarten Löffler

Center for Geometry, Imagingand 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

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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, ...

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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?

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

1st order

0th order

4th order

Higher order Delaunay triangulations

• Introduced by Gudmundsson, Hammar and van Kreveld (ESA 2000)

• Minimize local minima for 1st order:O(n log n) time

• Minimize local minima for kth order:O(k2)-approximation algorithm inO(nk3 + nk log n) time (hull heuristic)

This paper, results

• NP-hardness of minimizing local minima

• NP-hardness for kth order, k = (n)

• New flip heuristic: O(nk2 + nk log n) time

• Faster hull heuristic: O(nk2 + 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(nk2 + nk log n) time

flip18

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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

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order

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inim

aHull heuristic

Flip heuristic

Another realistic aspect

• Valleys continue

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normal edge ridge edge valley 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 60-70% 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