duality between pairs of incident cells pairs of incident cells have a symmetry in their structure...
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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
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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
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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
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Path TopologyPath Topology
( a ) an open path topology ( b ) a closed path topology
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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.
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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 )
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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.
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File Structures (1)File Structures (1)
• List Structures
S
L B K X E C
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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
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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
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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
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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
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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 )
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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
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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
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• Graphs
File Structures (2)File Structures (2)
ME
NHVT
MA
CT RI
A
BCD
E
( a ) ( b )
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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 )
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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
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• 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
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• 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
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• 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
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• 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
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• 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
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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
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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
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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
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Analog Digital
Zero-Dimensional
Objects
Two-Dimensional
Objects
One-Dimensional
Objects
point point
nodenode
line string
outline ring
arc chain
area area
region polygon
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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
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• 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
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• 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
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• 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
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• 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
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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.
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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?
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• 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)