L8  Delaunay triangulation L8 – Delaunay triangulation NGEN06(TEK230) – Algorithms in...

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L8  Delaunay triangulation
L8 – Delaunay triangulation
NGEN06(TEK230) –
Algorithms in Geographical Information Systems
L8  Delaunay triangulation
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.
L8  Delaunay triangulation
Background
Topological data structures such as the linknode 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 realtime.
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.
L8  Delaunay triangulation
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.
L8  Delaunay triangulation
Content
1. Topological versus proximity relationships
2. Applications of triangulation
3. Delaunay triangulation
4. Constrained Delaunay triangulation
5. Voronoi diagrams
L8  Delaunay triangulation
Topological versus proximity relationships
The topological relationships and the proximity relationships are both subsets of the spatial relationships.
L8  Delaunay triangulation
Topological versus proximity relationships
The topological relationships and the proximity relationships are both subsets of the spatial relationships.
L8  Delaunay triangulation
Data structure for explicitly storing spatial relationships
Topological relationshiplinknode data structureSimplicial complex
Proximity relationships Delaunay triangulation Voronoi diagram
L8  Delaunay triangulation
Triangulation
L8  Delaunay triangulation
Applications of triangulation
Storage of data Interpolation Data structure for proximity relationships
L8  Delaunay triangulation
1. Storage of dataFieldbased 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
L8  Delaunay triangulation
2. Interpolation
L8  Delaunay triangulation
3. Data structure for proximity analysis
A triangular data structure can be used to store proximity information explicitly.
L8  Delaunay triangulation
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
L8  Delaunay triangulation
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.
L8  Delaunay triangulation
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
L8  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 ijk ijk ≠ DelaunayTriangle⇒
end
end
end
end
E = Point set
O(n)4Naive algorithm
L8  Delaunay triangulation
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.
L8  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 endpoints
The input for CDT is a planar graph
Why is CDT interesting?
L8  Delaunay triangulation
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 endpoints 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 endpoints 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
L8  Delaunay triangulation
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
L8  Delaunay triangulation
Voronoi diagram
The Voronoi diagram is the dual to the Delaunay triangulation
L8  Delaunay triangulation
Applications of the Voronoi diagram
Interpolation
GIS analyses of airraid 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