Map Projections and Coordinate Systems

1
Map Projections and Coordinate Systems

• Jan-Van Sickle
• Khang-tsung Chang
• ESRI Using ArcMap

2
Coordinate Systems

• The location of a known point (Youghall, Colorado
  Mountains) is known on the ground
• Coordinates
• 1937 lat 40 25 33.504N Long 108 45
  55.378W
• 1997 lat 40 25 33.39258N Long 108 45
  57.78374W
• Elevation
• 1937 2658.2 m
• 1997 2659.6 m

3
How did this happen?

• Datum changed
• In 1937, the lat, long, elevation were based on
  the North American Datum 1927 (NAD27)
• In 1997, the lat, long, elevation it became based
  on the North American Datum 1983 (NAD83)

4
The example highlights the importance of

• Choice of Datum
• On what surface (ellipsoid) points lies
• From what datum (ellipsoid) heights are measured?
• Choice of Coordinate System

5
Datum

• A datum (spheroid) approximates the shape size
  of the Earth in the examined area
• Five Parameters were needed to define a datum
  (e.g., NAD27)
• Latitude of initial point (Meads Ranch)(39 13
  26.686)
• Longitude of initial point (Meads Ranch)(98 32
  30.506)
• Semi-major axis for Clarke 1866 ellipsoid
  (6,378,206.4 m)
• Semi-minor axis for Clarke 1866 ellipsoid (6,
  356,5836.6 m)
• An azimuth (75 28 09.64) from the initial
  point to a reference point (Waldo station)

6
Old New datums

• Old datum fixing the best suited ellipsoid to a
  region using a single point on Earth
• Problems
• Center of the datum did not coincide with center
  of Earth
• Old datum(s) (ellipsoid) work well regionally,
  but not globally
• New datum (WGS84) uses space-based geodesy
  (satellite measurements) to get precise
  measurements of distances to surface of the Earth
  and ultimately define a geocentric datum (center
  of ellipsoid coincides with center of Earth)

7
Datums - Ellipsoid/Spheroid
8
Datums
9
Examples

• NAD27 (North American Datum of 1927)
• Spheroid
• equatorial radius 6,378,206.4 m
• polar radius 6,356,583.8m
• Flattening 1/294.979
• Origin Meades ranch in Kansas

10
Geographic Coordinate Systems

• Based on angles, not distances
• Positions on the Earth surface are represented by
  latitudes longitudes
• Latitude-Longitude
• Degree-Minute-Second (DMS)
• 1 deg 60 min
• 1 min 60 sec
• Decimal Degrees (DD)
• 455230 45.875

11
Meridians - Parallels

• Meridians Lines of equal longitude
• Prime Meridian passes through Greenwich, England
  and has a reading of 0
• Using prime meridian as reference, we measure
  longitude of a point on Earths surface as 0 to
  180 east or west of the prime meridian
• Similarly N to S directions are measured using
  parallels (Equator 0 parallel).

12
Longitude - Latitude

• Longitude Angle between two planes plane passing
  by Prime Meridian and another Meridian passing by
  point of interest
• Latitude Angle between a plane passing by
  Equator and a line joining point of interest to
  center of Earth

13
Meridians - Parallels

• Meridians converge
• Parallels do not

14
Note

• As the poles are approached
• A degree in longitude shrinks (red)
• A degree of latitude expands slightly

15
Cartesian Coordinates (3-D)

• Basic X-Y-Z orthogonal coordinate system
  distances measured from an origin
• 3 Orthogonal Dimensions

16
Polar Coordinates (3-D)

• Allows 3-D representations
• 2 orthogonal angles and a distance
• The North or the East could be used as reference
  directions

17
Moving from 3-D to 2-D

• Location of spatial features on the globe are
  based on geographic coordinate systems (latitude
  and longitude)
• Whereas, locations of map features are based on
  plane coordinate system (x and y coordinates)

18
Cartesian Coordinate System

• Based on distances
• Basic X-Y coordinate system distances measured
  from an origin on a flat surface
• Two-dimensional Cartesian Coordinates are
  important elements in the vast majority of
  Coordinate Systems (e.g., UTM, state plane
  coordinates)

19
Conventions

• Based on Right Angle coordinates
• X Y axes similar units
• X axis east (ve)
• Y axis north (ve)

20
Question

• Does a flat Cartesian datum with two axes
  represent the real world (Earth) well?

21
Answer

• Only, if the Earth was flat
• Thus, across small areas, this approximation
  works well.
• If the area gets to be large, distortions will be
  problematic

22
Question

• Given the coordinates of two points (state Plane
  Coordinates Colorado North Zone)
• Youghall
• Northing Y1 2,414,754.47 ft
• Easting X1 2,090,924.62 ft
• Karns
• Northing Y2 1,418,088.47 ft
• Easting X2 2,091,064.07 ft
• Find the distance between them

23
Answer

• Distance ((X1-X2)2 (Y1-Y2)2)1/2
• Distance 3336.91 ft

24
Polar Coordinates

• Based on angles (measured counterclockwise), and
  distance from center point (origin)
• An origin has to be defined
• The North or the East could be used as reference
  directions

25
Representation
Angle measured from Polar Axis Example (100,120)
26
Differences in Representation

• Cartesian only one pair can represent a point
• Polar
• A point can be represented in many ways

27
Two Coordinate Systems (SPCS)

• Map Projections represent a portion of the actual
  Earth on a plane
• Can you flatten an
• Orange peel?

28
Only if

• The portion is very small
• A map on an orange peel will be distorted if
  attempts are made to flatten it

29
Examples

• State Plane Coordinate Systems (SPCS) Local
  Coordinate Systems
• Flat surface with Cartesian Axes
• Measured positions expressed in northings and
  eastings
• Tangent planes to the Earth surface
• Small section of the earth as with a small
  section of an orange conforms so nearly to a
  plane distortions are minimal

30
Can we peace together the small areas?

• Each plane has a unique coordinate system
• For each system, the scale orientation of axes
  is different
• Joining the peaces together will cause overlaps
  gaps
• Tangent plane map projections have no larger use

31
How can we work on regional global scales?

• We use media that can be flattened
• (2) Use one tangent plane

32
(1) Media that can be flattened Cylindrical
Media

• Start with a surface that is not deformed upon
  flattening

33
Conical Media