# Generating Realistic Terrains with Higher-Order Delaunay Triangulations - PowerPoint PPT Presentation

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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

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Generating Realistic Terrains with Higher-Order Delaunay Triangulations

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## 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

### 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

• Flip edges that might potentially remove a local minimum

• Preserve order k property

• O (n.k2 + n.k.logn)

• 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.logn)

• 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

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)

### 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