Map Projection---basic ideas

S. Arlinghaus.

BASIC CARTOGRAPHIC CONCERNS

Scale and map projection:

From Robinson, p. 55

"No matter how the spherical surface may be transformed to the plane surface the relationships on the spherical surface cannot be entirely duplicated on the plane. Because of the necessary scale alterations, a number of kinds of deformation involving angles, areas, distances, and directions must or may take place; any system of projection will involve some or all of the following deformations:

• Similar angles at different points on the earth may or may not be shown as similar on the map.
• The area of one section may or may not be enlarged or reduced in proportion to that of another region.
• Distance relationships among all points on the earth cannot be shown without distortion on the map.
• Directions among divergent points cannot be shown without distortion on the map.
There are many other specific spatial conditions which may or may not be duplicated in map projections, such as parallel parallels, converging meridians, perpendicular intersection of parallels and meridians, the poles being represented as points, and so on, which may assume great significance for certain maps…"

Preservation of angles, locally.

When angles are locally preserved, with local shape following, the method of projection is said to be conformal.
Example:  Mercator, 1, 2.

Preservation of area.

A map projection on which all areas of figures are represented in correct relative size is said to be an equal area projection.
Examples:  Behrmann, Mollweide, Sinusoidal

Compromise projections.
Examples:  Miller, Tobler, Robinson.

An interrupted projection.
Example:  Philbrick Sinumollweide

Grid transfer:
Example:  Lat/lon

BASIC GEOMETRIC AND TOPOLOGICAL CONCERNS

One Point Compactification Theorem

Stereographic projection of the sphere (from the north pole) to a tangent plane (tangent at the south pole) projects all of the sphere, except one point--the north pole--to the plane. Distortion is ensured; the sphere cannot be mapped precisely in the plane. It is however useful to have flat maps; they are easily portable---a globe is not. Thus, numerous compromises are made in transferring the surface of the sphere (also a compromise on earth-shape) to the plane.

Projection Choice: Work of D'Arcy Thompson

The manner in which a projection is made, and the choice of projection, can have profound implications for meaning of output. Indeed, the notion of projection need not be confined to geography. Consider the accompanying projections of fish, in which one species is transformed into another, simply by projecting one grid to another! This work of D'Arcy Thompson (On Growth and Form), as it parallels some existing ideas in geography, served also to stimulate further geographic work (Waldo Tobler, Map Transformations of Geographic Space; and, later work of Tobler and others).

Overview of projections

Because there is no perfect map of the globe in the plane, and because there never can be (by the one-point compactification theorem), there is an infinite number of possible projections of the sphere to the plane. A few of these, along with some of their characteristics, are given below. As with Thompson's fish, the pattern of distortion of the underlying grid gives strong clues as to the pattern of distortion in the surfaces draped over that grid--be they fish-flesh masses or land masses.

Project the earth-sphere (generating globe) onto a plane with point of tangency at the south pole.

Project from:

• The north pole--stereographic projection--projects all but the north pole…however, distortion is severe as one moves away from the south pole. Spacing between successive parallels increases gradually as one moves toward the limit of projection.
• The center of the earth---gnomonic projection--projects less than a hemisphere into the plane--distortion is severe away from the south pole. Spacing between successive parallels increases rapidly as one moves toward the non-attainable hemispherical limit.
• A point at infinity, along the line determined by the poles--orthographic projection--projects exactly one hemisphere--distortion is severe away from the south pole. Spacing between successive parallels decreases rapidly as one moves toward the hemispherical limit.
Any point might serve as a projection center; additionally, one might move the position of the tangent plane relative to the projection center, or one might allow the plane of the projection to move--to have more than one, or no, intersections with the earth-sphere.

Developable surfaces:

Developable surfaces are those which can be cut to unroll perfectly into the plane (and, a section of the plane can be rolled up into such a surface). A sheet of paper may be rolled up into a cylinder. One might then consider projecting the surface of the generating globe onto the surface of the cylinder, tangent at a great circle, and then unrolling it. There are many so-called cylindrical projections and classes of projections based on the idea of cylindrical projections.

There are other developable surfaces. A cylinder may be made into a torus (doughnut) by joining the circular top and bottom ends of the cylinder. Both of these surfaces may be unrolled into bounded portions of the plane and either might have a map projected upon it from a generating sphere.

Two other developable surfaces are formed as follows: take a rectangle and give it a half-twist--now join the ends as if to make a cylinder--what has been made instead, is a Moebius strip. Then, join the ends of the Moebius strip (as if to make a doughnut from it)…what is formed is called a Klein bottle. These maps are of interest in the theoretical realm; in the practical realm, cylindrical maps are interesting…many projections of the globe are fundamentally formed in this manner. The map on a torus is interesting…the issue of how many colors are necessary and sufficient to color a map was proved on the torus in advance of being proved in the plane. The map on the Moebius strip is interesting…on it, antipodal points are identified (glued together, topologically). Why a map on a Klein bottle might be interesting is difficult to imagine, because a Klein bottle is difficult to visualize…there is theoretical issue that is entirely open. Thus, developable surfaces, as mapping surfaces, are of interest in both theory and practice.

Projection to the cone:

Place a cone on the earth-sphere, with the apex of the cone lying on the polar axis of the generating globe (simple conic projection). Project from the center of the generating globe onto the surface of the cone. The cone is tangent at a small circle; the projection becomes increasingly distorted as one moves away from the circle of tangency in the map surface. One way to improve this situation is to allow the cone to intersect the surface of the earth--then distortion increases as one moves away from each of these small circles in this "secant" projection. These small circles are referred to as "standard parallels" and a brief description of any conic projection should tell you the location of the standard parallels. When the cone is positioned as above, distortion is reduced in east/west direction. So, this sort of projection is better suited to a landmass or a nation with greater east/west extent than north/south extent---such as the USA. This observation suggests what is critical in making good choices for projections:

LET THE PROJECTION FIT THE UNDERLYING AREA AS BEST AS IS POSSIBLE.

What "best" means will of course vary from project to project, depending on emphasis. There are an infinite number of modifications of all of the projections: from moving the center of projection, to altering the spacing of the parallels and meridians by passing the simple spacing through a sort of prism, with angular variation based on mapping needs.

A FEW REFERENCES

Thompson, Sir D'Arcy Wentworth. On Growth and Form. Abridged Edition, edited by J. T. Bonner. Cambridge at the University Press, 1961, first published, 1917.

Robinson, Arthur H. Elements of Cartography. 2nd Edition. New York, Wiley, 1962, copyright 1953.

Snyder, John P. Flattening the Earth: Two Thousand Years of Map Projections. Chicago, University of Chicago Press, 1993.