### Map Projection Distortions with Scale Factor and Tissot's Indicatrix

OBJECTIVE: The primary objective is to examine the distortions of relative area, shape, and distance across various map projections.

BACKGROUND: The Scale Factor (Robinson and Sale 1969, Chap 2, p 35), is one way to assess scale or distance distortion. It is the ratio:

(denominator of principle scale) / (denominator of actual scale)

expressed as a single number (e.g. 1.8). It assumes that you can calculate the actual scale between two points on the map (rememeber RF is (map-distance / earth-distance) reduced to a unitless ratio to one, 1:denomnator).

Another way that cartographers have characterized these distortions systematically is by using Tissot's Indicatrix (see pg 209 in Robinson and Sale 1969, or Wikipedia), which was presented in an apparently posthumous 1881 treatise on map projections. The basic notion is to consider the distortion of the shapes of infinitely small circles on the globe that are then projected to the map. The circles' distortions indicate the distribution of distortion over the map. The graticule's right angles are probably distorted as are the lengths of the circle's radii; the resulting figures are generally elliptical. Considering a set of such figures distributed across the map gives a graphic indication of the distribution of the distortions inherent in the projection. We will use web-based software to visually compare the distortions in several projections applied to the globe.

SOFTWARE: For this exercise we will use a Java applet written by Rogério Vaz de Almeida Jr, Jonas Hurrelmann, Konrad Polthier, and Humberto José Bortolossi and available at www.uff.br/mapprojections/mp_en.html . (N.B. Recent versions of windows will probably require that you configure Java to allow the applet to run. See the note, below. ). The applet shows three panels: the globe on the upper left, the projected map on the upper right, and a control panel at the bottom. To use it: (1) on the "Visualization" tab, check Coastline, MapProjection's Globe and Graticule; (2) on the "Position" tab, select two cities a hemisphere or so apart, and check the "Show loxodrome curve" box. (3) on the "Projections" tab, check the "Show Tissot's Indicatrices" box, and then select a projection via the drop down selection. (4) Examine the shapes of the indicatrices to assess the distortions. (The loxodrome and straight line on the map are just along for the ride, but you might note how they act on the various projections.)

Note The current (Fall 2014) Java plug-in seems to have been "hardened" to increase security; you probably need to configure it. On Windows, navigate to: "windows globe" -> All Programs -> Java -> Configure, then set the security to Medium, and add the "www.uff.br/mapprojections/mp_en.html" site to the list of permitted sites. Not sure on Mac or Linux.

Alternatively, you can use FlexProjector, by Bernhard Jenny of ETH Zurich, which is available at www.flexprojector.com . You'll need to download and install the software on your computer. To cut right to the exercise, choose the "Display" tab, uncheck "Show Flex Projection", check "Show Second Projection" (this is where you select which projection to show), and check "Tissot's Indicatrices".

Another alternative (though getting ahead of the schedule a little and not as informative) is to use QGIS or ArcGIS to examine different projections. If you know how, feel free.

Yet another (and poorer) alternative... you can use the Wikipedia entry on "List of Map Projection" to examine a number of projections. (This is less satisfactory for seeing representations of Tissot's Indicatrix and rhumb-lines (or loxodromes) but is less bothered by Java changes.)

### ASSIGNMENT and REPORT:

Use one of the Map Projection applets, or Wikipedia's entry for "Map Projection", (Or ArcGIS or QGIS if you are ready) to display and examine ten (10) different world map projections.

For each of them:

• note the name,
• note the overall shape of the projection,
• note the shapes of the meridians and parallels (straight or curved, converging or diverging, parallel/concentric)
• note which properties (distance, angle, area, shape, special/other) the projection seems to preserve/sacrifice (hopefuly citing what you notice about the indicatrix across each map).
either as a table, like:
Projection Name Overall Shape Meridians Shape Parallels Shape Properties Preserved or Compromised? Why?

or as an annotated list, like:
```
projection name

overall shape..., meridians do..., parallels do...,
preserves...  refer to Tissot's Indicatrix and or
the shape and sapcing of the graticule.

projection name

etc...
```
And then, for the heavier thinking part (one half page or so) describe the pros and cons you see of using Tissot's indicatrix as opposed to Scale Factor to understand map distortion. (Which should include succinct statements of what those are, right?)

Extra credit question... what projection does Google Earth use on the display?

### Other Sources

USGS web page on Map Projections that they use. Here.

USGS Map Projection poster in pdf. Here. And probably elsewhere at USGS iwith a little more looking.

Wikipedia on the Web Mercator projection used in GoogleMaps, open street map etc. is described here . (Between 85.051129 degrees north and south, it's a square, man.)