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Generating Realistic Terrains with Higher-Order Delaunay
Triangulations
Thierry de KokMarc van KreveldMaarten Löffler
Center for Geometry, Imagingand Virtual Environments
Utrecht University
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Overview
• Introduction• Results on local minima
– NP-hard– Two heuristics
• Results on valley components– A new heuristic
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Motivation
• Terrain modeling for geomorphological applications
• TIN as terrain representation• Realism necessary• Choice of triangulation is
important
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• Few local minima• Connected valley components• Wrong triangulation can introduce
undesirable artifacts
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Triangulations
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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)
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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
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Local Minima Results
• Computing optimal HODT for minimizing local minima is NP-hard
• Two heuristics • Experimental results comparing
the heuristics and analysing HODT
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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
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Flip Heuristic
• Start with Delaunay triangulation • Flip edges that might potentially
remove a local minimum• Preserve order k property• O (n.k2 + n.k.log n)
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• New edge must be “lower” than old edge• New triangles must be order k
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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.k2 + n.k.log n)
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• When adding an edge, compute the hull• Retriangulate the hull• Do not add any other edges
intersecting the hull
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Experiments on real Terrains
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• Quinn Peak• Elevation
data grid• 382 x 468• 1 data point
= 30 meter
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• Random sample
• 1800 vertices
• Delaunay triangulation
• 53 local minima
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• Hull heuristic applied
• Order 4 Delaunay triangulation
• 25 local minima
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0
10
20
30
40
50
60
0 1 2 3 4 5 6 7 8 9 10
order
local
min
ima
hull heuristic
flip heuristic
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Drainage on TIN
• Complex to model due to material properties
• Water follows path of steepest descent– Over edge – Over triangle
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Definitions
• Three kinds of edges:
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• Valley component: maximal set of valley edges s.t. flow from these edges reaches lowest vertex of the component
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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
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Isolated valley edge
• Try to remove it– No new valley edges
should be created– New triangle order k
• Otherwise try to extend it
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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
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• 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
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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)
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Results on a terrain
Number of Valley Components
140
160
180
200220
240
260
280
0 1 2 3 4 5 6 7 8
Order
Nu
mb
er o
f co
mp
on
ents
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Results compared to flip and hull
Number of valley components at order 8
0
50
100
150
200
250
300
350
DT flip-8 hull-8 valley-8 flip-8 +valley
hull-8 +valley
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Delaunay triangulation
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Flip-8
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Hull-8
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Valley-8
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Flip-8 + valley heuristic
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Hull-8 + valley heuristic
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Conclusions Local Minima
• Low orders already give good results
• Hull is often better than flip• Hull performed almost optimal
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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
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Future Work
• NP-hardness for small k• Other properties of terrains
– Local maxima– More hydrological features
(watersheds)• Different local operators for
valleyheuristic