Map projections and distortion (2024)

Map projections and distortion

Convertinga sphere to a flat surface results in distortion. This is the most profoundsingle fact about map projections—they distort the world—a fact that you willinvestigate in more detail in Module 4, Understanding and ControllingDistortion.

Imagine amap projection as an attempt to reconstruct your face in two dimensions. Somemaps will get the shapes of all your features just right, but not thesizes—your forehead and chin, for instance, may come out huge. Other maps willget the sizes right, but the shapes will be stretched—maybe your full, roundmouth will appear wide, thin, and rather mean.

Some mapspreserve distances. Measurements from the tip of your nose to your chin, ears,and eyes will be right, even though the size and shape of your features is wrong.Other maps preserve direction. Your features may look weird, and they may bescrunched up or set too far apart, but their relative positions will becorrect.

Finally,some maps are compromises—they get nothing exactly right but nothing too farwrong. In particular, compromise projections try to balance shape andarea distortion.

So the fourspatial properties subject to distortion in a projection are:

<![if !supportLists]>·<![endif]>Shape

<![if !supportLists]>·<![endif]>Area

<![if !supportLists]>·<![endif]>Distance

<![if !supportLists]>·<![endif]>Direction

Shape
If a map preserves shape, then feature outlines (likecountry boundaries) look the same on the map as they do on the earth. A mapthat preserves shape is conformal. Even on a conformal map, shapes are abit distorted for very large areas, like continents.

A conformalmap distorts area—most features are depicted too large or too small. The amountof distortion, however, is regular along some lines in the map. For example, itmay be constant along any given parallel. This would mean that features lyingon the 20th parallel are equally distorted, features on the 40th parallel areequally distorted (but differently from those on the 20th parallel), and so on.

Area
If a map preserves area, then the size of a feature ona map is the same relative to its size on the earth. For example, on an equal-areaworld map, Norway takes up the same percentage of mapspace that actual Norway takes up on the earth.

To look atit another way, a coin moved to different spots on the map represents the sameamount of actual ground no matter where you put it.

In anequal-area map, the shapes of most features are distorted. No map can preserveboth shape and area for the whole world, although some come close over sizeableregions.

Distance
If a line from a to b on a map is thesame distance (accounting for scale) that it is on the earth, then the map linehas true scale. No map has true scale everywhere, but most maps have at leastone or two lines of true scale.

An equidistantmap is one that preserves true scale for all straight lines passing througha single, specified point. For example, in an equidistant map centered on Redlands, California, a linear measurement from Redlands to any other point on the map wouldbe correct.

Direction
Direction, or azimuth, is measured in degrees of angle from north. Onthe earth, this means that the direction from a tob is the angle between the meridian on which a lies and the greatcircle arc connecting a to b.

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The azimuth of a to b is 22 degrees.

If theazimuth value from a to b is the same ona map as on the earth, then the map preserves direction from a to b.An azimuthal projection is one thatpreserves direction for all straight lines passing through a single, specifiedpoint. No map has true direction everywhere.

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A fewprojections with different properties. The Lambert Conformal Conic preservesshape. The Mollweide preserves area. (Compare therelative sizes of Greenland and South America in one and then theother.) The Orthographic projection preserves direction. The Azimuthal Equidistant preserves both distance anddirection. The Winkel Tripelis a compromise projection.

More about scale

Scale isthe relationship between distance on a map or globe and distance on the earth.

Suppose youhave a globe that is 40 million times smaller than the earth. Its scale is1:40,000,000. Any line you measure on this globe—no matter how long or in whichdirection—will be one forty-millionth as long as thecorresponding line on the earth. In other words, the scale is true everywhere.This is because the globe and the earth have the same shape (disregarding thecomplication of sphere versus spheroid).

Now supposeyou have a flat map that is 40 million times smaller than the earth. (See theproblem coming? Instead of comparing a big orange to a little orange, we'recomparing a big orange to a little wafer.) This map also has a scale of 1:40,000,000,but because the map and the earth are differently shaped, this scale cannot betrue for every line on the map.

The statedscale of a map is true for certain lines only. Which lines these are depends onthe projection and even on particular settings within a projection. We'll comeback to this subject in Module 4, Understanding and Controlling Distortion.

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Not all of the earth'scurves can be represented as straight lines at the same fixed scale. Some linesmust be shortened (and others lengthened).

Expressingmap scale
There are three common ways to express map scale:

Linearscales
Linear scales are lines or bars drawn on a map withreal-world distances marked on them. To determine the real-world size of a mapfeature, you measure it on the map with a ruler or a piece of string. Then youcompare the feature's length on the string to the scale bar.

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A typical scale bar.

Verbalscales
Verbal scales are statements of equivalent distances. For example, if a 4.8kilometer road is drawn as a 20 centimeter line on a map, a verbal scale wouldbe “20cm = 4.8km.” You could also formulate the scale (reducing both sides by20) as “1cm = .24km.”

Representativefractions
Representative fractions express scale as a fraction or ratio of map distanceto ground distance. For example, a scale of 1:24,000 (also written 1/24,000)means that one unit on the map is equal to 24,000 of the same units on theearth. Since the scale is a ratio, it doesn't matter what the units are. Youcan interpret it as 1 meter = 24,000 meters, 1 mile = 24,000 miles, or 1 handwidth = 24,000 hand widths (as long as it's the same hand).

Smallscale and large scale maps
It's easy to mix these terms up. Here's one way tokeep them straight: on a large-scale map, the earth is large (so not very muchof it fits on the map). On a small-scale map, the earth is small (so all ormost of it fits on the map). A map of your town, or your property, is going tobe a large-scale map. A continental or world map is a small-scale map.

Another way to think of the difference in terms of representativefractions. The larger the fraction, the larger the map's scale. Forexample, 1/10,000 is a larger fraction than 1/1,000,000. So a1:10,000 map is larger scale than a 1:1,000,000 map.

Measuring distortion using Tissot's Indicatrix

In thenineteenth century, Nicolas Auguste Tissot developed a method to analyze map projectiondistortion.

Aninfinitely small circle on the earth's surface will be projected as aninfinitely small ellipse on any given map projection. The resulting ellipse ofdistortion, or indicatrix, shows the amount and typeof distortion at the location of the ellipse.

Forexample, if an indicatrix is elongated from north tosouth, shape is correspondingly distorted at that location on the map. The samegoes for east–west stretching or oblique stretching. On a conformal map, the indicatrices are all circles, but they vary in size. On anequal area projection, the indicatrices have varying ellipticity, but the same area.

Tissot indicatrices for four projections.