1. Map projections
CS 128/ES Lecture 3a

2. The dilemma Maps are flat, but the Earth is not!
Producing a perfect map is like peeling
an orange and flattening the peel without distorting
a map drawn on its surface.
CS 128/ES Lecture 3a

3. For example: The Public Land Survey System
As surveyors worked north along a central meridian, the sides of the sections they were creating converged
To keep the areas of each section ~ equal, they introduced “correction lines” every 24 miles
CS 128/ES Lecture 3a

4. Like this Township Survey Kent County, MI 1885
CS 128/ES Lecture 3a

5. One very practical result
CS 128/ES Lecture 3a

6. Geographical (spherical) coordinates
Latitude & Longitude (“GCS” in ArcMap)
Both measured as angles from the center of Earth
Reference planes:
- Equator for latitude - Prime meridian (through Greenwich, England) for longitude
CS 128/ES Lecture 3a

7. Lat/Long. are not Cartesian coordinates
They are angles measured from the center of Earth
They can’t be used (directly) to plot locations on a plane
Understanding Map Projections. ESRI, 2000 (ArcGIS 8). P. 2
CS 128/ES Lecture 3a

8. Parallels and Meridians
Parallels: lines of latitude.
Everywhere parallel
1o always ~ 111 km (69 miles)
Some variation due to ellipsoid (110.6 at equator, at pole)
Meridians: lines of longitude.
Converge toward the poles
1o =111.3 km at 0o = “ at 45o
= “ at 90o
CS 128/ES Lecture 3a

9. The foundation of cartography
Model surface of Earth mathematically
Create a geographical datum
Project curved surface onto a flat plane
Assign a coordinate reference system
(leave for next lecture)
CS 128/ES Lecture 3a

10. 1. Modeling Earth’s surface
Ellipsoid: theoretical model of surface - not perfect sphere - used for horizontal measurements
Geoid: incorporates effects of gravity - departs from ellipsoid because of different rock densities in mantle - used for vertical measurements
CS 128/ES Lecture 3a

11. Ellipsoids: flattened spheres
Degree of flattening given by f = (a-b)/a (but often listed as 1/f)
Ellipsoid can be local or global
CS 128/ES Lecture 3a

12. Local Ellipsoids Fit the region of interest closely Global fit is poor
Used for maps at national and local levels
CS 128/ES Lecture 3a

13. Examples of ellipsoids
Local Ellipsoids
Inverse flattening (1/f)
Clarke 1866
Clarke 1880
N. Am. 1983
(uses GRS 80, below)
Global Ellipsoids
International 1924
297
GRS 80 (Geodetic Ref. Sys.)
WGS 84 (World Geodetic Sys.)
CS 128/ES Lecture 3a

14. 2. Then what’s a datum?
Datum: a specific ellipsoid + a set of “control points” to define the position of the ellipsoid “on the ground”
Either local or global
> 100 world wide
Some of the datums stored in Garmin 76 GPS receiver
CS 128/ES Lecture 3a

15. North American datums
Datums commonly used in the U.S.: - NAD 27: Based on Clarke 1866 ellipsoid Origin: Meads Ranch, KS - NAD 83: Based on GRS 80 ellipsoid
Origin: center of mass of the Earth
CS 128/ES Lecture 3a

16. Datum Smatum NAD 27 or 83 – who cares?
One of 2 most common sources of mis-registration in GIS
(The other is getting the UTM zone wrong – more on that later)
CS 128/ES Lecture 3a

17. 3. Map Projections Why use a projection?
A projection permits spatial data to be displayed in a Cartesian system
Projections simplify the calculation of distances and areas, and other spatial analyses
CS 128/ES Lecture 3a

18. Properties of a map projection
Area
Shape
Projections that conserve area are called equivalent
Distance
Direction
Projections that conserve shape are called conformal
CS 128/ES Lecture 3a

19. An early projection Leonardo da Vinci [?], c. 1514
CS 128/ES Lecture 3a

20. Two rules:
Rule #1: No projection can preserve all four properties. Improving one often makes another worse.
Rule #2: Data sets used in a GIS must be displayed in the same projection. GIS software contains routines for changing projections.
CS 128/ES Lecture 3a

21. Classes of projections
Cylindrical
Planar (azimuthal)
Conical
CS 128/ES Lecture 3a

22. Cylindrical projections
Meridians & parallels intersect at 90o
Often conformal
Least distortion along line of contact (typically equator)
Ex. Mercator - the ‘standard’ school map
CS 128/ES Lecture 3a

23. Transverse Mercator projection
Mercator is hopelessly distorted away from the equator
Fix: rotate 90° so that the line of contact is a central meridian (N-S)
Ex. Universal Transverse Mercator (UTM)
CS 128/ES Lecture 3a

24. Planar projections a.k.a Azimuthal Best for polar regions
CS 128/ES Lecture 3a

25. Conical projections Most accurate along “standard parallel”
Meridians radiate out from vertex (often a pole)
Poor in polar regions – just omit those areas
Ex. Albers Equal Area. Used in most USGS topographic maps
CS 128/ES Lecture 3a

26. Compromise projections
Robinson world projection
Based on a set of coordinates rather than a mathematical formula
Shape, area, and distance ok near origin and along equator
Neither conformal nor equivalent (equal area). Useful only for world maps
CS 128/ES Lecture 3a

27. More compromise projections
CS 128/ES Lecture 3a

28. What if you’re interested in oceans?
CS 128/ES Lecture 3a

29. “But wait: there’s more …”
All but upper left:
CS 128/ES Lecture 3a

30. Buckminster Fuller’s “Dymaxion”
CS 128/ES Lecture 3a