Compiled with assistance from Vicki Chmill, University of
California, Santa Barbara
NOTES
This unit needs many overhead illustrations. Of course,
the best figures are in commercially published books. To
avoid copyright infringements, we have not included the
masters for overheads needed here. Instead, you are directed
to the References which lists several basic texts with good
illustrations. We have included suggested overheads giving
references and specific page numbers to help you locate
suitable figures.
UNIT 27 - MAP PROJECTIONS
Compiled with assistance from Vicki Chmill, University of
California, Santa Barbara
A. INTRODUCTION
- a map projection is a system in which locations on the
curved surface of the earth are displayed on a flat sheet
or surface according to some set of rules
- mathematically, projection is a process of transforming
global location (j,l) to a planar position (x,y) or (r,q)
- for example, the transformations for Mercator
projection are:
x = l
y = loge tan(p/4 + j/2)
Relevance to GIS
- maps are a common source of input data for a GIS
- often input maps will be in different projections,
requiring transformation of one or all maps to make
coordinates compatible
- thus, mathematical functions of projections are
needed in a GIS
- often GIS are used for projects of global or regional
scales so consideration of the effect of the earth's
curvature is necessary
- monitor screens are analogous to a flat sheet of paper
- thus, need to provide transformations from the
curved surface to the plane for displaying data
B. DISTORTION PROPERTIES
- angles, areas, directions, shapes and distances become
distorted when transformed from a curved surface to a
plane
- all these properties cannot be kept undistorted in a
single projection
- usually the distortion in one property will be kept
to a minimum while other properties become very
distorted
Tissot's Indicatrix
- is a convenient way of showing distortion
- imagine a tiny circle drawn on the surface of the globe
- on the distorted map the circle will become an ellipse,
squashed or stretched by the projection
- the size and shape of the Indicatrix will vary from one
part of the map to another
- we use the Indicatrix to display the distorting effects
of projections
Conformal (Orthomorphic)
Reference: Mercator projection (Strahler and Strahler 1987,
p. 15)
- a projection is conformal if the angles in the original
features are preserved
- over small areas the shapes of objects will be
preserved
- preservation of shape does not hold with large
regions (i.e. Greenland in Mercator projection)
- a line drawn with constant orientation (e.g. with
respect to north) will be straight on a conformal
projection, is termed a rhumb line or loxodrome
- parallels and meridians cross each other at right angles
(note: not all projections with this appearance are
conformal)
- the Tissot Indicatrix is a circle everywhere, but its
size varies
- conformal projections cannot have equal area properties,
so some areas are enlarged
- generally, areas near margins have a larger scale
than areas near the center
Equal area (Equivalent)
Reference: Lambert Equal Area projection (Maling 1973, p.
72)
- the representation of areas is preserved so that all
regions on the projection will be represented in correct
relative size
- equal area maps cannot be conformal, so most earth angles
are deformed and shapes are strongly distorted
- the Indicatrix has the same area everywhere, but is
always elliptical, never a circle (except at the standard
parallel)
Equidistant
Reference: Conic Equidistant projection (Maling 1973, p.
151)
- cannot make a single projection over which all distances
are maintained
- thus, equidistant projections maintain relative distances
from one or two points only
- i.e., in a conic projection all distances from the
center are represented at the same scale
C. FIGURE OF THE EARTH
- a figure of the earth is a geometrical model used to
generate projections; a compromise between the desire for
mathematical simplicity and the need for accurate
approximation of the earth's shape
- types in common use
1. Plane
- assume the earth is flat (use no projection)
- used for maps only intended to depict general
relationships or for maps of small areas
- at scales larger than 1:10,000 planar representation
has little effect on accuracy
- planar projections are usually assumed when working with
air photos
2. Sphere
- assume the earth is perfectly spherical
- does not truly represent the earth's shape
3. Spheroid or ellipsoid of rotation
Reference: Ellipsoid of rotation (Maling 1973, p. 2)
- this is the figure created by rotating an ellipse about
its minor axis
- the spheroid models the fact that the earth's diameter at
the equator is greater than the distance between poles,
by about 0.3%
- at global scales, the difference between the sphere and
spheroid are small, about equal to the topographic
variation on the earth's surface
- with a line width of 0.5 mm the earth would have to
be drawn with a radius of 15 cm before the two
models would deviate
- the difference is unlikely to affect mapping of the
globe at scales smaller than 1:10,000,000
Accuracy of figures used
- the spheroid is still an approximation to the actual
shape
- the earth is actually slightly pear shaped, slightly
larger in the southern hemisphere, and has other
smaller bulges
- therefore, different spheroids are used in different
regions, each chosen to fit the observed datum of
each region
- accurate conversion between latitude and longitude
and projected coordinates requires knowledge of the
specific figures of the earth that have been used
- the actual shape of the earth can now be determined quite
accurately by observing satellite orbits
- satellite systems, such as GPS, can determine latitude
and longitude at any point on the earth's surface to
accuracies of fractions of a second
- thus, it is now possible to observe otherwise
unapparent errors introduced by the use of an
approximate figure for map projections
D. GEOMETRIC ANALOGY
Developable surfaces
- the most common methods of projection can be conceptually
described by imagining the developable surface, which is
a surface that can be made flat by cutting it along
certain lines and unfolding or unrolling it
Reference: Developable surfaces (Maling 1973, pp. 55-57)
- the points or lines where a developable surface touches
the globe in projecting from the globe are called
standard points and lines, or points and lines of zero
distortion. At these points and lines, the scale is
constant and equal to that of the globe, no linear
distortion is present
- if the developable surface touches the globe, the
projection is called tangent
- if the surface cuts into the globe, it is called secant
- where the surface and the globe intersect, there is
no distortion
- where the surface is outside the globe, objects
appear bigger than in reality - scales are greater
than 1
- where the surface is inside the globe, objects
appear smaller than in reality and scales are less
than 1
Reference: Projection equations (Maling 1973, pp. x-xi and
234-245)
- note: symbols used in the following:
l - longitude
j - latitude
c - colatitude (90 - lat)
h - distortion introduced along lines of
longitude
k - distortion introduced along lines of latitude
(h and k are the lengths of the minor and major
axes of the Indicatrix)
- commonly used developable surfaces are:
1. Planar or azimuthal
- a flat sheet is placed in contact with a globe, and
points are projected from the globe to the sheet
- mathematically, the projection is easily expressed as
mappings from latitude and longitude to polar coordinates
with the origin located at the point of contact with the
paper
- formulas for stereographic projection (conformal)
are:
r = 2 tan(c / 2)
q = l
h = k = sec2(c / 2)
References:
Azimuthal projections (Strahler and
Strahler 1987, p. 13, Robinson et al 1984, p. 102)
- stereographic projection
- gnomic projection
- Lambert's azimuthal equal-area projection
- orthographic projection
2. Conic
- the transformation is made to the surface of a cone
tangent at a small circle (tangent case) or intersecting
at two small circles (secant case) on a globe
- mathematically, this projection is also expressed as
mappings from latitude and longitude to polar
coordinates, but with the origin located at the apex of
the cone
Examples
- References: Conic projections (Strahler and Strahler
1987 p. 14, Maling 1973, p. 164)
- Alber's conical equal area projection with two
standard parallels
- Lambert conformal conic projection with two standard
parallels
- equidistant conic projection with one standard
parallel
3. Cylindrical
- developed by transforming the spherical surface to a
tangent or secant cylinder
- mathematically, a cylinder wrapped around the equator is
expressed with x equal to longitude, and the y
coordinates some function of latitude
- formulas for cylindrical equal area projection are:
x = l
y = sin(j)
k = sec(j)
h = cos(j)
Examples
- References: Mercator and Lambert projections
(Strahler and Strahler 1987, p. 15, Maling 1973, p.
72)
- note: Mercator Projection characteristics
- meridians and parallels intersect at right
angles
- straight lines are lines of constant bearing -
projection is useful for navigation
- great circles appear as curves
4. Non-Geometric (Mathematical) projections
- some projections cannot be expressed geometrically
- have only mathematical descriptions
Examples
- Reference: Non-geometric projections (Robinson et
al, 1984, p. 97)
- Molleweide
- Eckert
E. UNIVERSAL TRANSVERSE MERCATOR (UTM)
- UTM is the first of two projection based coordinate
systems to be examined in this unit
- UTM provides georeferencing at high levels of precision
for the entire globe
- established in 1936 by the International Union of Geodesy
and Geophysics
- adopted by the US Army in 1947
- adopted by many national and international mapping
agencies, including NATO
- is commonly used in topographic and thematic mapping, for
referencing satellite imagery and as a basis for widely
distributed spatial databases
Transverse Mercator Projection
- results from wrapping the cylinder around the poles
rather than around the equator
- the central meridian is the meridian where the cylinder
touches the sphere
- theoretically, the central meridian is the line of
zero distortion
- by rotating the cylinder around the poles
- the central meridian (and area of least distortion)
can be moved around the earth
- for North American data, the projection uses a spheroid
of approximate dimensions:
- 6378 km in the equatorial plane
- 6356 km in the polar plane
Zone System
Reference: UTM zones (Strahler and Strahler 1987, p. 18)
- in order to reduce distortion the globe is divided into
60 zones, 6 degrees of longitude wide
- zones are numbered eastward, 1 to 60, beginning at
180 degrees (W long)
- the system is only used from 84 degrees N to 80 degrees
south as distortion at the poles is too great with this
projection
- at the poles, a Universal Polar Stereographic
projection (UPS) is used
- each zone is divided further into strips of 8 degrees
latitude
- beginning at 80 degrees S, are assigned letters C
through X, O and I are omitted
Distortion
- to reduce the distortion across the area covered by each
zone, scale along the central meridian is reduced to
0.9996
- this produces two parallel lines of zero distortion
approximately 180 km away from the central meridian
- scale at the zone boundary is approximately 1.0003
at US latitudes
Coordinates
- coordinates are expressed in meters
- eastings (x) are displacements eastward
- northings (y) express displacement northward
- the central meridian is given an easting of 500,000 m
- the northing for the equator varies depending on
hemisphere
- when calculating coordinates for locations in the
northern hemisphere, the equator has a northing of 0
m
- in the southern hemisphere, the equator has a
northing of 10,000,000 m
Reference: UTM coordinates (Strahler and Strahler 1987, p.
19)
Advantages
- UTM is frequently used
- consistent for the globe
- is a universal approach to accurate georeferencing
Disadvantages
- full georeference requires the zone number, easting and
northing (unless the area of the data base falls
completely within a zone)
- rectangular grid superimposed on zones defined by
meridians causes axes on adjacent zones to be skewed with
respect to each other
- problems arise in working across zone boundaries
- no simple mathematical relationship exists between
coordinates of one zone and an adjacent zone
F. STATE PLANE COORDINATES (SPC)
- SPCs are individual coordinate systems adopted by U.S.
state agencies
- each state's shape determines which projection is chosen
to represent that state
- e.g. a state extended N/S may use a Transverse
Mercator projection while a state extended E/W may
use a Lambert Conformal Conic projection (both of
these are conformal)
- projections are chosen to minimize distortion over the
state
- a state may have 2 or more overlapping zones, each
with its own projection system and grid
- units are generally in feet
Advantages
- SPC may give a better representation than the UTM system
for a state's area
- SPC coordinates may be simpler than those of UTM
Disadvantages
- SPC are not universal from state to state
- problems may arise at the boundaries of projections
Use in GIS
- many GIS have catalogues of SPC projections listed by
state which can be used to choose the appropriate
projection for a given state
REFERENCES
Maling, D.H., 1973. Coordinate Systems and Map Projections,
George Phillip and Son Limited, London.
Robinson, A.H., R.D. Sale, J.L. Morrison and P.C. Muehrcke,
1984, Elements of Cartography, 5th edition, John Wiley
and Sons, New York. See pages 56-105.
Snyder, J.P., 1987. Map Projections - A Working Manual, US
Geological Survey Professional Paper 1395, US Government
Printing Office, Washington.
Strahler, A.N. and A.H. Strahler, 1987. Modern Physical
Geography, 3rd edition, Wiley, New York. See pages 3-8
for a description of latitude and longitude and various
appendices for information on coordinate systems.
DISCUSSION AND EXAM QUESTIONS
1. Define the three standard properties of map projections:
equal-area, equidistant and conformal. Discuss the relative
importance of each for different applications. What types
of applications require which properties?
2. What type of projection would you expect to be used in
the following circumstances, and why?
a. an airline pilot flying the North Atlantic between New
York and London.
b. a submarine navigating under the ice of the North Pole.
c. an agricultural scientist assembling crop yield data for
Africa.
d. an engineer planning the locations of radio transmitters
across the continental US.
3. What map projections would you choose in designing a
workstation to be used by scientists studying various
aspects of global environmental change?
4. By examining the list of SPC systems adopted by the
states, what can you deduce about the criteria used to
determine the projection adopted and the number of zones
used? You will need a map of the US showing the boundaries
of states. Are there any surprising choices?
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