Sunday, October 21, 2007

The Robinson Projection: Not shaped like an Egg

A Pleasing View of the World

The Robinson Projection came into being in 1963 and was introduced by Dr. Arthur H. Robinson.

This projection can be classified as a pseudo-cylindrical projection because of its straight parallels, along each of which the meridians are spaced evenly. The central meridian is also a straight line and all other meridians are curved.

The projection is neither equal-area nor conformal, therefore abandoning both for a compromise for creating what Dr. Robinson felt produced a better overall view of the world. This was the first map projection to be developed for commercial interests. Rand McNally felt that many of the map projections in use did not present the earth as a whole very well. With Mercator the poles were distorted. Robinson, was essentially contracted to develop a map projection that did not maintain angle, direction, or limit distortion, but was sanctioned to produce a map projection that "looked good" for books and atlases.

Remember maps are designed for one of the following 4 reasons:

  • Conformality - the shapes of places are accurate
  • Distance - measured distances are accurate
  • Area/Equivalence - the areas represented on the map are proportional to their area on the earth
  • Direction - angles of direction are portrayed accurately

Dr. Robinson specified the projection to be constructed by referring to a table of cartesian coordinate values at specific intersections of latitude and longitude. The intermediate locations are to be found by interpolation. Dr. Robinson developed the projection through a series of trials, continually iterating till he settled upon the meridian shapes and parallel spacing most pleasing to the eye. In comparison with other Map Projections, they are mainly developed or are formulated as mathematical equations.

Parallels are straight parallel lines, equally spaced between latitudes 38 degrees north and south. Space decreases beyond these limits. The Equator is 0.8487 times as long as the circumference of a sphere of equal area. The central meridian is a straight line 0.5072 as long as the Equator. Other meridians are equally spaced elliptical arcs and concave toward the central meridian. The scale is true along latitudes 38 degrees north and south, constant along any given latitude, and the same for the latitude of opposite sign (Robinson 1974; Snyder and Voxland 1989).

This map projection is also based on a sphere, not an ellipsoid. This is an important point to remember, as the earth is being modelled differently.

What is the true shape of the Earth?

The Earth, in actual fact, is shaped more like an egg, hence, even an ellipsoid is not the best model. In the past, when datum's were more local (such as NAD27, SAD69, Cape Datum, etc.), various ellipsoids were tied to local points on Earth. For NAD27, this was Meades Ranch, Kansas.

But back to the Earth's shape; it is very close to an oblate spheroid — a rounded shape with a bulge around the equator — although the precise shape (the geoid) varies from this by up to 100 metres.

The rotation of the Earth creates the equatorial bulge so that the equatorial diameter is 43 km larger than the pole to pole diameter.

What about height? A whole other story.

Interesting, eh? There are so many different ways to see the Earth. When we look at height systems and describe the geoid, we are describing an equipotential surface. But height is a whole other story, as gravity is involved and it is not as simple as getting the "Zed" or "Zee" measurement from your GPS.

I'll explain height later and how we can determine MSL (and where the "mean" actually comes from).


SharpGIS said...

What projection preserves distance? The only ones I can think of is equidistant projections but they are only accurate extending from one point, along a line or in one specific direction.

Terra ETL said...

You are correct, equidistance projections perserve some aspect of distance over some portion of the map. With equidistant projections, it is not possible to represent all distances correctly. As you accurately point out it is possible to produce a projection such that all distances from one or two points are true to scale or a projection on which all distances measured perpendicular to a standard line are true to scale. An example being, the Cylindrical Equidistant Projection where parallels of latitude are equally spaced along meridians and therefore the distance between parallels being equal to the arc length between parallels on the generating globe. Thereby any distance measured along the Equator (tangent case) or standard parallels (secant case) and distances measured along any meridian will be true to scale. All other distances are distorted. The cylindrical equidistant projection is also called an Equirectangular Projection or Plate Carre. Another example is the Two-Point Equidistant Projection which was first described by Hans Maurer in 1919. On this map projection any distances from any point on the map to two control points scale to the geodesic distances of the same points on the sphere. Hence it is sometimes used for flight maps depicting air routes.

nmm said...

Do you have any links to longitude/latitude conversion techniques for the Robinson projection? You mention the data tables and extrapolation techniques but I haven't found anything more specific than that (in particular, which tables to use and how to do the extrapolation).

Many thanks - Nils.

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