Lecture 8

Map Projections

NOTE: these notes are adapted from class overheads, but many additional illustrations will be shown in class. Do not rely only on these notes! You still need to come to class.

1. Definition

--- Map Projection: A method for transferring features from a sphere to the appropriate locations on a plane.

- Needed because flat maps are convenient, easy to store and print in books.
- Not significant for maps of small areas (less than 100 km across)
--- (all projections look the same for small areas)
- Earliest maps (over 3000 years ago) were of small areas.
- Projections first devised when maps of large areas were compiled.
- 2500 to 2000 years ago in Greece.
--- The world was already known to be a sphere.

2. Physical Projection Method

- Features are plotted on a transparent globe.
--- A light at a suitable location casts shadows of the features onto a screen.
--- The map is traced from the shadows.
--- Different light positions give different map shapes.

- The screen may be flat or wrapped around the globe.
--- It can be flat (a plane), or a cone or cylinder.
--- Cones and cylinders curve in only ONE direction.
--- They can be CUT and UNROLLED into a plane without distortion.
--- Different screen shapes give different map shapes.
--- The screen is called a Developable Surface. (means it can be made flat)

- CLUMSY and impractical but a useful analogy.
--- Some projection names describe the screen shape:
--- cylindrical, conic, plane

- Globe manufacture: reverse of physical map projections.
--- Globes are usually made by printing a map in small strips (gores).
--- the gores are outlined by meridians (north-south lines).
--- gores printed side by side are pasted over a blank globe.
--- look carefully and you can see joins between strips.

3. Mathematical Projection Method

- Feature locations are defined by latitude and longitude.
--- Equations transform latitude and longitude to X, Y coordinates on the map.
--- Two equations are needed: one for X, one for Y
--- Flexible, practical, especially for computers.
--- Allows new projections which have no physical analogy.

- Equations become complicated because many factors are included:
--- positions on globe
--- position of centre of map (can be anywhere)
--- desired characteristics of map
- (these correspond to the light position and screen shape in physical projections)

- In this course we don’t look at the equations.

4. Projections in GIS

- Geographic Information Systems combine multiple maps for analysis.
--- Maps must match exactly.
--- Every detail of map projections must match.

- Users may be asked to choose projection from menu.
--- Other information like centre of projection may need to be given.

- Projection specifications include assumptions about the shape of the world.
--- e.g.: North American Datum of 1983.
--- Sources of data should include all necessary information.

5. Distortion

- Cause:
--- A sphere curves in two directions at once.
--- north to south, and east to west.
--- flattening one causes the other to shrink or stretch.
--- Not a problem for small areas

- Partial solution:
--- Accept some distortion.
--- Minimize it in the most important areas of the map.
--- Trade off one kind of distortion for another.

- Types of Distortion:
--- Area
--- Shape

- We can trade one for the other:
--- perfect shapes but areas distorted unequally.
--- areas all true to scale but shapes distorted.
--- IMPOSSIBLE to combine true shapes and equal areas.

6. Classification of projections

- The many types of projection can be grouped in classes.

- by distortion type:
--- area, shape, or both are distorted.
--- we refer to a projection by its undistorted characteristic.

------ "an equal-area projection" (area is not distorted) (also called "equivalent")
------ "a conformal projection" (shape is not distorted)

- by geometry
--- plane, cone, cylinder, other

------ "a cylindrical projection"

------ "a conic projection" etc.

- Note: special property of planar projections (projected onto a plane):

--- the point where the plane touched the globe is the middle of the map.
--- from this point all features are in the correct azimuth (direction).
--- So plane (planar) projections are often called Azimuthal.

- Almost every combination is possible, so a projection can be:
--- "Equal Area Conic", "Conformal Conic" etc.
--- "Equivalent Cylindrical", "Conformal Cylindrical" etc.

7. SIMPLE MAP PROJECTION CLASSIFICATION

(names of projections are just examples - there are many more possibilities)
Developable
Surface:		 PLANE CONE CYLINDER NONE
Projection
Class:		 Planar
(Azimuthal)		 Conic Cylindrical Mathematical
DISTORTION
TYPE:
 - - - -
EQUAL AREA
(equivalent)
 Lambert's
Equivalent
Azimuthal		 Albers
Equal Area
Conic		 Cylindrical
Equivalent		 Mollweide
TRUE SHAPE
(conformal)
 Stereographic Conformal
Conic		 Mercator -
OTHER
 Orthographic Equidistant
(simple)
Conic		 Simple
Cylindrical		 -

Other classes:

-- Interrupted (not all gaps between gores are closed)
-- Condensed (areas not needed are omitted, needed areas are fitted closer together. e.g. oceans omitted from population map).

8. Choosing Projections

-- Many projections to choose from.
-- Select according to two needs:

acceptable type of distortion.

-- Conformal (angles correct): navigation charts, migration, wind, ocean current maps
-- Equivalent (areas correct): distribution maps (population, geology, vegetation etc.)
-- (rarely used) Equidistant (distances correct, but only from point or line where developable surface touched globe)
-----(used for travel distance maps)

location of area of interest.

-- Put the area of most interest in the area of least distortion:
-- Polar exploration: planar projection centred at a pole.
-- Population: cylindrical projection centred on the equator.
-- Topography of Australia: conic projection centred at the latitude of Australia.
-- Southern Canada: conic projection centred around 50 degrees North.
-- Canadian Arctic: conic projection centred around 70 degrees North.
-- United States (contiguous): conic, centred around 35 degrees North.

9. CONFORMALITY

'True shape'.

-- Only applies perfectly at a point.
-- Shapes of large areas like continents are distorted in conformal maps.

Two components of shape:

-- angles must be correct (latitude/longitude grid lines cross at 90 degrees).
-- scales must be equal in all directions (north-south AND east-west).

Some points are always excluded from a conformal map:

-- If a geometric point (e.g. pole) is stretched to visible size on the map, the degree of stretching is infinite.
----- for example:
-- Planar projection centred at North pole - South pole will be stretched to form the circumference of the map.
-- Cylindrical projection centred at equator - poles will be stretched into lines as long as the equator.
-- To make conformal, an infinite amount of north-south stretch would be needed at the poles.
-- This is impossible, so Conformal maps can never cover the whole world
-- (use 2 separate maps, one for each hemisphere, or map poles separately on another conformal projection). Check out these links for illustrations of many projections:

http://www.geometrie.tuwien.ac.at/karto/index.html

http://www.geography.hunter.cuny.edu/mp/