Cartographic Projections

Lloyd A. Treinish
lloydt@watson.ibm.com

IBM Thomas J. Watson Research Center
Yorktown Heights, NY

Introduction

 Cartography is an ancient art and science of methods to project -- mathematically transform all or part of the surface of a sphere (e.g., the earth) onto a two-dimensional, flat surface or plane. The process of map projection introduces distortions of the data and/or its geometry. The choice of a specific projection method in visualization is very important for the proper communication of information. It is very much dependent on the visualization task (e.g., exploration, analysis, presentation, decision support, etc.). Too often, a very popular projection, such as Mercator, or a simple rectilinear projection is employed without knowledge of the resultant distortion of the visualized data. As a very brief introduction to the subject, consider four families of map projections: cylindrical, conic, azimuthal and singular, some of which have been applied to a data set of coastlines and national boundaries consisting of 109,928 points on 1247 lines. Each of these lines consists of anywhere from two to 3949 points.

Cylindrical Projections

Cylindrical projections are based upon the projection of the earth onto a cylinder that can be then unrolled into a plane. The familiar Mercator projection is the best-known of these projections. The principal characteristics of cylindrical projections are that (if the usual aspect is seen) all meridians and parallels are straight lines, with the meridians parallel and equally spaced, and the parallels also parallel and perpendicular to the meridians. The spacing of the parallels is what distinguishes one projection from another in this family.

Conic or Conical Projections

Conical projections are based upon the projection of the earth onto a cone, which, like the cylinder, can be unrolled. They are rarely, if ever, used for the whole world and are best suited to maps covering small ranges of latitude. However, for such ranges they often give the most "natural" looking maps. The principal characteristics of conic projections are that (if the usual aspect is seen) all meridians are straight lines converging at the pole, and all parallels are concentric arcs of circles. As in the cylindrical family, the spacing of the parallels is the only distinction between one conic projection and any other.

Azimuthal or Zenithal Projections

Azimuthal projections are in fact the limiting case of conic projections as the cone becomes a plane. Unlike the conics, they are frequently used for large areas of the world. Although they resemble the conic projections in having circular parallels and straight-line meridians in the normal (polar) aspect, they are distinguished by the fact that the parallels in conic projections are arcs of less than 360 degrees, while those in azimuthal projections are 360 degree circles. However, a more important factor is that the azimuthal projections unlike the conic projections, are frequently used in other than normal aspect, and in those cases the meridians and parallels are less of a clue as to the type of projection.

Singular Projections

Singular projections include all projections not included in the three families described above. A few of these projections are the Bonne, Sanson- Flamsteed sinusoidal and Mollweide projections. (In fact, the sinusoidal is a special case of the Bonne, but it is mathematically different because of the effect of the substitution of straight lines for circles as the curvature goes to zero). The Werner Cardiform projection is a special case of the Bonne, and is obtained by letting the standard latitude be 90 degrees.

Characteristics of Projections

A summary of twenty-five common projections and their useful characteristics and limitations is provided.

Albers Equal-Area

This is a conic projection chosen to preserve equality of areas. It is one with very little distortion in the areas between the standard parallels, although there is little but taste to recommend it over the Lambert Equal-Area Conic in this respect.

Azimuthal Equidistant

The principal feature of this projection is that all distances measured from the pole of the projection are true. Like all azimuthal projections, this one also preserves direction from the pole of the projection. This projection is frequently used with its pole chosen to be other than at the earth's pole, when true distances from one specific point are desired.

Figure 1. Azimuthal Equidistant Projection of Coastlines and National Boundaries.

Bonne

 This is an equal-area projection which is particularly good for small areas, although it can be used for the whole world if desired. Parallels are circles, equally spaced, and distances measured along the parallels are true. Distances along one central meridian are also true. Shapes are relatively little distorted except far from the central meridian and standard parallel. Two special cases of this projection are the Werner (standard latitude 90 degrees) and sinusoidal (standard latitude 0 degrees); however, the sinusoidal is best considered as a separate case in its own right.

Cylindrical (General) Perspective

This is a rarely-used general perspective cylindrical projection.

Cylindrical Central-Perspective

This is a rarely-used projection which resembles the Mercator in its appearance.

Cylindrical Equidistant (Plate Carre)

This is the simplest cylindrical projection, and has the property that equal spaces on both meridians and parallels are to be found. Scale in the meridional direction is true; in other directions it varies with latitude.

Figure 2. Cylindrical Equidistant or Plate Carr National Boundaries.

De L'Isle Conic

 This is one of a number of conic projections with equally spaced parallels, which thus guarantees a precise scale on the meridians.

Gnomonic

This is an azimuthal projection with one major advantage: all straight lines on the map represent great circles on the globe. It cannot, however, represent even one-half of the globe, as size distortion increases rapidly as one gets further from the pole of the projection.

Lambert Azimuthal Equal-Area

This is one of a large number of projections bearing Lambert's name, and is the most common of them. As the name suggests, it is an equal-area projection and is also azimuthal; it and the Azimuthal Equidistant are the most commonly represented azimuthal projections. Shapes are greatly distorted in areas far from the pole of the projection, but it is one of only two azimuthal projections (the Azimuthal Equidistant is the other) that can picture the whole world.

Lambert Conformal Conic

The name of this projection is sufficient to describe it; it is conformal (preserves angles) except at the pole of the projection, and it is a conic with all the general characteristics of conic projections.

Lambert Cylindrical Equal-Area

As in all equal-area projections, this projection preserves equality of areas by compensating horizontal-scale increases with vertical scale decreases. In a cylindrical projection, this causes large shape distortions away from the equator of the projection. A transverse aspect of this projection is known as Cassini's projection and is frequently used for areas near a specific meridian which is taken for the equator of the projection.

Lambert Equal-Area Conic

 This is actually closely related to the Lambert Azimuthal Equal-Area, but adapted to a conic projection.

Mendeleev-Conic

This is an alternate name for the Simple Conic (see Simple Conic).

Mercator

This cylindrical projection is the most common one in use. Its main advantage is conformality, which means that shapes and sizes are true. Sizes, whether figured as distances or as areas, increase without limit as the position gets further from the equator. However, to preserve conformality, the scale at any point does not depend on the direction.

Figure 3. Mercator Projection of Coastlines and National Boundaries.

Miller

 This modification to the Mercator reduces the increase of size as distance from the equator increases by sacrificing absolute conformality. It is close enough to conformality to be used instead of the Mercator, but allows the entire world to be mapped, unlike Mercator which cannot include the poles. There are two versions of the Miller projection differing only in the degree of compression. The first type is nearer to the Mercator, the second is more extensively compressed.

Mollweide

The principal feature of this projection is that all the meridians are ellipses. The central meridian is a straight line and the 90 degree meridians are ellipes of eccentricity 0, or circular arcs. The equator and parallels are straight lines perpendicular to the central meridian. The central meridian and the equator are true length.

Figure 4. Mollweide Projection of Coastlines and National Boundaries.

Murdoch Minimum-Error Conic

 This conic projection was designed to compromise the distortions in size and shape of various conics. It is not exact in size or shape, but close in all respects.

Orthographic

This azimuthal projection is often used when it is desired to give a visual appearance, since it represents the actual appearance of a globe as viewed from far away. It cannot show more than half the world at a time.

Figure 5. North and South Polar Orthographic Projections of Coastlines and National Boundaries.

Plate Carre

 This is an alternate name for the Cylindrical Equidistant Projection (see Cylindrical Equidistant Projection).

Ptolemy Equidistant Conic

This conic projection is one of the oldest, as its name implies, from Ptolemy's atlases. It is, like the Simple Conic and several others, a projection with equally-spaced parallels, but is chosen to have a true scale along a standard parallel.

Sanson-Flamsteed

Also called the Sinusoidal Projection (see Sinusoidal Projection).

Simple Conic (Mendeleev Conic)

This, as its name implies, is the simplest of the conic projections. All parallels are equally spaced, and distances measured from the pole are true.

Sinusoidal (Sanson-Flamsteed)

This is an equal-area projection related to the Bonne. However, unlike the Bonne projection, the parallels are straight lines. Because the Sinusoidal Projection is most distorted far from the central meridian, it is frequently used in an "interrupted" mode in which the globe is divided into a number of sections (known as gores) centered on different meridians. Scales are true on all parallels and the central meridian (or central meridians, if interrupted) and so this projection, like the Bonne, combines equal area with equidistance along many lines.

Stereographic

This projection is the conformal member of the azimuthal family, that is to say, it preserves shapes and angles but not sizes. The distortion in size becomes significant at any distance from pole of the projection, but not as great as in the Gnomonic. It can represent all but one point on the world, but is frequently used for a circle representing only a relatively short latitude departure from the pole.

Werner

This projection is a special case of the Bonne (see Bonne). It can be used for the whole world, and when this is done, a heart-shaped map is obtained, hence the name Werner Cardiform projection. All the comments regarding the Bonne Projection apply to the Werner.

References

1. Mahling, D. H. Measurements from Maps: Principles and Methods of Cartometry. Pergamon Press, 1989.

2. Mahling, D. H. Coordinate Systems and Map Projections, Second Edition. Pergamon Press, 1992.

3. Pearson, Frederick II. Map Projections: Theory and Applications. CRC Press, 1990.

lloydt@watson.ibm.com