Download - L8 - Delaunay triangulation L8 – Delaunay triangulation NGEN06(TEK230) – Algorithms in Geographical Information Systems.

L8 - Delaunay triangulation

L8 – Delaunay triangulation

NGEN06(TEK230) –

Algorithms in Geographical Information Systems


Background

For simple GIS applications coordinates are stored independently in computer files.

This is not adequate in geographic analysis and many processes.

Information about relative positions between the objects is required.

If the positions of the objects are stored independently, it is possible to derive both topological and proximity relationships between the objects. But this is not suitable due to computational complexity, requirements of error free data, etc..

Spatial data structures that store topological and proximity relationships explicitly is needed.


Background

Topological data structures such as the link-node data structure, stores topological relationships explicitly.

But the topological data structures do not give any proximity information.

GIS analyses often require proximity information, this information can either be stored explicitly or derived in real-time.

Methods are required to enhance the efficiency.

The data structures Delaunay triangulation and Voronoi diagram can be used for storing proximity information in a GIS system.


Aim

Main aim

To present theory about spatial data structures that stores proximity information.

Specific aim

To learn definitions and applications of Delaunay triangulation and Voronoi diagram.


Content

1. Topological versus proximity relationships

2. Applications of triangulation

3. Delaunay triangulation

4. Constrained Delaunay triangulation

5. Voronoi diagrams


Topological versus proximity relationships

The topological relationships and the proximity relationships are both subsets of the spatial relationships.


Data structure for explicitly storing spatial relationships

Topological relationshiplink-node data structureSimplicial complex

Proximity relationships Delaunay triangulation Voronoi diagram


Triangulation


Applications of triangulation

Storage of data Interpolation Data structure for proximity relationships


1. Storage of dataField-based data (e.g. heights) are often stored in a grid structure or in a triangular structure (TIN)

TIN- It is possible to use the observed values- Many computations become more complex than corresponding computations on grid data

GRID- Resampling of the original observed values


2. Interpolation


3. Data structure for proximity analysis

A triangular data structure can be used to store proximity information explicitly.


Properties of the Delaunay triangulation I

1. The Delaunay triangulation is unique.

2. The external edges of the triangulation equal the convex hull of P

The external edges are marked with thick lines


Properties of the Delaunay triangulation II

3. If you create a circle by three points on its border (where the three points is a Delaunay triangle) then this circle does not contain any other points of the set P.

The circle induced by the three points a, b and c on its border does not contain any other point.


Properties of the Delaunay triangulation II

4. The triangles in a Delaunay triangulation are as equilateral as possible.

list all the angles (α1, α2, α3, α4, α5, etc.) in increasing order

Denote the smallest angle αmin

The equilateral property then gives that αmin is larger than corresponding smallest angle for any other triangulation

if a triangulation obeys one of the defining properties (3 or 4) then it is a Delaunay triangulation


Algorithms to compute Delaunay triangulation

for i ∈ E

for j E∈ j ≠ i

for k E∈ k ≠ i, k ≠ j

create circle through i, j, k

for l E∈ l ≠ i, l ≠ j, l ≠ k

Check if l is inside the circle i-j-k i-j-k ≠ DelaunayTriangle⇒

end

end

end

end

E = Point set

O(n)4Naive algorithm


There are Delaunay triangulation algorithms that run in O(n log n) time.

Algorithms to compute Delaunay triangulation

An often used method is to first compute the Voronoi diagram and from that diagram derive the Delaunay triangulation.


Constrained Delaunay triangulation CDT

In constrained Delaunay triangulation some input edges are forced to belong to the triangulation.

Edges are not allowed to intersect.

Each edge has two end-points

The input for CDT is a planar graph

Why is CDT interesting?


Edges and points must constitute a planner graph

Constrained Delaunay triangulation can be defined as follows:

In a CDT, for the three edges e in circle c, apart from the input edges in the planar graph G, the following requirements hold

1) The end-points of edge e are on the boundary of c.

2) If any point p of G is inside c then there is an edge in G that intersects at least one of the lines between p and the end-points of e.

To investigate if the triangle t is a CDT

The edges e1 and e3 are input edges

Now to test if the edge e2 is part of the triangulation

There is a point p inside the circle c but this point is allowed since a straight line

between points y and p intersects one of the input edges (e1) (property 2)

Triangle t obeys both of the properties and is then a CDT


Voronoi diagram

Is a common data structure to store proximity information in GIS (sometimes denoted Thiessen polygons).

Properties of the Voronoi diagram

All points in a polygon is closer to the centre point in that particular polygon than to the centre point in any other polygon


Voronoi diagram

The Voronoi diagram is the dual to the Delaunay triangulation


Applications of the Voronoi diagram

Interpolation

GIS analyses of air-raid shelters using Voronoi diagrams. The air raid shelters are symbolized with triangles and the buildings with rectangles

Proximity analyses

For data in ordinal and nominal scale