L1-Spatial Concepts L1 - Spatial Concepts

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1 L1 - Spatial Concepts NGEN06(TEK230) Algorithms in Geographical Information Systems

2 Aim Understand the relationship between spatial queries and mathematical concepts. Know how topological relationships are defined in GIS. Get knowledge about the relationshop between type of queries and suitable methods of storing geographic data.

3 Concept of Space Object model > set of objects (vector) Field-based model -> set of locations with properties (raster or grid) How do we store geographic data (geometry)?

4 Content 1. Spatial queries 2. Set-based queries 3. Topological queries 4. Graph-based queries 5. Metric queries 6. Euclidean queries 7. Storing relationships or deriving in real-time?

5 Spatial queries Set-based query: Is Uganda a country in Africa? Africa

6 Spatial Queries Topological queries Which countries are neighbours to Uganda? Africa Topological Relationships

7 Spatial Queries Graph-based queries How long is the traveling time from Uganda to Egypt? Africa Relationships between elements distance between elements

8 Spatial Queries Metric queries How long is the traveling distance from Uganda to Egypt? Africa Metrics

9 Spatial Queries Euclidean queries What is the area of Uganda?

10 Do we always need coordinates to answer spatial queries? L1-Spatial Concepts

11 We are not always relying on coordinates (or the Euclidean space) in GIS; by storing set-based, topological and graph-based data explicitely (without using coordinates) we can answer many spatial queries without considering coordinates.

12 Set-based queries Countries_in_africa ={Egypt, Uganda,...} Z={..., -2, -1, 0 1 2,...} R= the real numbers R 2 = R x R E= {x x R 2 0<x1<100, 0< x2<100 }

13 Set algebra Set Operations: Union Intersection Complement Logical Operators: OR AND NOT

14 Data structure to store set based data explicitely Is Uganda a country in Africa? Table: Countries_in_Africa Country Uganda Capital Kampala Etc. Egypt Cairo Nigeria Lagos Which standard SQL query can be used to answer this question?

15 Topological queries Topology is derived from the Greek and means the science of position. Topological Space: Set and a number of subsets (which follow certain rules)

16 Topological Relationships Using a rubber sheet (where all points, lines and areas are drawn), topological relationships are the properties that remain between the points, lines and areas for all possible kinds of deformation of the rubber sheet (except tearing). <- Rubber sheet transformation Examples: Point is inside a polygon Two lines intersect Not a topological relationship: an object is close to another (spatial relationship)

17 Topological transformation 1) There should be one-to-one correspondence between the elements in the original and transformed set (bijection). 2) Two points that are connected in the original set should also be connected in the transformed set.

18 Topological Relationships L1-Spatial Concepts

19 4-intersection model It is defined using the boundary and the interior or objects. This terminology is defined for cells (2 dimensional, connected sets without holes - closed) in R 2. A A A Connected Not Connected

20 4-intersection model Definitions of boundary and interiors of conected objects (A) in R 2 Boundary ( A ) Interior (A o ) Point Line Area The empty set The end points The line(s) that constitute the border of the area Point The line apart from the end points The area inside the border lines

21 Definitions of topological relationships (using the 4-intersection model) A disjoint B A contains B A meets B A covers B A inside B A equals B A coveredby B A overlaps B

22 Definitions of topological relationships (using the 4-intersection model) A B A 0 B 0 A B 0 A 0 B Topological relationship Ø Ø Ø Ø A disjoint B Ø Ø Ø Ø A meets B Ø Ø Ø Ø A equals B Ø Ø Ø Ø A inside B Ø Ø Ø Ø A coveredby B Ø Ø Ø Ø B inside A Ø Ø Ø Ø A covers B Ø Ø Ø Ø A overlaps B Ø = empty set Ø = not empty set

23 Other models of topological relationships 9-intersection model (DE-9IM) The 4-intersection model is actually not that suitable for line and/or point objects. DE-9IM was proposed to be an international standard by the International Standard Organization ISO Defines topological relationships using interior, exterior and boundary of objects. Program specific models

24 L1-Spatial Concepts Link-node structure

25 Graph-based queries Also called network queries They consider distances Shortest (fastest) route is a typical example. A graph space contains 2 things: A set Rules for distances between elements in the set.

26 Graph-based queries Traveling time between airports. The points (A, B,..., H) are airports (i.e. elements in the set airports). The edges denote that there are flight routes between the airports.

27 Graph-based and metric queries In a graph-based query there is no restriction on the distances between the elements in the set. Metric query is a sub-set of a graph-based query that set constrainst on the distances-> they must obey the rules of a metric.

28 Metric query A metric (d) is a distance measure between elements in a set. It has to obey the 3 following rules (where p,q, and r are elements in the set): 1. d(p,q)>=0, d(p,q)=0 p=q 2. d(p,q)=d(q,p) (symmetry) 3. d(p,q)<=d(p,r)+d(r,q) (triangle inequality)

29 Metric query There are an infinite number of metrics. Two of the most common metrics (in R 2 ) in GIS are: 1) Euclidean distance: 2 d ( p, q) = ( x - x ) + ( y p q p - y q ) 2 2) Manhattan distance: d(p,q)= xp- xq + yp- yq

30 Data structures to store graph-based data explicitely Graphs can be stored as matrixes. L1-Spatial Concepts Not recommended if there are much input data Relational databases Cannot handle recursive programming Lists and tree structures

31 Euclidean queries What is the distance between a building and a road? -> Require coordinates to be stored.

32 Storing relationships or deriving in real time? In some cases the relationships can be derived from stored coordinate data. Traveling distance (but not traveling time) Storing relationships: Advantages: It saves processing time Could enhance the quality of the answer Disadvantages: It takes more space in memory It entails redundancy (storing information twice)

33 Storing relationships or deriving in real time? A few things you should consider before you decide what relationships should be stored explicitely: What type of queries will the database serve? How will the database be mantained? Will the database will be connected to other databases? Will the deriving relationships be treated as own objects?