Map Projections

Basic Definitions and Concepts

"To project" means transferring Earth features to a suitable surface, like a tangent plane or a truncated secant cone. Regions nearer the surface are usually better presented.

Cones and cylinders can be useful intermediary surfaces, but the darker conversion paths unavoidably incur in distortion.

Maps, Globes and Projections

Any study in geography requires a reduced model of the Earth, like a globe or map.  Neither is perfect: a globe is seldom practical, and flat maps are never free from errors. Selecting or creating a good map involves interesting choices and trade-offs.

What's a Projection?

A cartographical map projection is a formal process which converts (mathematically speaking, maps) features between a spherical or ellipsoidal surface and a projection surface, often flat. Although many projections have been designed, just a few are currently in widespread use.  Some were once historically important but were superseded by better options, some are useful only in very specialized contexts, while others are little more than fanciful curiosities.

Projection Surfaces

The map's support, the projection surface is usually created, i.e., developed, conceptually touching the mapped sphere in one (surface is tangent) or more (surface is secant) regions. Intuitively, portions near the touching regions depart less from the original spherical shell and are more faithfully reproduced. Some projections are actually composites, fitting separate surfaces to different regions of the map: overall error is reduced at the cost of greater complexity.

Sometimes a conceptual auxiliary surface like a cone, open cylinder, ellipsoid or torus is employed: sphere features are transferred to that surface (many times by the laws of perspective), which is then flattened. Many projections are classified as "cylindrical" or "conic"; however, for most of them, the naming is just an analogy or didactic device, since they aren't actually developed on an intermediary surface; rather, the resulting map can be rolled onto a tube or a cone.

The Unavoidable Distortion

Two quasi-orthographic views of a sphere (plus polar axis) divided in eight equal sections, surrounded by four maps at the same scale using, clockwise from top right: azimuthal equidistant, Lambert's equal-area cylindrical, Maurer's equal-area star with four lobes, Winkel Tripel projections. A few selected projections illustrate how the same spherical data can be stretched, compressed, twisted and otherwise distorted in different ways. The azimuthal equidistant map has interesting properties regarding directions and distance from the central pole, but the outer hemisphere is greatly stretched: its pole becomes a circle. Both poles become lines in the equal-area cylindrical map, but it covers the same area as the original sphere; also, all octants have identical shape. This particular star projection has unequal octants and marked loss of continuity; however, it also preserves area. In the Winkel Tripel map, octants have different shapes, area is changed and poles are linear, but overall distortion is subjectively smaller. Finally, the orthographic views, projections themselves, show only part of the sphere. All projections suffer from some distortion; none is "best" for all purposes. Octants would assume even stranger shapes in oblique aspects.

No matter how sophisticated the projection process, the original surface's features can never be perfectly converted to a flat map: distortion, great or small, is always present in at least one region of planar maps of a sphere. Distortion is a false presentation of angles, shapes, distances and areas, in any degree or combination.

Every map projection has a characteristic distortion pattern. An important part of the cartographic process is understanding distortion and choosing the best combination of projection, mapped area and coordinate origin minimizing it for each job.

Cones and cylinders are developable surfaces with zero Gaussian curvature (in a nutshell, at every point passes a straight line wholly contained in the surface). Distortion always occur when mapping a sphere onto a cone or cylinder, but their reprojection onto a plane incurs in no further errors.

The Choice of Coordinate Origin

A particular projection may be employed in several aspects, roughly defined by the graticule lay-out and the sphere's region nearest the projection surface, commonly the center of a whole-world map:

• a polar map aligns the Earth axis with the projection system's, thus one of its poles lies at the map's conceptual center;
• an equatorial map is centered on the Equator, which is set across one of the map's major axes (mostly horizontally);
• an oblique map has neither the polar axis nor the equatorial plane aligned with the projection system.

• the most "natural" aspect of a projection, called normal, conventional, direct or regular, is ordinarily determined by geometric constraints; it often demands the simplest calculation and produces the most straightforward graticule. The polar aspect is the normal one for the azimuthal and conic groups of projections, while the equatorial is the normal for cylindrical and pseudocylindrical groups.
• the transverse aspect frequently resembles the normal one, except by a simple rotation of 90°.

Normal Transverse Oblique   (polar) (equatorial)       (equatorial)         (polar) (seldom used) (seldom used) Three (normal, transverse and oblique) aspects applied to three (azimuthal equal-area, Gall's stereographic cylindrical and Albers's conic) projections with different tangent projection surfaces in blue (just a few of infinitely many possible oblique maps are presented). Some projections may actually be derived via perspective geometry; for most, however, surfaces are only illustrative: the map may be laid on a developable surface, but is not calculated from it. Distinctive graticules of some projection groups (radially symmetric meridians in azimuthal and conic maps, rectangular grid in cylindrical maps) are only realized using their simple, normal aspects. Trivial rotations of the finished map, like turning it sideways or upside-down, leave both aspect and projection unchanged. On the other hand, modifying the aspect does not affect either represented area or the shape of the whole map.

• their properties may be less useful otherwise. E.g., many factors like temperature, disease prevalence and biodiversity depend on climate, thus roughly on latitude; for projections with constant parallel spacing, on equatorial aspects latitude is directly converted to vertical distance, easing comparisons.
• several projections whose graticules in normal aspects comprise simple curves, like straight lines and arcs of circle, were originally defined by geometric construction. Since many non-normal aspects involve complex curves, they were not systematically feasible before the computer age (indeed, mapping was an important motivation for calculation shortcuts like logarithms).

www.progonos.com/furuti    June 26, 2006