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Zero Dimensional

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					 Cartographic Objects

 Digital Images of a Map


Vector Data   Raster Data
   Model         Model
Vector Model vs Raster Model
 Vector Data Model       Raster Data Model

• Mapping space in a    • Mapping space is
plane is a continuous   filled by a discrete set
set of points           of points called a two-
                        dimensional lattice
• The basic unit of
observation             • The basic unit is a
corresponds to a line   unit of space within a
on a map                mesh
  Requirements for Definition of
   Digital 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 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)
A zero-dimensional object
that specifies an absolute
location in a two-
dimensional space
    Endpoint (0-D)
A point that marks the
terminus of a one-
dimensional positional
object.
  Lattice Point (0-D)
A zero-dimensional object
that specifies an absolute
location in a tessellation
of two-dimensional space
         Line (1-D)
A locus of points
that forms a
nonintersecting
curve in a two-
dimensional
space terminating
at two endpoints.
     Outline (1-D)
A line whose
two endpoints
have the same
absolute
location.
Straight Line Segment (1-D)
A locus of points
that does not
change its
orientation in a
two-dimensional
space terminating
at two endpoints.
Alias: line segment
          String (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.
         Ring (1-D)
A sequence of
line segments
that intersect
once and only
once at each
line segment
endpoint.
  Directed Objects (1-D)
Directed 1-D objects
imply movement from
the start point to the    Right   Left
terminus point. The       Side    Side
left and right sides of
a directed object is
uniquely defined.
        Area (2-D)
The interior
of a
continuous
2-D object
(may
include
rings).
       Region (2-D)
An area having
one or more
outer outlines
and zero or
more
nonintersecting
inner outlines.
 Background region (2-D)
The
complement
to the set of
all regions.
       Polygon (2-D)
An area
bounded by
one outer ring
and zero or
more
nonintersecting
inner rings.
Background Polygon (2-D)
The
complement
to the set of
all polygons.
         Pixel (2-D)
A regularly
shaped 2-D
picture element
that is the
smallest
nondivisible
element of an
image.
            Cell (2-D)
A 2-D object that
represents an
element of a regular
tessellation of space.
The most common
cells are rectangles,
squares, triangles,
and hexagons.
Basic Analytic Geometry
Because analytic
geometry can describe
the locus of the
movement of points, it is
widely used in digital
cartography.
              Number Scale
     The set of all real numbers
     the set of all points on a number scale

                                           P

-3       -2                                r
 Cartesian 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).
 Cartesian Coordinate System
The two axes divide the                  +
coordinate plane into
four quadrants (I, II, III,   Qua dra nt     Quadrant
                                  II             I
and IV). There is a one-
to-one relationship         -                           +
between the position of                    (0,0)
a point on a plane and        Qua dra nt     Qua dra nt
a pair of real numbers             III           IV
as Cartesian
coornidates.
                                       -
Hardware Space                      P1
                       Rm
 The resolution of     Rm + 1
                                    P2
 any computer is
                      R2 m
 limited. And for     R2m+ 1

 the display                        P3
                      R3 m
 device, it is also
                      R(n-1)m + 1
 the case.                          Pn
                      Rnm
         Vector
A vector is an n-dimensional
force emanating from an
origin point and having a
direction and a fixed length
or magnitude.
Some Example Vectors
  Y


                q

               Dire c tion


        p

  0                 X
Halfplane Partition of a Plane
      Y

          Counte r
          c loc kwise
          Ha lfpla ne     q

          Le ftha nd


            p

                        Rightha nd


                                     X
     Component Values of a Vector
    Y
y2                       Q(x2 , y2 )       x   x2 -x1
                                           y   y2 -y1
y


    y1        P( x1 , y1 )



(0,0)                    x2
         x1                            X
                 X
Vector Starting from Origin
                         Y

                                      p 1 (x1 , y 1 )
  p 2 (x2 , y2 )
                              Y1
                    y2

                             (0, 0)          X
                   X2          X1
          Questions 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?