Duality between Pairs Duality between Pairs of Incident Cellsof 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

A Primal Set of Objects A Primal Set of Objects and Its Dualand Its Dual

A

B

C

D

a

b

c

d

e

f

1

2

3

4

Diagrammatical Representation Diagrammatical Representation of Adjacencyof Adjacency

S S

s1 s1s2 s2s3 s3s4 s4

( a ) ( b )

The Adjacency between the set object and each element

Path TopologyPath Topology

( a ) an open path topology ( b ) a closed path topology

Network TopologyNetwork 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.

Decomposition of a Disconnected Decomposition of a Disconnected Set into Connected SubsetsSet into Connected Subsets

x1x2

x3x4

y1

y2

y3

S

XY

X Y

S

X Y

( a ) ( b )

( c )

Arrangement of Objects Arrangement of Objects within a Data Structurewithin 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.

File Structures (1)File Structures (1)

• List Structures

S

L B K X E C

Operations for ListsOperations 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

Stacks, Queues and DequesStacks, Queues and Deques

L B K X E C

L B K X E C

L B K X E C

Insert

Delete

Delete

DeleteDelete

Insert

Insert

Insert

( a ) an example stack

( b ) an example queue

( c ) an example deque

Sequential AllocationSequential Allocation （（存存储）储）

Vertex Attributes

s + n

s + 2n

s + 6n

s + 5n

s + 4n

s + 3n

L PL

B PB

K PK

X PX

E PE

C PC

A list Stored in a

Sequential Allocation

Random Access AllocationRandom Access AllocationVertex Attributes

L PL

B PB

K PK

X PX

E PE

C PC

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

Linked AllocationLinked 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 )

Circular ListsCircular ListsL

B

K

H

X

E

C

s + n

s + 2n

s + 3n

s + 4n

s + 5n

s + 6n

s + 7n

Ls + 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

Double ListsDouble 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 sl c l c

k l

ø e

c x

h b

b ø

e h

x k

k l

ø e

c h

h b

b ø

e h

e k

First Address Last Address

Successor Address

Predecessor Address

After Deletion

Before Deletion

• Graphs

File Structures (2)File Structures (2)

ME

NHVT

MA

CT RI

A

BCD

E

( a ) ( b )

File Structures (3)File Structures (3)ELDEST NEXT

VERTEX PREDECESSOR SUCCESSOR SIBLING

A ø B øB A E CC A G DD A ø øE B ø FF B H GG C ø øH F ø II F ø ø

A

B C D

E F G

H I

• Trees

( a ) ( b )

CartogrCartographic Objects aphic Objects and Their Neighborhoods (1)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-

An Area A Region

• 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

CartogrCartographic Objects aphic Objects and Their Neighborhoods (2)and Their Neighborhoods (2)

Interior Region Point

• An exterior region point or boundary point

is a point whose neighborhood lies partially outside the region

CartogrCartographic Objects aphic Objects and Their Neighborhoods (3)and Their Neighborhoods (3)

Exterior Region Point

• 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

CartogrCartographic Objects aphic Objects and Their Neighborhoods (4)and Their Neighborhoods (4)

Exterior Outline

Interior Outline

• 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

CartogrCartographic Objects aphic Objects and Their Neighborhoods (5)and Their Neighborhoods (5)

Arc

Interior Arc Point

• An Exterior Arc Point is one whose neighborhood lies partially outside the domain of the arc and is more commonly called a Node

CartogrCartographic Objects aphic Objects and Their Neighborhoods (6)and Their Neighborhoods (6)

Exterior Arc Point

Or Node

IslandsIslands• If one region completely surrounds

another region or regions, the surrounded region(s) is called an island

R1

Interior Island

R2

R2

Exterior Island

ChainsChains• 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

1112

N1

N2p1 P1

P2

P3Chain C Chain: C

Segment List: 11, 12

Point List: N1, p1, N2

PolygonsPolygons

• 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

Analog Digital

Zero-Dimensional

Objects

Two-Dimensional

Objects

One-Dimensional

Objects

point point

nodenode

line string

outline ring

arc chain

area area

region polygon

Cobounding and Adjacent Relations (1)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.

CpP

sP

P (x, y)

pL sL

• 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 Ci, node N is cobounded by a line segment Li and is adjacent to node Ni

Cobounding and Adjacent Relations (2)Cobounding and Adjacent Relations (2)

N

N1N2

N3

C1C2

C3L1

L3

L2

• 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)Cobounding and Adjacent Relations (3)

L

PP

SP

C

pL sL

• 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)Cobounding and Adjacent Relations (4)

rP

lP

L1

L2 L3

L4

p1

p2

p3

pN sN

lC

rC

C

• 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)Cobounding and Adjacent Relations (5)

P1

P2

P3

P4

C1

C2

C3

C4C5

C6

L1L2

L3

L4L5L6L7

L8

SummarySummary

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.

Questions for Review (1)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?

• 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)Questions for Review (2)