1. Spatial Databases - Topology
Spring, 2015
Ki-Joune Li

2. Spaghetti Model The Most Simple Representation by Vector Model
Point Table Line Table
P1 (30,70) L1
P4 (20,60) L4 L5 L3
P3 (20,50)
P2 (40,50) L2
Point# x y p1 30 70 p2 40 50 p3 20 p4 60
Line # x1 y1 x2 y2 L1 30 70 40 50 L2 20 L3 60 L4 L5

3. Spaghetti Model Example: DXF Program Assignment #1
A CAD Data Format by Spaghetti Model
Representation of geometric properties not spatial or geographic ones
Program Assignment #1
Decode given data file in DXF
Define schema of Oracle spatial DB
Insert objects into DB
Programming Environment: JDBC + Oracle SDO
Due data: by April 23th

4. Problem of Spaghetti Model: Example
When P1 moves to (40,80),
Ambiguity
P1 (40,80)
P1 (30,70) L4 L1
P4 (20,60) L5 ? L3
40, 80
P3 (20,50)
P2 (40,50) L2
Point# x y p1 30 70 p2 40 50 p3 20 p4 60
Line # x1 y1 x2 y2 L1 30 70 40 50 L2 20 L3 60 L4 L5
40, 80

5. Problem of Spaghetti Model: Example
To make it clear
Relationship between Point and Line
Line # ps pe L1 p1 p2 L2 p3 L3 p4 L4 L5
Point# x y p1 30 70 p2 40 50 p3 20 p4 60 p5
40, 80
No change of line table
P2 (40,50)
P3 (20,50)
P4 (20,60) L5 L2 L3 L1 L4
P1 (40,80)
P5 (30,70)

6. Problem of Spaghetti Model: Example
No clear information about the connection without additional data
L1:L3 ? or L1:L2 P2
L1 : 3,300V P1
L2 : 220V P3 P5
L4 : 220V
L3 : 3,300V P4
Point# x y p1 30 70 p2 40 50 p3 20 p4 60 p5 80
Line # x1 y1 x2 y2 L1 30 70 40 50 L2 L3 20 L4 60
Point# x y p1 30 70 p6 p2 40 50 p3 20 p4 60 p5
Line # ps pe L1 p1 p2 L2 p6 p5 L3 p4 L4 p3
 Need Topological Information

7. Topology Topology Topological Equivalence (Homeomorphism)
Invariant properties during elastic transformation (rubber sheeting)
Relationship between Spatial Objects
Topological Equivalence (Homeomorphism) l1 l2 l3 l4 l5 l1 l2 l3 l4 l5

8. Some Definitions of Topology
Mathematical Definition based on Point Set Theory
An area: Defined as an infinite set of points
Neighbor of p: set of all points within a unit disc of p
Boundary: Set of all points that have neighbors within A and neighbors within AC at the same time
Interior and Exterior
Neighbor of p p
Boundary of A
neighbors in A,
neighbors in AC A AC

9. Examples of TopologiesB A B A B
Disjoint(A,B)
Meet(A,B)
Overlap(A,B) A B B
A, B A
Some complicated cases ?
Cover(A,B) or
CoveredBy(B,A)
Contain(A,B) or
Inside(B,A)
Equal(A,B) A
8 Relationships B
Topology between Lines ?

10. Representation of Topology by 9-IM
9-Intersection Model
Example
Extensions of 9-IM
A + B +
A 0  B +
A -  B +
A + B 0
A 0  B 0
A -  B 0
A+  B -
A 0  B -
A -  B -
R(A,B)= 1 A B
R(A,B)=

11. Built-In Topology: Topology Levels
By VPF (Vector Product Format): A Military Standard
VPF  NATO Standard  CEN/TC287  ISO/TC211
Level 0: No topology (Spaghetti Model)
Level 1
Level 2
Level 3: Face topology
Less Topology
More Topology

12. Topology Level 1: Connectivity
Description of connectivity between lines
l1 is connected with l2.
Example
Pipeline network
No Planarity restriction l1 l2 l3 p1 p2 p3

13. Topology Level 2: Connectivity + Planarity
Topology Level 1 + Planarity Condition
Planarity
Only ONE object at a position
No overlapping is allowed
Example: ? p3 p3
l21
l12 p2 p2
l11 p5 l1 p1 p1 l2
l22 p4 p4
Level 2
Level 1 (not Level 2)

14. Topology Level 3: Face Topology
Adjacency between faces
Left / Right Face of a line
Example B C A
Original Boundary B C A B C A
Without Face Topology
With Face Topology

15. How to maintain the Planarity of Faces
Example 1
Example 2
Original Faces F2
New Face to respect
the planarity F1 F3 F2 F1
New Inserted Line

16. Simple Data Modeling of Topology
Starting Pt 1..1
Point (Node)
(X,Y)
Line (Edge)
Point String
Connected Line 1..*
Ending Pt 1..1
Connected Line 1..*
Topology Level 1, 2
Left Face 1..1
Right Face 1..1
Face
Boundary 1..*
Boundary 1..*
Topology Level 3
Face with Holes ?
Outer Ring
Inner Ring