1. Duality between Pairs of Incident Cells Pairs of incident cells have a symmetry in their structure called duality Every occurrence of a primal object or relation can be replaced by its corresponding dual object or relation A 0-cell is a dual to a 2-cell and a 1- cell is a dual to a 1-cell

2. A Primal Set of Objects and Its Dual A B C D a b c d e f 1 2 3 4

3. Diagrammatical Representation of Adjacency SS s1s1 s1s1 s2s2 s2s2 s3s3 s3s3 s4s4 s4s4 ( a )( b ) The Adjacency between the set object and each element

4. Path Topology ( a ) an open path topology( b ) a closed path topology

5. Network Topology For a connected set S, if one subset is adjacent to three or more other subsets, then the collection forms a network topology on the set S.

6. Decomposition of a Disconnected Set into Connected Subsets x1x1 x2x2 x3x3 x4x4 y1y1 y2y2 y3y3 S X Y XY S XY ( a )( b ) ( c )

7. Arrangement of Objects within a Data Structure The arrangement of objects within a data structure is based in part on the bounding, cobounding and adjacency relations that exist between pairs of objects in a set.

8. File Structures (1) List Structures S LBKXEC

9. Operations for Lists a) Access to the jth vertex b) Search the list with a certain value c) Determine the number of vertices d) Make a copy of the list e) Insert a new vertex before the jth f) Delete the jth vertex g) Merge two or more lists into one h) Split a list into two or more i) Sort the vertices based on some values

10. Stacks, Queues and Deques LBKXEC LBKXEC LBKXEC Insert Delete Insert ( a ) an example stack ( b ) an example queue ( c ) an example deque

11. Sequential Allocation （存储） Vertex Attributes s + n s + 2n s + 6n s + 5n s + 4n s + 3n LPLPL BPBPB KPKPK XPXPX EPEPE CPCPC A list Stored in a Sequential Allocation

12. Random Access Allocation Vertex Attributes LPLPL BPBPB KPKPK XPXPX EPEPE CPCPC b c e k l x l b k x e c s + 1 s + 2 s + 3 s + 4 s + 5 s + 6 A List Stored in a Random Access Allocation

13. Linked Allocation L B K H X E C s + n s + 2n s + 3n s + 4n s + 5n s + 6n s + 7n b c e k l x h l b k h x e c B C E K L X H s + 1 s + 2 s + 3 s + 4 s + 5 s + 6 s + 7 b c e k l x h B C E K L X H s l b k h x e c ø ( a ) ( b ) ( c )

14. Circular Lists L B K H X E C s + n s + 2n s + 3n s + 4n s + 5n s + 6n s + 7n L s + 8n b c e k l x h l b k h x e c B C E K L X H s + 1 s + 2 s + 3 s + 4 s + 5 s + 6 s + 7 ls + 8 b c e k l x h B C E K L X H s l b k h x e c l

15. Double Lists b c e k l x h b c e k l x h B C E K L X H B C E K L X H s slclc kl ø e cx hb b ø eh xk kl ø e ch hb b ø eh ek First AddressLast Address Successor Address Predecessor Address After Deletion Before Deletion

16. Graphs File Structures (2) ME NHVT MA CTRI A BCD E ( a )( b )

17. File Structures (3) ELDESTNEXT VERTEXPREDECESSORSUCCESSORSIBLING AøBø BAEC CAGD DAøø EBøF FBHG GCøø HFøI IFøø A BCD EFG HI Trees ( a )( b )

18. Cartographic Objects and Their Neighborhoods (1) An Area (an open 2-cell) is an open set of points on a manifold ( 族 ) having a graph topology A Region (a closed 2-cell) is the closure of this set of points S-S- S-S- An AreaA Region

19. The Point Neighborhood for a given point in a region is its ∊ -ball on a 2-D surface An Interior Region Point is one whose neighborhood is completely contained within the region Cartographic Objects and Their Neighborhoods (2) Interior Region Point

20. An exterior region point or boundary point is a point whose neighborhood lies partially outside the region Cartographic Objects and Their Neighborhoods (3) Exterior Region Point

21. An Exterior Outline is a circular list of boundary points on the outer extremity of the region An Interior Outline is a circular list of boundary points on an inner extremity of the region Cartographic Objects and Their Neighborhoods (4) Exterior Outline Interior Outline

22. An Arc (1-cell) is the list of exterior points formed by the nonempty intersection of two regions An Interior Arc Point is one whose neighborhood is completely contained in the domain of the arc Cartographic Objects and Their Neighborhoods (5) Arc Interior Arc Point

23. An Exterior Arc Point is one whose neighborhood lies partially outside the domain of the arc and is more commonly called a Node Cartographic Objects and Their Neighborhoods (6) Exterior Arc Point Or Node

24. Islands If one region completely surrounds another region or regions, the surrounded region(s) is called an island R1 Interior Island R2 Exterior Island

25. Chains There are an infinite number of points in an arc, an arc is caricaturized in digital representation by a finite list of line segments called a chain 11 12 N1 N2 p1 P1 P2 P3 Chain C Chain: C Segment List: 11, 12 Point List: N1, p1, N2

26. Polygons The caricaturized representation of a region is called a polygon which consists of at least one exterior ring and zero or more interior rings and will be adjacent to other polygons

27. AnalogDigital Zero- Dimensional Objects Two- Dimensional Objects One- Dimensional Objects point node linestring outlinering arcchain area regionpolygon

28. Cobounding and Adjacent Relations (1) A simple point p contained within chain C is cobounded by a predecessor line segment pL and a successor segment sL. C pP sP P (x, y) pL sL

29. A node N is cobounded by a circular list of chain that can be sequenced in a counter-clockwise direction around it. For each chain C i, node N is cobounded by a line segment L i and is adjacent to node N i Cobounding and Adjacent Relations (2) N N1 N2 N3 C1 C2 C3 L1 L3 L2

30. A Line segment L is contained within chain C. It is also cobounded by a predecessor segment pL and point PP and a successor segment sL and point SP Cobounding and Adjacent Relations (3) L PP SP C pL sL

31. A chain C is alternatively equivalent to a list of line segment or a list of points. It is cobounded by a precessor node pN and a successor node sN Cobounding and Adjacent Relations (4) rP lP L1 L2 L3 L4 p1 p2 p3 pN sN lC rC C

32. A ring R bounds polygon P; as one moves clockwise along ring R, polygon P always lies to its right, vice versa. Cobounding and Adjacent Relations (5) P1 P2 P3 P4 C1 C2 C3 C4 C5 C6 L1 L2 L3 L4L5 L6 L7 L8

33. Summary These cartographic objects and their topological relations form the basis for the representation of space in different vector data model. These data models are translated into data structures for organizing the data elements of a geographic base map in a machine environment. The following section examines alternative topological models and their corresponding data structures.

34. Questions for Review (1) What is the diagrammatical representation for the relationship of adjacency? What are the operations facilitated by the list structure? What is the advantage of the random access allocation compared to the sequential allocation? How is the process implemented when one inserts a vertex in a linked list?

35. What is the relations among lists, trees and graphs? How can one represent the data model of a tree in data structure? What are the denotations of 0-cell, 1- cell, and 2-cell objects in analog and digital environments? Questions for Review (2)