CartographiCartographic Objectsc Objects

Digital Images of a Map

Vector Data Model

Raster Data Model

Vector Model vs Raster ModelVector Model vs Raster Model

Vector Data Model Raster Data Model

• Mapping space in a plane is a continuous set of points

• The basic unit of observation corresponds to a line on a map

• Mapping space is filled by a discrete set of points called a two-dimensional lattice

• The basic unit is a unit of space within a mesh

Requirements for Definition of Requirements for Definition of Digital Cartographic ObjectsDigital Cartographic Objects

• Can combine the spatial properties of absolute location and relative location

• Must be modular so that lower dimensional objects can be used to define higher dimensional objects

• Can be studied in planar, hyperbolic and elliptic geometry

• Must be expandable at a later date

Definitions from NCDCDSDefinitions from NCDCDS• Zero-Dimensional Objects

- Point, Endpoint, Lattice Point• One-Dimensional Objects

- Line, Outline, Straight Line Segment, String, Ring

• Two-Dimensional Objects

- Area, Region, Background Region, Polygon, Background Polygon, Pixel, Cell

Point (0-D)Point (0-D)

A zero-dimensional object that specifies an absolute location in a two-dimensional space

Endpoint (0-D)Endpoint (0-D)

A point that marks the terminus of a one-dimensional positional object.

Lattice Point (0-D)Lattice Point (0-D)

A zero-dimensional object that specifies an absolute location in a tessellation of two-dimensional space

Line (1-D)Line (1-D)

A locus of points that forms a nonintersecting curve in a two-dimensional space terminating at two endpoints.

Outline (1-D)Outline (1-D)

A line whose two endpoints have the same absolute location.

A locus of points that does not change its orientation in a two-dimensional space terminating at two endpoints.

Alias: line segment

Straight Line Segment (1-D)Straight Line Segment (1-D)

A sequence of line segments that intersect once and only once at each line segment endpoint excluding two segment endpoints that form the endpoints of the string.

String (1-D)String (1-D)

A sequence of line segments that intersect once and only once at each line segment endpoint.

Ring (1-D)Ring (1-D)

Directed 1-D objects imply movement from the start point to the terminus point. The left and right sides of a directed object is uniquely defined.

Directed Objects Directed Objects (1-D)(1-D)

Left Side

Right Side

The interior of a continuous 2-D object (may include rings).

Area (2-D)Area (2-D)

An area having one or more outer outlines and zero or more nonintersecting inner outlines.

Region (2-D)Region (2-D)

The complement to the set of all regions.

Background region (2-D)Background region (2-D)

An area bounded by one outer ring and zero or more nonintersecting inner rings.

Polygon (2-D)Polygon (2-D)

The complement to the set of all polygons.

Background Polygon (2-D)Background Polygon (2-D)

Pixel (2-D)Pixel (2-D)A regularly shaped 2-D picture element that is the smallest nondivisible element of an image.

A 2-D object that represents an element of a regular tessellation of space. The most common cells are rectangles, squares, triangles, and hexagons.

Cell (2-D)Cell (2-D)

Basic Analytic GeometryBasic Analytic Geometry

Because analytic geometry can describe the locus of the movement of points, it is widely used in digital cartography.

Number ScaleNumber Scale

The set of all real numbers

the set of all points on a number scale

P

r-2-3

Cartesian CoordinatesCartesian Coordinates

If two number scales are drawn at a right angle with respect to each other, these number scales are called coordinate axes – one for X-axis and another for Y-axis. Any point on a 2-D plane has Cartesian coordinates (x,y).

The two axes divide the coordinate plane into four quadrants (I, II, III, and IV). There is a one-to-one relationship between the position of a point on a plane and a pair of real numbers as Cartesian coornidates.

Cartesian Coordinate SystemCartesian Coordinate System

+-

+

-

(0,0)

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Hardware SpaceHardware Space

The resolution of

any computer is

limited. And for

the display

device, it is also

the case.

Rm

Rm+1

R2m

R2m+1

R3m

R(n-1)m+1

Rnm

P1

P2

P3

Pn

VectorVector

A vector is an n-dimensional force emanating from an origin point and having a direction and a fixed length or magnitude.

Some Example VectorsSome Example Vectors

Direction

p

q

X

Y

0

Halfplane Partition of a PlaneHalfplane Partition of a Plane

X

Y

CounterclockwiseHalfplane

p

q

Lefthand

Righthand

Component Values of a VectorComponent Values of a Vector

x

y

x -x2 1

y -y2 1

X

Y

(0,0) x1x2

X

y1

y2

y

P x y( , )1 1

Q(x , y )2 2

Vector Starting from OriginVector Starting from Origin

X

Y

(0, 0)

X1

Y1

X2

y2

p (x , y )1 1 1

p (x , y )2 2 2

Questions for ReviewQuestions for Review• How many basic data models are there to represent

the digital images on a map?(a vector and raster data model )

• According to NCDCDS, which objects belong to the 0-D objects?

• Which objects belong to the 1-D objects?• Which objects belong to the 2-D objects?• Can you explain the incorrespondency between the

real numbers and the positions of the points in the hardware space?