Map Projections

Useful Map Properties: Distances and Scale

In this equidistant cylindrical map, all green lines are equally long; however, actual world distances (in km, actually measured along the red geodesics) between the ends of each line vary enormously. Four blue graphic scales are supplied; for this projection, each horizontal one is only useful along its specific parallel (i.e., if it were a ruler, one could slide it horizontally only and still take precise measurements) or the one symmetric opposite the Equator; the vertical scale is valid anywhere. None will give meaningful results if rotated.

Can Distances be Accurately Measured?

Any practical map, spherical or flat, shrinks Earth features down to a manageable size. The general rate of reduction, the scale, can be expressed graphically or numerically. The latter case is usually a fraction; for instance, on a 1:100000-scale map, two points separated by 1 unit could represent two cities actually 100000 units apart; 1 centimeter on the map means 1000 meters on Earth. With a scale 10 times "larger", 1:10000, one inch on the map corresponds to 10000 inches on Earth, and so on. Therefore, if the same region is presented in different maps, smaller-scale versions can be more portable but probably less precise and detailed.

A graphic scale is, at a minimum, a small "ruler" tagged with absolute distances and directly superimposed on the map. Although a distance evaluation requires two measurements (one for the ruler, another for the map itself), it has an important advantage: the map may be reproduced at different sizes with no modifications, while a numerical scale must be recomputed for each size. Graphic scales are especially useful for electronic devices like computer screens (whose resolution/physical size ratios are seldom correctly calibrated) and presentation slides (where the display's dimensions depend on the distance to the projection screen).

On the cylindrical equidistant map on the left, point A seems 41% farther from C than from B. Actually, every point on the red meridian is equally far from A. The distance between D and C is zero, since both lie on a pole. The orthographic map on the right puts those facts in a better perspective, in more than one sense.

Either graphic or numeric, only on a true globe a scale can be directly applied to any pair of points on its surface. In flat maps, most likely the reduction rate is not constant, changing with direction and location. Especially, the rate may be nonlinear: the length ratio of two segments of a line on the map is not the same for the corresponding segments on Earth. Finally, the shortest distance between two points on a sphere is rarely represented by a straight line on a flat map, and measuring distances along an arbitrary (not always marked) curve is not a straightforward procedure.

Thus, depending on the range of the mapped region (the larger, the greater will be distance distortions), neither numbers nor rulers can be reliably used without knowledge of the projection's properties; they both can be just misleading.

Standard Lines and Equidistant Maps

Some standard lines (marked in proportion 1:2:3) in selected projections

In an equatorial Sanson-Flamsteed (or sinusoidal) map, all parallels are standard lines: if segment A is twice the length of B, corresponding lines on Earth follow the same proportion. Although the straight vertical distance between parallels is true to scale, distances along meridians are not, except for the central one. This equatorial cylindrical equidistant map is roughly the reciprocal of the sinusoidal: only meridians and the Equator preserve a constant scale. Since all parallels are equally long in the map, horizontal scale increases quickly towards the top and bottom, reaching infinity at the poles. The set of standard lines is tied to the map's aspect, not to the mapped features. In the transverse aspect of the equidistant cylindrical, also known as Cassini's projection, the standard lines remain the same, but do not in general coincide with meridians. An azimuthal equidistant map preserves distances along any lines passing through the central point. In the polar aspect, those lines are the meridians. Standard lines are not necessarily straight. Those in Werner's cordiform projection are concentric circular arcs. In the normal aspect, they are parallels centered on a pole, and the central meridian is also standard. Not all straight lines are standard. Parallels in van der Grinten III projection have nonlinear scale: meridians near the central one are slightly closer together than those near the boundary.

Lines on the map, straight or not, with constant, linear scale (with length proportional to that of corresponding lines on Earth) are lines of true scale or standard lines. Map projections with a well-defined, nontrivial set of standard lines are sometimes referred to as equidistant. Some well-known examples are the equidistant cylindrical (in the normal aspect, all meridians and one or two parallels are standard), azimuthal equidistant (in the normal aspect all meridians are standard), and the conic equidistant (in the normal aspect, all meridians and one or two parallels are standard).

A common source of gross mistakes is assuming a circle drawn on a map represents the same range of distances as the corresponding shape on Earth. This only happens if the all straight lines radiating from the circle's center are standard lines.

www.progonos.com/furuti    October 15, 2012