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Title: An introductory lecture on Geographic Location and Map Projections

An introductory lecture on Geographic Location
and Map Projections
  • Waldo Tobler
  • Professor Emeritus
  • Geography department
  • University of California
  • Santa Barbara, CA 93106-4060
  • http//
  • Spatial Perspectives on Analysis for Curriculum
  • California Summer Institute, July

This presentation covers procedures used to
identify locations on the earth.
  • Some parts of this presentation will get a little
  • But my intention is to present an overview which
    augments conventional treatments given in
    standard textbooks.
  • This means that there may be more detail than is
    normal for an undergraduate course even though it
    is not enough for a graduate course in the
    subject. The internet is a good source for more
  • You are welcome to use some of these slides in
    your course.

Many local systems of location identification do
not recognize the shape of the earth.
  • For example city street addresses.
  • When a larger national or international area is
    included the earths shape must be considered.
  • Special methods have been devised for this
  • One type applies directly on the ground.
  • The U.S. Public Land Survey is one such system.
  • Real estate property description using metes and
    bounds is another.
  • Another approach works by using coordinates
    applied to maps.
  • The State Plane Coordinate (SPC) system and the
    Universal Transverse Mercator (UTM) system are

Of course there are many ways of specifying
geographic location.
  • For example, here is my address if you wish to
    correspond with me.

Newer systems include
  • Telephone numbers.
  • These identify a location to approximately one
    meter, the length of the telephone cord,
  • and use area codes.
  • If the area codes are similar the places are far
    apart, to avoid mistakes.
  • Many systems contain redundancies of this sort.
  • E-mail addresses locate people in IP space.

A Geographic Locations Conversion Table Sixteen
Common Geographic Locational Aliases and
In addition to locational conversions there are
important data conversion problems
  • For example between, say, population counts by
    census tracts and information needed by school
  • Source data is by census tracts
  • Target data needed by school districts
  • This extremely common set of important problems
    is addressed in my power point presentation on
    Geographical Interpolation at
  • http//
  • and in my paper on Pycnophylactic Interpolation
    which appeared in the Journal of the American
    Statistical Association in 1979.

Some other conversions
  • Conversions between data storage formats.
  • For example between ASCII and GIS systems sych as
  • Conversions between vectors and rasters.
  • For example line drawings and gridded data.
  • Analytical conversions - scalars to gradients.
  • And many more.
  • These will not be discussed here.

The Public Land Survey of the United States.
Location on a round earth simplified.
The Public Land Survey In use in the Western
United States
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Identifying places has long been done using
geographic coordinates
Where am I?
Wouldnt this be handy?
Here is one way to get latitude and longitude
The shape of the earth adds complication to the
  • It involves a choice of an earth model
  • which depends on purpose and the required
    accuracy. It may also depend on the specific
    country or part of the world.
  • It may also involve a map projection.

The Mapping Process Common Surfaces Used in
cartography. Different ellipsoids are used in
different parts of the world,
The surface of the earth is two dimensional.
  • This is why only (but also both) latitude and
    longitude are needed to pin down a location. Many
    authors refer to it as three dimensional. This is
  • All geographic maps preserve the two
    dimensionality of the surface. The Byte magazine
    cover from May 1979 shows how the graticule rides
    up and down over hill and dale. Yes, it is
    embedded in three dimensions, but the surface is
    a curved, closed, and bumpy, two dimensional
  • Geographic maps may, or may not, preserve other
    properties such as areas, directions, angles, or

The Surface of the Earth Is Two-Dimensional
A cardboard box can be unfolded to lay flat - the
surface is two-dimensional.
Think of latitudes and longitudes as graph paper
covering the earth.
  • It is somewhat similar to the use of polar
  • The current system was invented in circa 300 BC,
    and works very well.
  • But other kinds of graph paper could be used.
  • For example, hyperbolic coordinates are possible
    and could be used for hyperbolic navigation
  • Often isometric or authalic coordinates are used.

Isometric coordinates - the latitude spacing is
forced to equal the longitude spacing. This is
equivalent to using a Mercator map on the sphere.
Authalic coordinates - the latitude spacing is
forced to yield equal areas. This is equivalent
to using a Lambert cylindrical map on the sphere.
Many analytical problems can be solved directly
in geographic coordinates
  • This is often easy when the earth is considered
  • It is more difficult to work with an ellipsoidal
  • Some people like to work in plane, Euclidean,
    coordinates. Then a map projection is needed.
  • Of course the projection must be suited to the

Sphere or Ellipsoid?
  • The departure of the earth from a sphere is
    approximately one part in three hundred.
  • This is 3/10ths of one percent.
  • This can be used as a rule of thumb
  • Is your work accurate to better than one

Sphere or Map?
  • This is equivalent to asking whether you want to
    work in latitude and longitude or plane
  • Programs exist, for example, to convert from
    street address to Lat/Lon. There are also
    programs to convert from Lat/Lon to X, Y, and
    visa versa.
  • Many kinds of analysis are very simple on a
  • This includes such things as distance, direction,
    or area computation.
  • A plane is a sufficiently good approximation to a
    sphere for a small area.
  • You can glue a postage stamp, without wrinkling
    it, on a 20 cm globe.

Map projections are necessary when it is desired
to make a map on a flat surface.
  • Or to provide a graphical method for solving
    geographical problems on a flat surface.
  • Or to work in plane, Euclidean, coordinates.

There are many map projections
  • Theoretically there are infinitely many.
  • About 300 have names, often associated with their
  • Only a dozen or so are commonly used.
  • Many GIS packages handle the most common

Detail on maps made on different map projections
will not agree in position, or size
  • Thus it is usually important to know the
    projection on which you are working.
  • In particular, when converting geographic
    information from a map to a digital file, or visa
    versa, the name and details of the projection
    must be noted.
  • Along with the information date and map scale.

In order to chose a map projection a map purpose
must be specified
  • Equal area maps for distributions, for example,
    Albers equal area for statistical maps of the
  • Conformal maps for movement related to contours
    or gradients.
  • Azimuthal equidistant for items relating to a
  • Stereographic to show spherical circles.
  • If in doubt chose one of the common ones.

From Globe to Map
The general procedure for producing map
  • Locations on the earth are identified by latitude
    (f) and longitude (?), in the form of numbers.
  • Positions on paper (or CRT) are identified by X
    and Y names.
  • A pair of equations is introduced to associate
    the earth and paper locations.
  • Think of strings connecting points on a globe
    with locations on paper, establishing a 1-to-1

Identifying the association between earth and
map is done using equations, in Eulers notation
  • XF(f,?), YG(f,?).
  • f represents latitude, ? longitude
  • F and G are usually different functions. They
    may be simple or complicated.
  • A simple example
  • XR?,
  • YRf,
  • where R is the earth radius, assumed spherical
    and usually taken to be one unit. This is the
    rectangular or Plate Carée projection.

Derivation of a map projection
  • The map projection properties are obtained by
    setting the partial differential equation
    describing the property on a sphere (or
    ellipsoid) equal to the differential equation
    describing this same property on a plane.
  • Then specify boundary conditions and solve the
  • For example, in the case of equal area
    projections, require that
  • spherical area map area, that is
  • df d? cos(f) dx dy.
  • This differential equation has many solutions.
  • Consequently additional conditions are specified.

Map Projection Properties Some of which are
  • Equal area (a.k.a. equivalent) - all map areas
    are proportional to their area on the earth.
  • Conformal - the scale is the same in all
    directions at any point but differs at every
    point. Local angles are preserved.
  • Equidistant - distances are correct, to scale,
    generally from one point only, but occasionally
    from two points or from a line.
  • Azimuthal - directions from one (or two) points
    is correct.
  • A variety of more specialized properties can be
  • On many maps no special properties obtain.
  • They may be happy compromises.

The least understood property is conformal.
  • Conformality is perhaps best visualized by
    imaging that you are looking at a globe through a
    microscope on wheels. These wheels are connected
    to the magnification system. Every time you move
    the microscope on the globe the wheels force the
    magnification to change slightly. Everything
    looks perfectly fine except that the scale, or
    area size, is different everywhere, and you can
    only see a little piece at a time. The latter
    property suggests local shape invariance and that
    local angles are preserved.

Tissots indicatrix measures distortion It is a
rather technical specification
  • It is based on the four partial derivative of
    the defining transformation X F(f,?), Y
    G(f,?), namely
  • ?x/?f, ?y/?f, ?x/??, ?y/??.
  • As such it is a tensor function of location. It
    varies from place to place, and reflects the fact
  • the instantaneous map scale is different in every
  • at a location, unless the map is conformal.
  • Tissots indicatrix is used to specify local
    properties of a map such as angular, areal, or
    linear distortion. In books on map projections it
    is often shown as distortion ellipses.

A simplistic classification of map projections
  • is found in numerous textbooks.
  • It is based on the idea of a geometric projection
    onto a surface such as a cylinder, cone or plane.
  • Cylindric X F(?), Y G(f)
  • World maps have a rectangular form
  • Conic r F(?), ? G(n ?) in polar coordinates
  • World maps have a fan-like form
  • Planar r F(?), ? G(?) in polar coordinates
  • World maps have a circular form
  • also polyconic and polycylindric
  • World maps have a rather bent form

Mercators Projection Was designed in 1569 for
navigation at sea should not be used for other
  • This projection is often depicted as being
    projected geometrically from a globe to a
  • It is actually produced, in the spherical case,
    using the equations
  • X ?, Y Ln Tan (p/4 f/2).
  • The easy way to demonstrate that Mercators
    projection cannot be obtained as a true
    perspective is to draw lines from the latitudes
    on the projection to their occurrence on a
    sphere, represented by an adjoining circle. The
    rays will not intersect in a point.

Mercators projection is not perspective
  • It is defined by a pair of equations

Here is a polycylindrical development. From three
cylinders to infinitely many, resulting in a
continuous map.
Plane coordinate systems are based on map
  • The two most important ones are
  • The Universal Transverse Mercator System (UTM)
  • The State Plane Coordinate system.
  • The equations for both are complicated and
    based on an ellipsoid.
  • You cannot find these marked on the ground like
    the Public Land Survey system. But most GPS and
    maps have them.
  • The equations, parameters, and specifications
    are available
  • free in the form of computer programs from the
  • Therefore virtually all Geographic Information
    Systems include them.

Heres how its done Add rectangular coordinates
on top of a map.
First an accurate map is made. Then a rectangular
grid is superimposed.
The transverse Mercator projection
  • The military uses the term Universal, thus the
  • Within an area of about 300 km it is a good
    approximation to the earth, in area, distances,
    and directions.
  • 60 separate but overlapping North-South zones are
    used, each 6 degrees in width, to cover the
  • The coordinates are shown on recent USGS maps.
  • A different system is used for the polar areas.
  • It is not simple. The equations are

UTM Equations
The UTM System
UTM Zones in the United States The systen is
designed to cover the entire earth.
Local plane coordinates
  • Each state in the US also has one or more local
    coordinate systems.
  • These have legal standing for property
  • They are known as State Plane Coordinates.
  • In all there are about 111 particular systems,
    depending on the shape of the state, in order to
    be accurate to one part in ten thousand. They are
    based on an ellipsoid used for the US.
  • They use several different projections, the
    most common being the Lambert Conformal Conic and
    the transverse Mercator (not quite the same as
    the UTM!).
  • The coordinates are shown on USGS maps.
  • Virtually all countries of the world have similar
    local systems, printed on their topographic

State Plane Coordinate Zones
A map projection for quick analysis or display
  • Want to analyze some geographic data or
    display it on a computer screen? Here's a quick
    simple map projection that will do the job nicely
    for a modest sized region, away from the poles.
    The data are assumed to be given in latitude and
    longitude coordinates. The main parameter is the
    average latitude of the region in question, and
    this can be computed by the program. The average
    longitude is also needed, to center the
    projection. The projection uses the Gaussian mean
    radius sphere at the average latitude on the
    Clarke ellipsoid of 1866. An alternative is to
    use the WGS83 ellipsoid. The resulting X, Y
    coordinates are in kilometers, centered on the
    mean location, and can be used for analysis or

The equations used are X R cos(?o) ?? -
sin(?o) ?? ?? Y R ?? 0.5
sin(?o) cos(?o) ?? ?? , where R is in
kilometers per degree on the mean radius sphere
(computed by my program). ?? is the latitude
minus the average latitude ?o , and ?? is the
longitude minus the average longitude ?o. The X
and Y coordinates are given in kilometers.
  • The simplicity of the system can be seen by
    rewriting it as X a01 ?? a12 ?? ?? Y
    a10 ?? a22 ?? ??. The distortion is also
    easily calculated from these equations.

Two different map projections or locations on two
different maps
  • Will be represented by two different pairs of
  • X F1(f,?) and Y G1(f,?) for one and
  • U F2(f,?) and V G2(f,?) for the
  • Where X,Y are rectangular coordinates on one map,
    and U,V
  • are rectangular coordinates on the other.
  • F1 F2 differ as do G1 G2.
  • In foreign areas the ellipsoidal basis of the
    maps may also differ.

To inter-convert between two projections
  • Either
  • Go from X, Y to Lat/long, using the inverse
    equations, if known F-1 and G-1. Then proceed to
    the other map projection.
  • Or
  • Inter-convert directly, which is usually
  • Most mapping and GIS packages include use
    instructions and inversion conversion routines,
    usually taken from free US government

When the equations are not known
  • A number of empirical procedures are used. These
    include fitting bivariate polynomials, spline
    fitting, and rubber sheeting.
  • These techniques are also used to fit satellite
    images to maps.
  • The techniques require the identification of
    comparable landmarks in each space.

Reconciling images in map matching. Example
Map and Image
The difference between the map and the image
Shown as discrete vectors
A table showing the Map to Image Displacements
  • Coordinates
  • Map Image
  • x y u v
  • 25 11 18 03
  • 74 28 59 29
  • 21 51 12 47
  • 52 86 30 92
  • 63 12 49 10
  • 58 37 42 38
  • 83 51 68 55
  • 86 68 69 75
  • 73 19 61 20

Difference Vectors by themselves, without the grid
The scattered vectors can be interpolated to
yield a Vector Field
  • Inverse distance, krieging, splining, or other
    forms of interpolation may be used.
  • Smoothing or filtering of the scattered
    vectors or of the vector field can also easily be
    applied. This is done by applying the operator to
    the individual vector components.
  • Or treat the vectors as complex numbers
    with the common properties of numbers.

Interpolated Vector Field
Great Lakes Displaced The grid has been pushed
by the interpolated vector field
The coastlines may be drawn using the warped grid
  • Observe that either the map, or the image, can
    be considered the independent variable in this
    bidimensional regression.
  • Relating two sets of coordinates (the map and
    the image) requires a bidimensional regression,
    instead of a regular unidimensional regression.
    The bidimensional regression can be linear or
  • Converting between map projections is very
    similar to this.
  • W. Tobler, 1994, Bidimensional Regression,
    Geographical Analysis, 26 (July) 186-212

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All map projections result in distorted maps
  • Since the time of Ptolemy the objective has
    been to obtain maps with as little distortion as
  • Most Geographic Information Systems and
    government mapping agencies take this point of
  • But then Mercator changed this by introducing the
    idea of a systematic distortion to assist in the
    solution of a problem.
  • Mercators famous anamorphose helps solve a
    navigation problem.
  • It is not to be used for visualization.
  • His idea caught on.
  • Anamorphic projections are used to solve
    problems and are not primarily for display.

One way to use map projections
  • It is useful to think of a map projection like
    you are used to thinking of graph paper.
  • Semi logarithmic, logarithmic, probability plots,
    and so on, are employed to bring out different
    aspects of data being analyzed.
  • Map projections may be used in the same way. Just
    like graph paper they can bring out different
    facets of your data.
  • This is not a common use in Geographic
    Information Systems.

. Actually, but not shown, there is a small
hole in the middle of the map since the logarithm
of zero is minus infinity.
  • In studying migration about the Swedish city of
    Asby, Hägerstrand used the logarithm of the
    actual distance as the radial scale for a map.
    This enlarges the scale in the center of Asby,
    near which most of the migration takes place,
    providing focus for his study.

Hägerstrands Logarithmic Map
Another example Conventional Way of Tracking
Satellites Satellite tracks are curves. The
coverage areas are circles on the earth.
Instead of straight meridians and parallels with
curved satellite tracks, as on the previous map,
let us bend the meridians so that the satellite
track becomes a straight line. This is convenient
for automatic plotting of the satellite tracks.
What this looks like can be seen on the map
designed for a satellite heading southeast from
Cape Canaveral. Observe that the satellite does
not cross over Antarctica which is therefore not
on the map.The track is a sawtooth line, first
South, then North, then South again.
Bend the meridians instead
Area Cartograms
Area cartograms are also anamorphoses - a form of
map projection designed to solve particular
problems. They represent map area proportional to
some distribution on the earth, through a
uniformization. This property is useful in
studying distributions.
  • The equations show that equal area projections
    are a special case of area cartograms.
  • Area cartograms can also be displayed on a globe.

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A map projection to solve a special problem
The next illustration shows the U.S. population
assembled into one degree quadrilaterals We
would like to partition the U.S. into regions
containing the same number of people There
follows a map projection (anamorphose) that may
be useful for this problem
US population by one degree quadrilaterals
Now use the Transform-Solve-Invert paradigm
  • Transform the graticule, and map, into areas of
    equal population.
  • Then position a hexagonal tesselation on the map.
  • Then take the inverse transformation.
  • W. Tobler, 1973, A Continuous Transformation
    Useful for Districting, Annals, N.Y Academy of
    Sciences, 219215-220.

The lat/lon grid in the two spaces Left, the
usual grid. Right, transformed according to
US map in the two spaces Left, the usual map.
Right, the transform.
The inversion On the right are uniform hexagons
in the transformed space. On the left is the
solution The inverse transformation partitions
the US into cells of equal population
  • W. Tobler, 1973, "A Continuous Transformation
    Useful for Districting", Annals, New York Academy
    of Sciences, 219 215-220

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Some Recent References
  • L. Bugayevshiy, Snyder, J., 1995, Map
    Projections A Reference Manual, Taylor
    Francis, London.
  • F. Canters, 2002, Small-Scale Map Projection
    Design, Taylor Francis, London.
  • D. Maling, 1992, Coordinate Systems and Map
    Projections, 2nd ed., Pergamon,, London
  • J. Snyder, 1982, Map Projections used by the
    Geological Survey, Prof. Paper 1532, GPO,
    Washington D.C.
  • J. Snyder, 1987, Map Projections A Working
    Manual, USGS Prof. Paper 1395, GPO, Washington
  • J. Snyder, Steward, H., 1988, Bibliography of
    Map Projections, USGS Bulletin 1856, GPO,
    Washington D.C.
  • J. Snyder, Voxland, P, 1989, An Album of Map
    Projections, USGS Prof. Paper 1453, GPO,
    Washington D.C
  • J. Snyder, 1993, Flattening the Earth Two
    thousand Years of Map Projections, University of
    Chicago Press, Chicago
  • Q. Yang, Snyder, J., Tobler, W., 2000, Map
    Projection Transformation, Taylor Francis,

Thank You For Your Attention
  • You are now prepared to have fun with map

The Santa Barbaran View A cube root distance
azimuthal projection