The Cartographic Process: Data Manipulation

What I really need to know about Map Projections

Terms to know:

Geoid (mathematical model of earth’s shape)
Spheroid (slightly flattened geoids, eg., Clarke 1866, 1880)
Datum (a frame of reference, a baseline or point)
Geographic coordinate system (GCS) (uses spheroids to calculate position)

Need more? Glossary and Summary of Projection Properties

Definition of Projections

Because the earth is round and maps are flat, getting information from a curved surface to a flat one involves a mathematical formula called a map projection, or simply a projection:

A projection: is a mathematical formula that transforms feature locations between the earth’s curved surface and a map’s flat surface.

The Fundamental Question:

Why does Google Earth 0o line not line up with Greenwich?

Click this link to launch Google Earth

Turn on Lat-Long Grid under the "View" menu in GE

So what is going on here? Shouldn't they line up as they are both zero degrees?!

The short answer is: "Google Earth uses the WGS-84 coordinate system which puts the prime meridian 102.5 meters to the east"

But in order to understand this answer we need some more tools!

It has to do with projections of the earth, and the "datum" they are in


It is very typical that one dataset will not line up properly with another

Let's see why!

Google Earth uses the WGS-84 coordinate system which puts the prime meridian 102.5 meters to the east

The Prime Meridian at Greenwich

Here is another example:

The projections do not match up!

Cool T-shirts!

Basic definitions we need:

Projection: a transformation of the three dimensional curved earth to a two dimensional map

Spheroid: a sphere-like shape, the earth for example. Based on a model of the elliptical shape, called an ellipsoid

Georeferencing: spatial data tied into a coordinate system

Coordinate system: Cartesian coordinates (x,y) on a plane, or geographic coordinates (lat, long). Have a point of origin.


Next question: why do data not line up:

  • Sometimes we get data from other data sources (Census, internet, GIS Data Clearing Houses)
  • 99.9% of the time, this data is usually "in" a certain map projection already
  • Data may not match up with other data

Therefore when this happens we need to know what to do about it!


Georeferenced data are tied to a place or into a coordinate system (eg lat-long)

Non georeferenced or aspatial data is not tied it is useless for GIS!

How are they tied in? Tied in with a GIS model of the earth's size and shape called a spheroid (or oblate ellipsoid)

Since earth rotates, it bulges at the equator. It's not quite spherical in other words.

At different times in the past people have tried to mathematically model the earth's true shape...or come as close as they can. This was a big question in the nineteenth century... it was called the "lemon vs. orange" problem, referring to the shape of the earth


The earth bulges a bit: it's a spheroid

Local vs. global referencing

In making maps of small areas (large scale maps, or zoomed in maps) people used a local reference system.

Examples of local reference systems are: "Go five miles up the Interstate and take the next exit"

The directions are local or relative to a "control point."

Or, "A plot of land extending 10 miles west of the Mississippi"

Maps made on these local systems work well locally, they fit the terrain and the territory, but do not fit together on a smaller scale

Example: North American Datum 1927 (NAD27), based on Meades Ranch, Kansas (was the geodetic center of the USA, but not the earth!)

Today we need global referencing!


Meades Ranch, Kansas

More Definitions:

Flattening: how much the earth's sphere is flattened, eg., 1 in 300

Geoid: Approximation of earth's shape using equal gravitational attraction

Datum: A reference point or surface from which positional measurements are be made, based on a spheroid. (More on datums, Mike Price ESRI)


Spheroids and Flattening

Today we need global referencing!

Why? Because you need a global system of coordinates to tie into when you map your territory.

During 19th century world became more internationalised and global... timezones.

Today we have cruise missiles, satellites in orbit, car navigation systems, GPS, Google Earth, etc.

So a more global referencing system was developed in the 19th Century

The British Empire... work of Sir Alexander Ross Clarke resulted in a spheroid in 1866 (Clarke Spheroid) later updated in 1880

Clarke calculated the world was flattened by about 1 in 294.9787 or 1 in 300

Flattening (f) = 1 - b/a, where a, b are the radii of the sphere

In the 20th and 21st century we increasingly use remote sensing and satellite data to more accurately model the earth. Also, during the 1950s the USA and Soviet Union needed to accurately target their intercontinental ballistic missiles (ICBMs)!

These efforts include:

  • North American Datum (NAD) of 1927 (NAD27). Based on Clarke.
  • North American Datum (NAD) of 1983 (NAD83). Based on GRS80. [Good for USA maps]
  • World Geodetic System (WGS) of 1984 (WGS84) [Good for world maps]

NAD83 was based on NAD27, which in turn was based on Clarke.

Old data: NAD27
New data: NAD84, WGS84. Based on a new spheroid GRS80


WGS84 (and NAD83) are a global datum, an overall best fit, so it can be "off" in certain places... Greenwich! This is called a "datum shift."

Also, you can bet that some data is still in the old scheme, NAD27, and some in the new scheme, NAD84 (which itself is only good through 2010).

Textbooks give values for WGS84 of:

a = 6,378,137 meters (6378 kilometers)
b = 6,356,752.3 meters (6357 kilometers)

so f = 1/298.257

(Circumference is 2 pi R or about 40,000 kilometers)

Sir Alexander Ross Clarke, geodesist

Map of the British Empire 1886 (Mercator projection)










Old data. Comparison of 1866 Clarke spheroid, earth surface and geoid



New data. NAD83 based on a new ellipsoid model, replacing Clarke.

Optional ArcCatalog Practice

The Clarke spheroid lives on in ArcCatalog and navigate to:

and look for "world30.shp"

Click on it and examine its metadata

What is its datum?
What is its ellipsoid?
What is its flattening ratio?
Is this the same as the value given for Clarke above?
What is the ratio for WGS84?



Projecting the earth's sphere to flat surface or GIS layer

Many, many possible projections. About 200 "popular" ones:

  • Mercator
  • Alber's equal area
  • Goode's homosoline (p. 44)
  • Peters projection (featured on West Wing)

Here are the main ones: USGS Map Projections Poster

They have some useful summaries of which ones to use

What you need to know: The Four Distortions

Which one is best depends on what kind of distortion you want to minimise. There are four types of distortion: distance, area, shape, direction.

If a projection preserves...


It is known as...


equidistant projection


equal area projection


conformal projection


azimuthal projection

No projection can preserve all these properties; as a result, all flat maps are distorted to some degree.

Distances, MM pp. 104-5

Shape (conformality), MM pp. 102-3

Area (equivalence), MM pp. 100-1

Direction, MM pp. 104-5

MM = Making Maps, by Krygier & Wood

Give examples of situations where you want to preserve these properties:

These distortions can be graphically portrayed with Tissot's Indicatrix:

Monsieur Tissot invented a way to show what happens to a circle on the earth's surface after it has been projected. The size or shape (or both) may be distorted.

Click Mercator to see his projection

Mercator projection with Tissot circles

Robinson projection with Tissot circles


Optional Projections Exercise
How to use projections in ArcGIS

  1. Open ArcMap and add the "continent.sdc" data from the data folder
  2. Double click the Layers in the Table of Contents (ToC) and then select the Coordinate System tab
  3. Write down the name of the Coordinate System
  4. Notice that there is a folder called "Predefined" with "Projected Coordinate Systems" inside

I need to know: Almost always in GIS you will use projected coordinate systems ("named" projections). Otherwise calculations based on area, distance, direction will be incorrect.

Okay, let's look inside the Projected Coordinate Systems folder and see what's available. Inside the World folder try:

  • Mercator
  • Robinson
  • Albers equal area (under the Continental/North America folder)

Some of them look odd!

To see just how odd, switch to Layout view. Set page to Landscape orientation.

Under Properties of the Dataframe select the Grids tab. We'll make a grid (or graticule). Set graticule interval at 30 degrees.


  • preserving area: equal area projections
  • preserving shape: conformal projections
  • but, cannot preserve both area and shape in same projection
  • preserving distance: equidistant projections
  • but, cannot preserve both area and distance in same projection
  • preserving direction: azimuthal projections
  • but, can only preserve direction and area OR shape OR distance


 Need more? Glossary and Summary of Projection Properties