1. Projections

2. Review The Earth is a complex shape called a geoid
Ellipsoids are models that approximate the shape of the Earth
Ellipsoids are used in place of the geoid because they are much simpler mathematically
Datums link the geoid (real shape) to the ellipsoid (modeled shape)

3. Map Projections
The systematic transformation of points on the Earth’s surface to corresponding points on a plane surface
In other words: Translating the Earth (3D) to a flat map (2D)
All projections distort the Earth in one or more way(s)
Selection of a projection is done to minimize distortion for the particular application

4. Why do we need a projection?
Creating maps
we must choose an appropriate projection for the map to communicate effectively
part of good cartographic design
Analyzing geographic data
along with the datum (and the associated ellipsoid) and the coordinate system we must know the map projection in which the data are stored
identical projections are required for data to overlay correctly

5. Common types of projections
Imagine the light source (light bulb in picture) shining through a clear globe onto a piece of paper (either flat, rolled into a cylinder, or made into a cone). The image that shines on the paper becomes the map.
There are many more types of projections than these 3
(a) Azimuthal (b) Cylindrical (c) Conic

6. Views of projected surfaces
When unrolled the lines of latitude and longitude on the projected maps look something like this

7. Azimuthal Projections
Common for polar areas
meridians (lines of longitude) are straight lines, radiating regularly spaced from the central point
parallels (lines of latitude) are complete concentric circles
One on the right = Lambert's Azimuthal Equal-area Projection

8. Cylindrical projections
                                                                                                                                                  
Lines of latitude and longitude are straight.
Commonly used for tropical regions (near the equator) where distortion is the least problematic with a regular aspect & tangent to the equator)
You cut the cylinder along any meridian and unroll it to produce your base map. The meridian opposite the cut is called the central meridian (the red line). (ESRI Press.)

9. Cylindrical projections (Cont.)
                                                                                                                                                                      
Note that distortion increases towards the poles
The light source's origin for the map projection is also the origin of the spherical coordinate system, so simply extending the degree lines until they reach the cylinder creates the map projection. The poles cannot be displayed on the map projection because the projected 90 degree latitude will never contact the cylinder. (ESRI Press)

10. Conic Projections
Conic projections are often used for countries in mid latitudes (between the tropics & the poles, i.e., between ~23 & ~66 degrees north & south)

11. Standard Lines and Points
Location(s) on a projected map at the exact point or line where the surface (cylinder, cone, plane) touches the globe
Standard lines and points are free of all distortion
Distortion becomes more pronounced with increased distance from the standard line or point

12. Additional Projection Features
Projections get a lot more complicated because we can:
1) Change the aspect
2) Move the light source
3) Change where the paper touches the globe

13. Projection Aspects cylindrical conical planar North on Top in graphics
“Normal” is a synonym for “regular”
planar

14. Light source options
Orthographic (light source infinitely far away – think of the sun)
Stereographic (the point opposite of the point of tangency of the projection)
Vertical (how the earth would look from space)
Gnomonic (center of earth)
Draw graphic on board

15. Change where the paper touches the globe
Tangent case – the paper rests against the surface of the globe
Secant case – the paper goes into and back out of the globe (intersecting at 2 standard lines)
Standard line
Standard line
Standard line

16. Example of tangent and secant azimuthal projections
Note that the location of distribution changes in relation to the standard lines

17. Preservation of Properties
Map projections always introduce some sort of distortion. How to deal with it?
Choose a map projection that preserves the globe properties appropriate for the application
Note: The preservation of properties offers an alternative -- perhaps more meaningful -- way to categorize projections

18. Map projections distortion
Projections cause distortion. The projection process will distort one or more of the four spatial properties listed below. Distortion of these spatial properties is inherent in any map.
Shape
Area
Distance
Direction

19. Preservation of properties
Conformal projections
-preserve shape
shape preserved for local (small) areas
(angular relationships are preserved at each point)
sacrifices preservation of area away from standard point/lines
Equivalent/Equal-Area projections
-preserve area
all areas are correctly sized relative to one another
sacrifices preservation of shape away from standard point/lines

20. Equidistant projections
-preserve distance
scale is correct from one to all other points on the map, or along all meridians
however, between other points on map, scale is incorrect
Azimuthal projections
-preserve direction
azimuths (lines of true direction) from the center point of the projection to all other points are correct

21. A Few Common Map Projections Used For Display
                                                                                                                                                     
The Mercator (conformal) projection maintains shape and direction.
The Sinusoidal and Equal-Area Cylindrical projections both maintain area, but look quite different from each other.
The Robinson projection does not enforce any specific properties but is widely used because it makes the earth’s surface and its features "look right.“ (ESRI Press)

22. Map Projections Commonly Used For GIS Applications
Mercator: True compass directions are maintained (lines of latitude and longitude are at right angles to each other), but area is distorted toward the poles
This is a cylindrical projection
We often use Universal Transverse Mercator (UTM), which is a coordinate system applied to a Mercator projection
We’ll discuss coordinate systems more next week

23. Map Projections Commonly Used For GIS Applications
In the US these are also common projections
Lambert Conformal Conic – a conic projection that preserves shape
Albers Equal Area – a conic projection that preserves area
Remember that these projections (including UTM) can be associated with the same datum (e.g., NAD 1983, NAD 1927, etc.), which is in turn associated with a corresponding ellipsoid (e.g., NAD 1983 uses the GRS 1980 ellipsoid)
We’ll discuss coordinate systems more next week

24. True Direction & Distance
Cylindrical projection – Mercator (conformal) projection maintains shape and direction, but not distance (see flight to Japan) or size:
Greenland = 2,166,086 km^2
South America = 17,840,000 km^2

25. Tissot’s Indicatrix
The Tissot indicatrix is a figure that shows how a projection changes the geometry.   
It does so in a simple manner: by showing what a circle would look like on the map.
This is an equal area projection.
Blue circles are the projected circles (here, ellipses).
Grey circles are reference circles.
Radii are for reference regarding distance distortion.

26. Area scale An indicator of distortion on projected maps.
s = "area scale" = product of semi-axes of circle/ellipse.

27. Conformal vs. Equal-area projections

28. Examples of projections
Do the following examples clear up some myths we have grown to believe?

29. Equidistant example Azimuthal Equidistant projection
- planar with standard point centered on North Korea
Note: equidistant maps preserve distances from a the central point, not from one point to another.
Myth of where Korea is… most people in our country think it’s much further south than it is, in actuality the Korean Peninsula is similar in latitude than the Mid-Atlantic states and as close to the North Pole as it is to the Equator

30. True Direction & Distance
Gnomonic
- planar with standard point located at NYC
The great circle direction is the shortest distance between points on the Earth, but due to distortion from map projections this appears not to be the case.
myth: transatlantic flights go “out of their way”
The Mercator projection is misleading because lines of longitude actually converge at the poles, but they don’t appear that way
Other examples include flights from the US to Japan which typically fly near the north pole.
...compare with Mercator projection:

31. Example: What projection might this map be in?
Most published maps of the US won’t identify the projection
We can tell this map is NOT in a cylindrical projection because the lines of latitude are not straight (see 49th parallel - US-Canadian boarder in the western US).
But beyond that it’s hard to tell. It could be conic (maps of temperate countries frequently are) or some form of azimuthal. Realistically it doesn’t matter much for particular map, but it does distort what point is the farthest north (Maine and Washington seem to be the farthest, but it’s actually the Lake of the Woods area in northern Minnesota)

32. Compromise projections
...don’t preserve any properties completely, but achieve compromise between them
Example: Robinson projection - designed for world maps
Maps that “look right”

33. Wrap-Up Projections are a large and complicated topic to study
There are many more that we didn’t talk about
In practice, we generally stick with just a few common ones (e.g., whatever one our office typically uses)
Only when a new and different task comes up do we start to investigate IF a different projection is needed and WHAT that projection should be
If you’re still curious about them, this is a pretty good site

34. Odds and ends What night is good for an open lab?
Openings Monday, Tuesday, Wednesday