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 nondegenerate pointsets is NP-hard
too, when using order k DT
• Reduction from maximum nonintersecting 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.k2 + 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.k2 + 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 468
• 1 data point
= 30 meter
• Random
sample
• 1800
vertices
• Delaunay
triangulation
• 53 local
minima
• Hull
heuristic
applied
• Order 4
Delaunay
triangulation
• 25 local
minima
60
hull heuristic
local minima
50
flip heuristic
40
30
20
10
0
0
1
2
3
4
5
order
6
7
8
9
10
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
Number of components
Number of Valley Components
280
260
240
220
200
180
160
140
0
1
2
3
4
Order
5
6
7
8
Results compared to flip
and hull
Number of valley components at order 8
350
300
250
200
150
100
50
0
DT
flip-8
hull-8
valley-8
flip-8 +
valley
hull-8 +
valley
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