Generating Realistic Terrains with Higher-Order Delaunay

Triangulations

Thierry de KokMarc van KreveldMaarten Löffler

Center for Geometry, Imagingand 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.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

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

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 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

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

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