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 shape of an infinitely small circle on a globe that is then projected to the map. The graticule's right angles are probably distorted as are the lengths of the circle's radii, and the resulting figure is 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 (see note below) 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 . 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 (Fall 2014): The current 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 projedtion to show), and check "Tissot's Indicatrices".

Another alternative... you can use the Wikipedia entry on "List of Map Projection" to examine the projections. (This is less satisfactory for seeing representations of Tissot's Indicatrix and rhumb-lines (or loxodromes) but is less reliant on 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 map projections.

For each of them:

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...,
presereves...  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?)