Projections: Theory and Geometry

Objectives of lecture:

1. Preparation for Lab: field work on a sunny day!
2. Reminder of basic tradeoff in map projections
3. Parameters involved for each type of projection
4. Distribution of error and Tissot's Indicatrix

Surveying issues:

You do not just make "measurements". It all depends on relationships (hence the need to "adjust" geodetic survey results.

Readings

Snyder selection weaves back and forth between projections and description of ellipsoids. Sections 2 and 3 sort of duplicate the geodesy readings.
The bit on auxillary latitudes is quite bizarre, and not used much anymore...)

Snyder personal history;

The big tradeoff:

Snyder (page 4) reviews basic goals of map projections;

• Equal area
• Conformal (shape preserving)
• Equidistance (only from certain points)

Other concerns:

scale (not preserved completely by ANY projection)
direction (conformal preserves LOCAL direction; azimuthal for some points)
Rhumb lines (Mercator); Great circles (gnomonic)

State Plane chooses conformal projections; UTM is too. Hence angles do not need scaling, but distances have a "grid to ground" conversion [local scale factor from projection surface to ground location]; areas even more complicated (Do the GISs convert area from grid to ground?)

Types of Projection

Developable surfaces: plane, cone, cylinder
and mathematical projections that don't actually work on a geometric basis...

Take a tour of the graphics from the MICROCAM site...

or a tour of the Hunter site (not recently updated, links may fail)

Dana's Projections section of the Geographer's Craft Project

Projection parameters:

The parameters play out the geometry of the developable surface choice.

Modification of "Normal Aspect":

Location of projection geometry

Error in projections:

Snyder has other books about projections and distribution of error

Tissot's Indicatrix deformation from ellipse created by tiny circles projected [1881]
basically obtained by a differential of mapping equations [yes, calculus]
While Tissot had a geometric connection, Laskowski (1989) in Accuracy of Spatial Databases shows how to obtain more easily through Singular Value Decomposition [SVD] (obtained from Taylor expansion of mapping equations) (Laskowski's personal favorite)

See some examples of the Indicatrix for standard projections (MICROCAM);

research on projection error in global change (Karen Mulcahy)

Controversy about Projections

Arno Peters contends that "normal" projections for world maps under-represent the less-developed world (which happens to be equatorial). Proposes his "Peter's" Projection. An equal area projection probably originally developed by Gall in the 1830s.
Certain cartographers take exception to Peters, some smoke... little fire?
National Geographic Society adopted Robinson's projection, which distorts every property (hand drawn, not mathematical) from 1988 to 1998; now they have noved to another compromise, the Winkel Tripel. (Their site deals with the schematic of the tradeoff...)
Monmonier deals with the Peters issue in his cartocontroversies book...

Resources on Projections

Hunter College Map Projection pages

Geographer's Craft Project

Dana's Coordinate Systems pages have a Geodetic Datums section and a Projections section

Stephan Voser, great list of standard projections for Europe and elsewhere. (with ESRI parameters for most of them...)

Another summary with some specifics on European solutions

Bob Burth list of links on datums and coordinates (some links to specific country solutions)

the chatter it often takes to figure out that Switzerland is an oblique Mercator (1997 email, all links dead...)!

Global tesselations:

Global Grids research (conference at Santa Barbara in 2000)
Oregon State approach (done for EPA)
Geoff Dutton's (Spatial Effects pages listed in the prior lecture...)

US Geological Survey: order form for USGS publications, UTM factsheet, (poster not on web!)...

Official Coordinates for Washington State (RCW (the server) , Text (partially converted of Title 58 Chapter 20)