------Using GIS-Introduction to GIS
Lecture 9:
Introduction to Projections and
Coordinate Systems
By Austin Troy, University of Vermont,
with sections adapted from ESRI’s online course on projections
Introduction to GIS
The Earth’s Shape and Size
•It is only comparatively recently that we’ve been able
to say what both are
•Estimates of shape by the ancients have ranged from a
flat disk, to a cube to a cylinder to an oyster.
•Pythagoras was the first to postulate it was a sphere
•By the fifth century BCE, this was firmly established.
•But how big was it?
Introduction to GIS
The Earth’s Size
•It was Posidonius who used the stars to determine the
earth's circumference. “He observed that a given star
could be seen just on the horizon at Rhodes. He then
measured the star's elevation at Alexandria, Egypt, and
calculated the angle of difference to be 7.5 degrees or
1/48th of a circle. Multiplying 48 by what he believed
to be the correct distance from Rhodes to Alexandria
(805 kilometers or 500 miles), Posidonius calculated
the earth's circumference to be 38,647 kilometers
(24,000 miles)--an error of only three percent.”
More info
-source: ESRI
Introduction to GIS
So, what shape IS the earth?
•Earth is not a sphere, but an ellipsoid, because the
centrifugal force of the earth’s rotation “flattens it out”.
Source: ESRI
•This was finally proven by the French in 1753
•The earth rotates about its shortest axis, or minor axis,
and is therefore described as an oblate ellipsoid
Introduction to GIS
And it’s also a….
•Because it’s so close to a sphere, the earth is often
referred to as a spheroid: that is a type of ellipsoid that
is really, really close to being a sphere
Source: ESRI
•These are two common spheroids used today: the
difference between its major axis and its minor axis is
less than 0.34%.
Introduction to GIS
Spheroids
•We have several different estimates of spheroids
because of irregularities in the earth: there are slight
deviations and irregularities in different regions
•We must use a different spheroid for different regions
to account for irregularities, or we get positional errors
•The International 1924 and the Bessel 1841 spheroids
are used in Europe while in North America the GRS80,
and decreasingly, the Clarke 1866 Spheroid, are used
•In Russia and China the Krasovsky spheroid is used
and in India the Everest spheroid
Introduction to GIS
Spheroids
•Note how two different spheroids given slightly
different major and minor axis lengths
Source: ESRI
One more thing about spheroids: If your mapping scales are smaller
than 1:5,000,000 (small scale maps), you can use an authalic sphere
to define the earth's shape to make things more simple
Introduction to GIS
The Geographic Graticule/Grid
•This is a location reference system for the earth’s
surface, consisting of:
•Meridians: lines of longitude and
•Parallels: lines of latitude
•Prime meridian is at
Greenwich, England (that is
0º longitude)
•Equator is at 0º latitude
Source: ESRI
Introduction to GIS
The Geographic Graticule/Grid
•This is like a planar coordinate system, with an origin
at the point where the equator meets the prime meridian
•The difference is that it is not a Grid because grid lines
must meet at right angles; this is why it’s called a
graticule instead
•Each degree of latitude represents about 110 km,
although, that varies slightly because the earth is not a
perfect sphere
Introduction to GIS
The Geographic Grid/Graticule
•Latitude and longitude can be measured either in
degrees, minutes, seconds (e.g. 56° 34’ 30”); minutes
and seconds are base-60, like on a clock
•Can also use decimal degrees (more common in GIS),
where minutes and seconds are converted to a decimal
•Example: 45° 52’ 30” = 45.875 °
Introduction to GIS
The Geographic Grid/Graticule
•Latitude lines form parallel circles of different sizes,
while longitude lines are half-circles that meet at the poles
•Latitude goes from 0 to 90º N or S and longitude to 180 º
E or W of meridian; the 180 º line is the date line
Source: ESRI
Introduction to GIS
Map Projection
•This is the method by which we transform the earth’s
spheroid (real world) to a flat surface (abstraction),
either on paper or digitally
•Because we can’t take our globe everywhere with us!
•Remember: most GIS layers are 2-D
3D
Think about
projecting a seethrough globe onto
a wall
Source: ESRI
2D
Introduction to GIS
Map Projection
•The earliest and simplest map projection is the plane
chart, or plate carrée, invented around the first century;
it treated the graticule as a grid of equal squares,
forcing meridians and parallels to meet at right angles
•If applied to
the world as
mapped now,
it would look
like:
Introduction to GIS
Map Projection-distortion
•The problem with map projection is that it distorts one
or several of these properties of a surface:
•Shape
•Area
•Distance
•Direction
•Some projections specialize in preserving one or
several of these features, but none preserve all
Introduction to GIS
Map Projection-distortion
•Shape: projection can distort the shape of a
feature. Conformal maps preserve the shape of
smaller, local geographic features, while general
shapes of larger features are distorted. That is, they
preserve local angles; angle on map will be same
as angle on globe. Conformal maps also preserve
constant scale locally
Introduction to GIS
Map Projection-distortion
•Area:projection can distort the property of equal
area (or equivalent), meaning that features have the
correct area relative to one another. Map projections
that maintain this property are often called equal area
map projections.
•For instance, if S America is 8x larger than
Greenland on the globe will be 8x larger on map
•No map projection can have conformality and equal
area; sacrifice shape to preserve area and vice versa.
Introduction to GIS
Map Projection-distortion
•Distance: Projection can distort measures of true
distance. Accurate distance is maintained for only
certain parallels or meridians unless the map is
very localized. Maps are said to be equidistant if
distance from the map projection's center to all
points is accurate. We’ll go into this more later.
Introduction to GIS
Map Projection-distortion
•Direction:Projection can distort true directions between
geographic locations; that is, it can mess up the angle, or
azimuth between two features; projections of this kind
maintain true directions with respect to the map
projection's center. Some azimuthal map projections
maintain the correct azimuth between any two points. In a
map of this kind, the angle of a line drawn between any
two locations on the projection gives the correct direction
with respect to true north.
Introduction to GIS
Map Projection-distortion
•Hence, when choosing a projection, one must take into
account what it is that matters in your analysis and what
properties you need to preserve
•Conformal and equal area properties are mutually
exclusive but some map projections can have more than
one preserved property. For instance a map can be
conformal and azimuthal
•Conformal and equal area properties are global (apply to
whole map) while equidistant and azimuthal properties are
local and may be true only from or to the center of map
Introduction to GIS
Map Projection-distortion
•Some examples:
•Mercator maintains shape and direction, but sacrifices
area accuracy
•The Sinusoidal and Equal-Area Cylindrical projections
both maintain area, but look quite different from each
other. The latter distorts shape
•The Robinson projection does not enforce any specific
properties but is widely used because it makes the earth’s
surface and its features look somewhat accurate
Introduction to GIS
Map Projection-distortion
robinson
Mercator—goes on forever
sinusoidal
Introduction to GIS
Map Projection-Distortion
•Tissot’s indicatrix, made up of ellipses, is a method
for measuring distortion of a map; here is Robinson
Introduction to GIS
Map Projection-Distortion
•Here is Sinusoidal
Area of these ellipses
should be same as
those at equator, but
shape is different
Introduction to GIS
Map Projection-General Types
•Cylindrical projection: created
by wrapping a cylinder around a
globe and, in theory, projecting
light out of that globe; the
meridians in cylindrical
projections are equally spaced,
while the spacing between parallel
lines of latitude increases toward
the poles; meridians never
converge so poles can’t be shown
Source: ESRI
Introduction to GIS
Map Projection-General Types
•In the simplest case, the cylinder is North-South, so it is
tangent (touching) at the equator; this is called the
standard parallel and represents where the projection is
most accurate
•If the cylinder is smaller than the circumference of the
earth, then it intersects as a secant in two places
•A north-south cylindrical Projections cause major
distortions in higher latitudes because those points on
the cylinder are further away from from the
corresponding point on the globe
Introduction to GIS
Cylindrical map distortion
•Think of the problem with this cylindrical equatorial projection
•Scale is constant in north-south direction and in east west
direction along the equator for an equatorial projection but non
constant in east-west direction as move up in latitude
Introduction to GIS
Cylindrical map distortion
•Why is this? Because meridians are all the same length, but
parallels are not.
•This sort of projection forces parallels to be same length so it
distorts them
•As move to higher latitudes, east-west scale increases (2 x
equatorial scale at 60° N or S latitude) until reaches infinity at the
poles; N-S scale is constant
Introduction to GIS
Cylindrical map distortion
•If such a map has a scale bar (see map in 104 Aiken),
know that it is only good for those places and directions in
which scale is constant—the equator and the meridians
•Hence, the measured distance between Nairobi and the
mouth of the Amazon might be correct, but the measured
distance between Toronto and Vancouver would be off; the
measured distance between Alaska and Iceland would be
even further off
Introduction to GIS
Map Projection-General Types
•Conic Projections: projects a globe onto a cone
•In simplest case, globe touches cone along a single
latitude line, or tangent, called standard parallel
•Other latitude lines are projected onto cone
•To flatten the cone, it must be cut along a
line of longitude (see image)
•The opposite line of longitude
is called the central meridian
Source: ESRI
Introduction to GIS
Map Projection-General Types
•Conic Projections:
•Projection is most accurate where globe and cone
meet—at the standard parallel
•Distortion generally increases north or south of it,
so poles are often not included
•Conic projections are typically used for midlatitude zones with east-to-west orientation. They
are normally applied only to portions of a
hemisphere (e.g. North America)
Introduction to GIS
Map Projection-General Types
•Planar or Azimuthal Projections: simply
project a globe onto a flat plane
•The simplest form is only tangent at one point
•Any point of contact may be used but the
poles are most commonly used
•When another location is used, it is generally
to make a small map of a specific area
•When the poles are used, longitude lines look
like hub and spokes
Source: ESRI
Introduction to GIS
Map Projection-General Types
•Planar or Azimuthal Projections:
•Because the area of distortion is circular around the
point of contact, they are best for mapping roughly
circular regions, and hence the poles
Introduction to GIS
Map Projection-Specific Types
•Mercator: This is specific type of cylindrical
projection
•Invented by Gerardus Mercator during the 16th
Century
•It was invented for navigation because it
preserves azimuthal accuracy—that is, if you
draw a straight line between two points on a map
created with Mercator projection, the angle of
that line represents the actual bearing you need to
sail to travel between the two points
Source: ESRI
Introduction to GIS
Map Projection-Specific Types
•Mercator: Of course the Mercator projection is not so
good for preserving area.
Take a look at how
it enlarges high
latitude features like
Greenland
Antarctica and
shrinks mid latitude
features
Introduction to GIS
Map Projection-Specific Types
•Transverse Mercator: Invented by Johann Lambert in
1772, this projection is cylindrical, but the axis of the
cylinder is rotated 90°, so the tangent line is
longitudinal, rather than the equator
•In this case, only the central longitudinal meridian and
the equator are straight lines
All other lines are
represented by complex
curves: that is they can’t
be represented by single
section of a circle
Source: ESRI
Introduction to GIS
Map Projection-Specific Types
•Transverse Mercator:
•Transverse Mercator projection is not used on a global
scale but is applied to regions that have a general northsouth orientation, while Mercator tends to be used more
for geographic features with east-west axis.
•It is used in commonly in the US with the State Plane
Coordinate system, with north-south features
Introduction to GIS
Map Projection-Specific Types
•Lambert Conformal Conic:invented in 1772, this is a
form of a conic projection
•Latitude lines are unequally spaced arcs that are portions
of concentric circles. Longitude lines are actually radii of
the same circles that define the latitude lines.
Source: ESRI
Introduction to GIS
Map Projection-Specific Types
•The Lambert Conformal Conic projection is very good
for middle latitudes with east-west orientation.
•It portrays the pole as a point
•It portrays shape more accurately than area and is
commonly used for North America.
•The State Plane coordinate system uses it for east-west
oriented features
Introduction to GIS
Map Projection-Specific Types
•The Lambert Conformal Conic projection is a slightly
more complex form of conic projection because it
intersects the globe along two lines, called secants, rather
than along one, which would be called a tangent
•There is no distortion along those two lines
•Distortion increases as move away
from secants
Source: ESRI
Introduction to GIS
Map Projection-Specific Types
•Albers Equal Area Conic projection: Again, this is a
conic projection, using secants as standard parallels but
while Lambert preserves shape Albers preserves area
•It also differs in that poles are not represented as points,
but as arcs, meaning that meridians don’t converge
•Latitude lines are unequally spaced concentric circles,
whose spacing decreases toward the poles.
•Developed by Heinrich Christian Albers in the early
nineteenth century for European maps
Introduction to GIS
Map Projection-Specific Types
•Albers Equal Area Conic: It preserves area by making
the scale factor of a meridian at any given point the
reciprocal of that along the parallel.
•Scale factor is the ratio of local scale a point on the
projection to the reference scale of the globe; 1 means the
two are touching and
greater than 1 means
the projection surface
is at a distance
Introduction to GIS
Other Selected Projections
• More Cylindrical equal area: (have straight meridians
and parallels, the meridians are equally spaced, the
parallels unequally spaced)
• Behrmann cyclindrical equal-area: single standard
parallel at 30 ° north
•Gall’s stereographic: secant intersecting at 45°
north and 45 ° south
•Peter’s: de-emphasizes area exaggerations in high
latitudes; standard parallels at 45 or 47 °
Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at
Boulder for links
Introduction to GIS
Other Selected Projections
• Azimuthal projections:
•Azimuthal equidistant: preserves distance
property; used to show air route distances
•Lambert Azimuthal equal area: Often used for
polar regions; central meridian is straight, others
are curved
•Oblique Aspect Orthographic
•North Polar Stereographic
Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at
Boulder for links
Introduction to GIS
Other Selected Projections
• More conic projections
•Equidistant Conic: used for showing areas near to,
but on one side of the equator, preserves only
distance property
•Polyconic: used for most of the early USGS quads;
based on on an infinite number of cones tangent to
an infinite number of parallels; central meridian
straight but other lines are complex curves
Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at
Boulder for links
Introduction to GIS
Other Selected Projections
• Pseudo-cylindrical projections: resemble cylindrical
projections, with straight, parallel parallels and equally
spaced meridians, but all meridians but the reference
meridian are curves
•Mollweide: used for world maps; is equal-area; 90th
meridians are semi-circles
• Robinson:based on tables of coordinates, not
mathematical formulas; distorts shape, area, scale,
and distance in an attempt to make a balanced map
Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at
Boulder for links