1
An introductory lecture on
Geographic Location and Map Projections
Waldo ToblerProfessor Emeritus
Geography departmentUniversity of California
Santa Barbara, CA 93106-4060http://www.geog.ucsb.edu/people/tobler.htmCenter for Spatially Integrated Social Science
Summer Institute Santa Barbara, CA, July 2002
2
Identifying places has long been done using coordinates
3
Here is one way to get latitude and longitude
4
Geographic Locations Transformation TableSixteen Cases
5
Common Geographic Locational Aliases and Conversions
6
There are many ways of specifying location
For example, here is my address if you wish to correspond with me
7
Some other conversions
Conversions between data storage formatsfor example between ASCII and ArcInfo
Conversions between vectors and rastersfor example line drawings and gridded data
Analytical conversions - scalars to gradientsAnd many more
8
The shape of the earthgenerates more conversions
This generally involves a choice of an earth modelwhich depends on purpose and the required accuracy.
It may also involve a map projection.
9
The Mapping ProcessCommon Surfaces Used in cartography
10
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 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.
Maps also use a flat two dimensional surface.
11
The Surface of the Earth Is Two-Dimensional
12
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.
13
Isometric coordinates - the latitude spacing is forced to equal the longitude spacing.
This is equivalent to using a Mercator map on the sphere.
14
Authalic coordinates - the latitude spacing is forced to yield equal areas.
This is equivalent to using a Lambert cylindrical map on the sphere.
15
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.
16
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?
17
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.
18
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.
19
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.
20
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.
21
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.
22
The general procedure for producing map projections:
Locations on the earth are identified by latitude (φ) and longitude (λ), 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.
23
Identifying the association between earth and mapis done using equations, in Euler’s notation:
X=F(φ,λ), Y=G(φ,λ).
φ represents latitude, λ longitude
F and G are usually different functions. They may be simple or complicated.
A simple exampleX=Rλ, Y=Rφ,
where R is the earth radius, assumed spherical and usually taken to be one unit. This is the rectangular or Plate Carée projection.
24
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 isdφ dλ cos(φ) = dx dy.
This differential equation has many solutions. Consequently additional conditions are specified.
25
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.
26
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 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.
27
Tissot’s indicatrix measures distortion
It is based on the four partial derivative of the transformationX = F(φ,λ), Y = G(φ,λ), namely
x/ , y/ , 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.Tissot’s 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.
28
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( )World maps have a rectangular form
Conic: r = F( ), = G(n ) in polar coordinatesWorld maps have a fan-like form
Planar: r = F( ), = G( ) in polar coordinatesWorld maps have a circular form
also polyconic and polycylindricWorld maps have a rather bent form
29
Mercator’s Projection
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 = λ, Y = Ln Tan(π/4 + φ/2).
The easy way to demonstrate that Mercator’s 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.
30
Mercator’s projection is not perspective
It is defined by a pair of equations
31
Here is a polycylindrical development.From three cylinders to infinitely many, resulting in a continuous map.
32
Plane coordinate systems are based on map projections
The two most important ones are The Universal Transverse Mercator System.The State Plane Coordinate system.The equations for both are complicated and based on an ellipsoid.
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.
33
First an accurate map is made.Then a rectangular grid is superimposed.
34
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 theearth, 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
35
UTM Equations
36
UTM Zones in the United States
37
Local plane coordinatesEach 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..
38
State Plane Coordinate Zones
39
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.
40
The equations used areX = R { cos(ϕo) *∆λ - sin(ϕo) * ∆ϕ *∆λ }Y = R { ∆ϕ + 0.5 * sin(ϕo) * cos(ϕo) * ∆λ * ∆λ },
where R is in kilometers per degree on the mean radius sphere (computed by my program). ∆ϕ is the latitude minus the average latitude ϕo , and ∆λ is the longitude minus the
average longitude λo. The X and Y coordinates are given in kilometers.
The simplicity of the system can be seen by rewriting it asX = a01 * ∆λ + a12 * ∆ϕ * ∆λY = a10 * ∆ϕ + a22 * ∆λ * ∆λ.The distortion is also easily calculated from these equations.
41
Two different map projections
Will be represented by two different pairs of equations.X = F1(φ,λ) and Y = G1(φ,λ) for one andU = F2(φ,λ) and V = G2(φ,λ) for the other.
Where X,Y are coordinates on one map, and U,V are plane coordinates on the other.
F1 & F2 differ as do G1 & G2.
In foreign areas the ellipsoidal basis of the maps may also differ.
42
To inter-convert between two projections
EitherGo from X, Y to Lat/long, using the inverse equations, if
known: F-1 and G-1. Then proceed to the other map projection.
OrInter-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.
43
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.
44
Reconciling images in map matching.Example: Map and Image
45
The difference between the map and the image Shown as discrete vectors
46
A table showing theMap to Image Displacements
CoordinatesMap Image x y u v25 11 18 0374 28 59 2921 51 12 4752 86 30 9263 12 49 1058 37 42 38
47
Difference Vectorsby themselves, without the grid
48
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.
49
Interpolated Vector Field
50
Great Lakes DisplacedThe grid has been ‘pushed’ by the interpolated vector field
51
The coastlines may be drawn using the warped grid
Observe that either the map, or the image, can be considered theindependent 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
52
53
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.
Mercator’s 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.
54
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, areemployed 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.
55
.
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
56
Hägerstrand’s Logarithmic Map
57
Another example:Conventional Way of Tracking Satellites
58
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, thenNorth, then South again.
59
Bend the meridians instead
60
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.
61
62
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
63
US population by one degree quadrilaterals
64
Now use the
Transform-Solve-Invert paradigmTransform 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, 219:215-220.
65
The lat/lon grid in the two spacesLeft, the usual grid. Right, transformed according to population.
66
US map in the two spacesLeft, the usual map. Right, the transform.
67
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
68
69
Some Recent ReferencesL. 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,,
LondonJ. 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.CJ. Snyder, 1993, Flattening the Earth: Two thousand Years of Map Projections,
University of Chicago Press, ChicagoQ. Yang, Snyder, J., Tobler, W., 2000, Map Projection Transformation, Taylor &
Francis, London
70
Thank You For Your Attention
You are now prepared to have fun with map projections.
71
The Santa Barbaran ViewA cube root distance azimuthal projection
