An introductory lecture on Geographic 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

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.

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.

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

Cases

Common Geographic Locational Aliases and

Conversions

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.

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.

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

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.

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

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.

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

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.

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

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.

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?

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.

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

inventor. - Only a dozen or so are commonly used.
- Many GIS packages handle the most common

projections.

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

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.

From Globe to Map

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.

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

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.

Map Projection Properties Some 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.

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

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.

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

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.

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

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.

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

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

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

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..

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

display.

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

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.

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.

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

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.

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

population.

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

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

Thank You For Your Attention

- You are now prepared to have fun with map

projections.

The Santa Barbaran View A cube root distance

azimuthal projection