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)
and 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 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.)

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".

ASSIGNMENT:

Use either of the Map Projection applets to display and examine ten different projections of the world. For each of them

  1. note the name,
  2. note the overall shape of the projection,
  3. are meridians and parallels are straight or curved, do they converge or not
  4. note which properties (distance, angle, area, shape, special/other) the projection seems to preserve and to sacrifice (perhaps citing what you notice about the indicatrix across each map)
    5. .

Projection Name Overall Shape Meridians Form Parallels Form Properties Preserved or Compromised? Why?
         
         
         
         
         
         
         
         
         
         

THE REPORT:

Include the filled-in table.

Write a description (one half page max) summarizing the utility of Tissot's indicatrix as used in this applet.

For the heavier thinking part... (one half page or so) (how) could you use this applet to estimate the range (or the worst) Scale Factors for each of your ten projections? (The applet does not give it directly, but how could you get at it?)