Relative Location of a Point with Relative Location of a Point with Respect to a Straight LineRespect to a Straight Line

(0,0) 5

5

(2, 2)

(4, 5)(0, 5)

(6, 3)-3x

+ 2y

+2

= 0

s = A xt + B yt + C

s < 0

s > 0

Perpendicular Distance between Perpendicular Distance between a Point and a straight Linea Point and a straight Line

(0,0) 5

5

(2, 2)

(4, 5)

(6, 3)

-3x

+ 2y

+ 2

= 0

(4, 0)

d

-3x

+ 2y

+ 1

2 =

0

L L’

s = C - C’ = 2 - 12

MidpointMidpoint

(0,0) 5

5

p (x1 y1)

q (x2 y2)

r (xr yr) xr

yr

(x1 + x2) / 2

(y1 + y2) / 2=

Perpendicular Bisector of Perpendicular Bisector of a Line Segmenta Line Segment

(0,0) 5

5

p (x1 y1)

q (x2 y2)P

X

Y

x1 – x2

y1 – y2

½ [ (x22 + y2

2) – (x12 + y1

2) ]

A

B

C

=

Intersection of Two LinesIntersection of Two Lines

(0,0) 5

5

-3x

+2y

+2 =

0

2x + 4y – 28 = 0

(4, 5)

X

Y

Ax

+ B

y +

C =

0

Ex + Fy + G = 0

(GB – FC) / (FA – EB)

(CE – AG) / (FA – EB)

xi

yi=

Using linear equation form requires a two-step process: first, the line intersection is calculated and then checks are made to determine if this point lies on each line segment

Intersection of Two Intersection of Two Line Segments (1)Line Segments (1)

For two segments, S and S’, connecting points (x1, y1) to (x2, y2) and (x1’, y1’) to (x2’, y2’) respectively, their intersection is determined by solving for h = D1 / D and

h’ = D2 / D, where

D = (x2’ - x1’) (y1 - y2) – (x2 - x1) (y1’ - y2’)

D1 = (x2’ - x1’) (y1’ - y1) – (x1’ - x1) (y2’ - y1’)

D2 = (x2 - x1) (y1’ - y1) - (x1’ - x1) (y1 - y2)

Intersection of Two Intersection of Two Line Segments (2)Line Segments (2)

Angle Bisector of Angle Bisector of Two Intersected LinesTwo Intersected Lines

(0,0)

Y

X

Ax

+ By

+ C

= 0

Ex + Fy + G = 0

Rx + Sy + T = 0

θ

=

R

S

T

A’ + E’

B’ + F’

C’ + G’

Different Angle Bisectors Different Angle Bisectors Depending on the Direction of Depending on the Direction of

the Intersecting Linesthe Intersecting Lines

(0, 0)

(0, 0)(0, 0)

(0, 0)(a)

(b)

(c)

(d)

Measurement of the Length Measurement of the Length of a Stringof a String

Length of a String:

212

11

21 ])()([ ii

n

iii yyxxLength

(x1 y1)

(x2 y2)

(x11 y11)

Measurement of the Area of a Measurement of the Area of a Polygon – Using Triangular Polygon – Using Triangular

DecompositionDecomposition

)()2/1( 11

1 ii

n

iii yxyxArea

(0,0)

Y

X

Vector Product (Vector Product (Review)Review)• Let a, b, c be vectors, a and b

have the same startpoint• Let | c | = | a | | b | sinθ

The direction of c is perpendicular to the plane decided by a and b, and according to the right hand law

• c = a x b = (xi+1 i + yi+1 j) x (xi i + yi j) = xi+1 yi – xi yi+1 where i, j, are the basic unit vectors, i x i = j x j = 0, i x j = k, j x i = -k, k = 1

(0, 0)

(xi+1 yi+1)

(xi yi)

θ

a

b

Measurement of the Area of a Measurement of the Area of a Polygon – Using Trapezoidal Polygon – Using Trapezoidal

DecompositionDecomposition

(0,0)

Y

X

n

iiiii yyxxArea

111 )()(2/1

Area Measurement of a Area Measurement of a Complex PolygonComplex Polygon

(0,0)

Y

X

Area Calculation in a Area Calculation in a Tessellation(Tessellation( 剖分剖分 ) ) of Spaceof Space

Number of Areal Units = (1/2) b + c -1

(a) (b)

SummarySummaryWe have focused on the basics of analytical geometry that comprise the primitive tasks within more complex algorithms in digital cartography. However, the emphasis has been on the coordinate properties of maps as absolute locations and not their relative position within a digital image. In the following chapter, the data organization of digital maps is discussed and topological properties are added to the locational ones.

Questions for Review (1)Questions for Review (1)

(0,0)

Y

X

p (5, 4)

A symbol with the representative point p (5, 4) needs to be rebuilt upside down, reduced in its scale to the half and the point p needs to be placed at a new location (3, 1). Please give the sequence of the matrices for point transformations.

• A straight line was built in such a way that it passes through two points p (2, 2) and q (4, 5) sequentially. In which halfplane does another point t (1, 4) lie with respect to this straight line? What is the perpendicular distance between point t and the line pq?

• We know that one can also build an equation for a perpendicular bisector （中垂线） of a line segment using the midpoint and the negative reciprocal number of the slope of the given line segment. Compared with the method in this class, which one is better? Why?

Questions for Review (2)Questions for Review (2)

Questions for Review (3)Questions for Review (3)

(2, 5)5

5

(5, 4)

(6, 2)

(4, 1)

(3, 2)

(2, 1)

(1, 3)

Please cauculate the perimeter and the area of this polygon.

Data StructuresData Structures• The central object in a digital cartographic

processing system is the digital representation

• Real World → Data Model → Data Structure → Hardware Storage

• Data structures form the basis of software design

• File structures are representation of the objects in hardware storage

• Topology: a branch of geometry, concerned with a particular set of geometrical properties – those that remain invariant under topological transformation

• Topological Transformation: the transformation induced by stretching on a rubber sheet is called a topological transformation or homeomorphism

Basic Topological Concepts (1)Basic Topological Concepts (1)

Basic Topological Concepts (2)Basic Topological Concepts (2)

A topological space is a set of objects, T, together with a collection of open subsets of T, denoted as { t }, called a topology on T, that satisfy the following axioms:

1) the intersection of any finite collection of open subsets is open

2) the union of any collection of open subsets is open

NeighborhoodNeighborhoodA subset S is a neighborhood of any object, s, that is an element of the set T, if there is a member R that is an element of { t } such that s is an element of R and R is a subset of S.

R

ST

s

{ t }

E

Neighborhood of an Neighborhood of an ee Ball BallIn Euclidean Rn space, a subset S is a special neiborhood of the point p called an ee-ball-ball if all points of if all points of Rn within an epsilon distance ( a very small distance) of p are contained in S. An ee-ball -ball associated with aassociated with any point in Hubei Province would also only ny point in Hubei Province would also only contain points that are part of Hubei. contain points that are part of Hubei.

Set S and Its ComplementSet S and Its Complement

S

S’

Closure SClosure S--

S S-

The closure S- of a subset S is the intersection of all closed subsets that contain S.

Interior of SInterior of S

S°

The interior of a subset S, denoted as S°, is the set of all the elements of S for which S is a neighborhood.

Boundary of SBoundary of S

jSThe boundary of a subset S, denoted as jS, is the set of all points in the intersection of the closure of S and the complement S’.

Near PointNear Point

Let S be a topological space. Then S has a set of neighborhoods associated with it. Let X be a subset of points in S and x an individual point in S. Define x to be near X if every neighborhood of x contains some point of X.

Near to C

Neighborhood seperating from C

Not near to C

Open unit circle C

Connected and Connected and Disconnected SetsDisconnected Sets

Let S be a topological space and X be a subset of points of S. Then X is connected if whenever it is partitioned into two non-empty disjoint subsets, A and B, then either A contains a point near B or B contains a point near A, or both.

( a ) ( b ) ( c )

CellsCells

Connected subsets are also called cells. In a 2-D surface, the primary connected subsets are 0-cells (points), 1-cells (lines) and 2-cells (areas).

0-cell 1-cell 2-cell

Topological Topological Incident RelationshipIncident Relationship

A pair of these objects having different dimensions is incident with each other if their intersection is not a null set.

NE1

E2

E3

E4

A

E1E2E3

( a )

( b )

Topological Topological Adjacent RelationshipAdjacent Relationship

Two objects of the same dimension are adjacent to each other if they share a bounding object.

E

N1

N2 EA1

A2

( a ) ( b )

Topological Topological Inclusive RelationshipInclusive Relationship

If the union of set A and set B equals set A, then set A includes set B. The dimension of set A is equal or greater than set B.

A

B1B2

B3

• What is topology ? What is a topological transformation? What is a topological space? What is a topological property?

• What is the concept of neighborhood of a subset S to an object s?

• What is the definition of a near point x to a subset X?

• Do the definitions of connected and disconnected sets accord with our intuition?

Questions for Review (4)Questions for Review (4)

• Between which kinds of objects does the incident relationship exist? Which kind of object bounds which kind of objects? What does the word ‘cobound‘ mean?

• Between which kinds of objects does the adjacent relationship exist?

• If set A includes set B, which has the dimension 1, what kinds of dimension can set A be?

Questions for Review (5)Questions for Review (5)