An introductory lecture on Geographic Location and Map Projections

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An introductory lecture onGeographic Location
and Map Projections

• Waldo Tobler
• Professor Emeritus
• Geography department
• University of California
• Santa Barbara, CA 93106-4060
• http//www.geog.ucsb.edu/tobler
• Spatial Perspectives on Analysis for Curriculum
  Enhancement
• California Summer Institute, July

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This presentation covers procedures used to
identify locations on the earth.

• Some parts of this presentation will get a little
  technical.
• 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
  information.
• You are welcome to use some of these slides in
  your course.

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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
  purpose.
• 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
  examples.

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Of course there are many ways of specifying
geographic location.

• For example, here is my address if you wish to
  correspond with me.

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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.

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A Geographic Locations Conversion TableSixteen
Cases
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Common Geographic Locational Aliases and
Conversions
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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
  districts.
• 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//www.geog.ucsb.edu/tobler
• and in my paper on Pycnophylactic Interpolation
  which appeared in the Journal of the American
  Statistical Association in 1979.

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Some other conversions

• Conversions between data storage formats.
• For example between ASCII and GIS systems sych as
  ArcInfo,
• 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.

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The Public Land Surveyof the United States.
Location on a round earth simplified.
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The Public Land SurveyIn use in the Western
United States
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Identifying places has long been done using
geographic coordinates
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Where am I?
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Wouldnt this be handy?
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Here is one way to get latitude and longitude
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The shape of the earth adds complication to the
conversions

• 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.

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The Mapping ProcessCommon Surfaces Used in
cartography. Different ellipsoids are used in
different parts of the world,
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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
  incorrect.
• 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
  surface.
• Geographic maps may, or may not, preserve other
  properties such as areas, directions, angles, or
  distances.

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The Surface of the Earth Is Two-Dimensional
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A cardboard box can be unfolded to lay flat - the
surface is two-dimensional.
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Think of latitudes and longitudes as graph paper
covering the earth.

• It is somewhat similar to the use of polar
  coordinates.
• 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
  systems.
• Often isometric or authalic coordinates are used.

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Isometric coordinates - the latitude spacing is
forced to equal the longitude spacing. This is
equivalent to using a Mercator map on the sphere.
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Authalic coordinates - the latitude spacing is
forced to yield equal areas. This is equivalent
to using a Lambert cylindrical map on the sphere.
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Many analytical problems can be solved directly
in geographic coordinates

• This is often easy when the earth is considered
  spherical.
• It is more difficult to work with an ellipsoidal
  earth.
• Some people like to work in plane, Euclidean,
  coordinates. Then a map projection is needed.
• Of course the projection must be suited to the
  problem.

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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
  percent?

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Sphere or Map?

• This is equivalent to asking whether you want to
  work in latitude and longitude or plane
  coordinates.
• 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
  sphere.
• 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.

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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.

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There are many map projections

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

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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.

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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
  USA.
• Conformal maps for movement related to contours
  or gradients.
• Azimuthal equidistant for items relating to a
  center.
• Stereographic to show spherical circles.
• If in doubt chose one of the common ones.

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From Globe to Map
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The general procedure for producing map
projections

• 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
  correspondence.

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Identifying the association between earth and
mapis 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.

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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
  equation(s).
• 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.

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Map Projection PropertiesSome of which are
incompatible.

• 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
  defined.
• On many maps no special properties obtain.
• They may be happy compromises.

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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.

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Tissots indicatrix measures distortionIt 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
  that
• the instantaneous map scale is different in every
  direction
• 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.

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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

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Mercators Projection Was designed in 1569 for
navigation at sea should not be used for other
purposes.

• This projection is often depicted as being
  projected geometrically from a globe to a
  cylinder.
• 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.

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Mercators projection is not perspective

• It is defined by a pair of equations

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Here is a polycylindrical development.From three
cylinders to infinitely many, resulting in a
continuous map.
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Plane coordinate systems are based on map
projections

• 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
  government.
• Therefore virtually all Geographic Information
  Systems include them.

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Heres how its doneAdd rectangular coordinates
on top of a map.
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First an accurate map is made.Then a rectangular
grid is superimposed.
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The transverse Mercator projection

• The military uses the term Universal, thus the
  UTM.
• 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
  world.
• The coordinates are shown on recent USGS maps.
• A different system is used for the polar areas.
• It is not simple. The equations are

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UTM Equations
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The UTM System
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UTM Zones in the United StatesThe systen is
designed to cover the entire earth.
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Local plane coordinates

• Each state in the US also has one or more local
  coordinate systems.
• These have legal standing for property
  descriptions.
• 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
  maps..

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State Plane Coordinate Zones
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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
  display.

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The equations used areX 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 asX a01 ?? a12 ?? ??Y
  a10 ?? a22 ?? ??.The distortion is also
  easily calculated from these equations.

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Two different map projectionsor locations on two
different maps

• Will be represented by two different pairs of
  equations.
• X F1(f,?) and Y G1(f,?) for one and
• U F2(f,?) and V G2(f,?) for the
  other.
• 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.

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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
  difficult.
• Most mapping and GIS packages include use
  instructions and inversion conversion routines,
  usually taken from free US government
  publications.

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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.

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Reconciling images in map matching.Example
Map and Image
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The difference between the map and the image
Shown as discrete vectors
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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

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Difference Vectorsby themselves, without the grid
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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.

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Interpolated Vector Field
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Great Lakes DisplacedThe grid has been pushed
by the interpolated vector field
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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
  curvilinear.
• 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
  possible.
• Most Geographic Information Systems and
  government mapping agencies take this point of
  view.
• 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.

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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.

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. 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.

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Hägerstrands Logarithmic Map
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Another exampleConventional Way of Tracking
SatellitesSatellite tracks are curves. The
coverage areas are circles on the earth.
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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.
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Bend the meridians instead
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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
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US population by one degree quadrilaterals
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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.

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The lat/lon grid in the two spacesLeft, the
usual grid. Right, transformed according to
population.
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US map in the two spacesLeft, the usual map.
Right, the transform.
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The inversionOn 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
  D.C.
• 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,
  London

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Thank You For Your Attention

• You are now prepared to have fun with map
  projections.

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The Santa Barbaran ViewA cube root distance
azimuthal projection