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3. Scale and Projections

Scale and projections are two fundamental features of maps that usually do not get the attention

they deserve. Scale refers to how map units relate to real-world units. Projections deal with the

methods and challenges around turning a three-dimensional (and sort of lumpy) earth into a two-

dimensional map.

This section will introduce you to…

 Scale and ways of telling the map user what the map is measuring on the ground

 Projection mechanics, type of projections, and their characteristics

By the end of this chapter, you should be able read map scales and identify common projections

along with their basic features and uses.

3.1 Scale

The world is vast. The earth’s surface has an area of over 500 million km2 and any picture of the

earth that you can easily carry can only show general outlines of continents and countries. When

we visually represent a region of the world on a map, we must reduce its size to fit within the

boundaries of the map. Map scale measures how much the features of the world are reduced to fit

on a map; or more precisely, map scale shows the proportion of a given distance on a map to the

corresponding distance on the ground in the real world.1

Map scale is represented by a representative fraction, graphic scale, or verbal description.

Representative fraction.

The most commonly used

measure of map scale is the

representative fraction (RF),

where map scale is shown as

a ratio. With the numerator

always set to 1, the

denominator represents how

much greater the distance is

in the world. Figure 3a

shows a topographic map

with an RF of 1:24,000,

which means that one unit on

the map represents 24,000

units on the ground. The

representative fraction is

accurate regardless of which

units are used; the RF can be

measured as 1 centimeter to 24,000 centimeters, one inch to 24,000 inches, or any other unit.

Figure 3a. Representative fraction and scale bars from a United States

Geological Survey (USGS) topographic map

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Graphic scale. Scale bars are graphical representations of distance on a map. Figure 3a has scale

bars for 1 mile, 7000 feet, and 1 kilometer. One important advantage of graphic scales is that

they remain true when maps are shrunk or magnified.

Verbal description. Some maps, especially older ones, use a verbal description of scale. For

example, it is common to see “one inch represents one kilometer” or something similar written

on a map to give map users an idea of the scale of the map.

Map makers use scale to describe maps as being small-scale or large-scale. This description of

map scale as large or small can seem counter-intuitive at first. A 3-meter by 5-meter map of the

United States has a small map scale while a UMN campus map of the same size is large-scale.

Scale descriptions using the RF provide one way of considering scale, since 1:1000 is larger than

1:1,000,000. Put differently, if we were to change the scale of the map with an RF of 1:100,000

so that a section of road was reduced from one unit to, say, 0.1 units in length, we would have

created a smaller-scale map whose representative fraction is 1:1,000,000.

When we talk about large- and small-scale maps and geographic data, then, we are talking about

the relative sizes and levels of detail of the features represented in the data. In general, the larger

the map scale, the more detail that is shown (Figure 3b).2

Figure 3b. These two satellite images depict the pyramids in Giza, Egypt. The image on the left is zoomed

out or appears to be taken from far above the earth. This is a small-scale map. The map on the right is a

larger-scale map. This distinction between larger-scale and smaller-scale is not intuitive, so here is an

easy tool to remember it: if the buildings are larger, then the scale is larger.

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3.2 Extent vs. Resolution

The extent of a map describes the area visible on the map, while resolution describes the smallest

unit that is mapped. You can think of extent as describing the region to which the map is

zoomed. The extent of the map in Figure 3c is national as it encompasses the contiguous United

States, while the resolution is the state, because states are the finest level of spatial detail that we

can see.

Figure 3c. Map showing a national extent and a state resolution.

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We often choose mapping resolutions intentionally to make the map easier to understand. For

example, if we tried to display a map with a national extent at the resolution of census blocks, the

level of detail would be so fine and the boundaries would be so small that it would be difficult to

understand anything about the map. Balancing extent and resolution is often one of the most

important and difficult decisions a cartographer must make. Figure 3d offers two more examples

of the difference between extent and resolution.

Figure 3d. Maps showing an extent of the Pacific Northwest, the left with a spatial resolution of county

and

the right with a spatial resolution of census tracts

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3.3 Projections

This section will introduce you to projections, the term

for turning a three-dimensional globe into a two-

dimensional map. We will discuss the process of how

objects on a 3-dimensional surface (the earth) come to

be represented on a flat piece of paper or computer

screen. Our emphasis will be on the properties that

different projections distort or maintain – area, shape,

and distance.

Projection is the process of making a two-dimensional

map from a three-dimensional globe. We can think of the earth as a sphere. In reality, it is more

of an ellipsoid with a few bulges, but it is fine to think of it as a sphere. To get a sense of how

difficult this process can be, imagine peeling the skin from an orange and trying to lay the skin

flat (Figure 3e).

As you peel and flatten the skin, you will encounter several problems:

 Shearing – stretching the skin in one or more directions

 Tearing – causing the skin to separate

 Compressing – forcing the skin to bunch up and condense

Cartographers face the same

three issues when they try to

transform the three-dimensional

globe into a two dimensional

map (Figure 3f). If you had a

globe made of paper, you could

carefully try to ‘peel’ it into a

flat piece of paper, but you

would have a big mess on your

hands. Instead, cartographers

use projections to create useable

two-dimensional maps.

Figure 3e. Flattened orange peel 3

Figure 3f. Shearing, tearing, and compression on a globe.

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3.4 Mechanics of Projection

The term “map projection” refers to both the process and product of transforming spatial

coordinates on a three-dimensional sphere to a two-dimensional plane.

In terms of actual mechanics, most projections use mathematical functions that take as inputs

locations on the sphere and translate them into locations on a two-dimensional surface.

It is helpful to think about projections in physical terms (Figure 3g). If you had a clear globe the

size of a beach ball and placed a light inside this globe, it would cast shadows onto a surrounding

surface. If this surface were a piece of paper that you wrapped around the globe, you could

carefully trace these shadows onto the paper, then flatten out this piece of paper and have your

projection!

Figure 3g. Thinking of projections in physical terms – a clear globe, a light bulb, and tracing paper. 4

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Most projections transform part of the globe to one of three “developable” surfaces, so called

because they are flat or can be made flat: plane, cone, and cylinder. The resultant projections are

called planar, conical, and cylindrical (as seen in Figure 3f). We use developable surfaces

because they eliminate tearing, although they will produce shearing and compression. Of these

three problems, tearing is seen as the worst because you would be making maps with all sorts of

holes in them! As we see below, however, there are times when you can create maps with tearing

and they are quite useful.

The place where the developable surface touches the globe is known as the tangent point or

tangent line (Figure 3h). Maps will most accurately represent objects on the globe at these

tangent points or lines, with distortion increasing as you move farther away due to shearing and

compression. It is for this reason that cylinders are often used for areas near the equator, cones

used to map the mid-latitudes, and planes used for polar regions.

For beginning mapmakers, understanding the exact mechanics of projections doesn’t matter as

much as knowing which map properties are maintained or lost with the choice of projection – the

topic of the next section.

Figure 3h. Red marks the tangent line/point. The flat surface touches the globe and it is the point on

the projected map which has the least distortion. 5

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3.5 Types of Projections

Projections must distort features on the surface of the globe during the process of making them

flat because projection involves shearing, tearing, and compression. Since no projection can

preserve all properties, it is up to the map maker to know which properties are most important for

their purpose and to choose an appropriate projection. The properties we will focus on are:

shape, area, and distance.

Conformal

Conformal projections preserve shape and angle, but strongly distort area in the process. For

example, with the Mercator projection (Figure 3i), the shapes of coastlines are accurate on all

parts of the map, but countries near the poles appear much larger relative to countries near the

equator than they actually are. For example, Greenland is only 7-percent the land area of Africa,

but it appears to be just as large! Conformal projections should be used if the main purpose of the

map involves measuring angles or representing the shapes of features. They are very useful for

navigation, topography (elevation), and weather maps.

Figure 3i. Mercator projection

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Equal Area

On equal-area projections, the size of any area on the map is in true proportion to its size on

the earth. In other words, countries’ shapes may appear to be squished or stretched compared to

what they look like on a globe, but their land area will be accurate relative to other land masses.

For example, in the Gall-Peters projection (Figure 3j), the shape of Greenland is significantly

altered, but the size of its area is correct in comparison to Africa.

This type of projection is important for quantitative thematic data, especially in mapping density

(an attribute over an area). For example, it would be useful in comparing the density of Syrian

refugees in the Middle East or the amount of cropland in production.

Figure 3j. Gall-Peters projection

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Equidistant

Equidistant projections, as the name suggests, preserve distance. This is a bit misleading

because no projection can maintain relative distance between all places on the map. Equidistant

maps are able, however, to preserve distances along a few clearly specified lines. For example,

on the Azimuthal Equidistant projection (Figure 3k), all points are the proportionally correct

distance and direction from the center point.

This type of projection would be useful visualizing airplane flight paths from one city to several

other cities or in mapping an earthquake epicenter.

Figure 3k. Azimuthal Equidistant projection

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Distortion and Developable Surfaces

Note that distortion is not necessarily tied to the type of developable surface but rather to the way

the transformation is done with that surface. It is possible to preserve any one of the three

properties using any of the developable surfaces, as Figure 3l shows when using a cone.

Conformal

Equal Area

Equidistant

Figure 3l. Three maps created using a conic surface each of which preserves a different map property

(Lambert conformal conic, Albers equal area conic, and Schjerning north polar equidistant conic)6

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Compromise Projections

Some projections, including the Robinson projection (Figure 3m), strike a balance between the

different map properties. In other words, they do not preserve shape, area, or distance, but

instead try to avoid extreme distortion of any of these properties. This type of projection would

be useful for a general purpose world map.

Interrupted Projections

Other projections deal with the challenge of making the 3D globe flat by tearing the earth in

strategic places. Interrupted projections such as the Interrupted Homolographic (Figure 3n)

represent the earth in lobes, reducing the amount of shape and area distortion near the poles.

Figure 3m. Robinson Projection 7

Figure 3n. Interrupted Homolographic. Pseudocylindrical equal-area (National Geographic Society

1904)

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Artistic projections

There are also a large number of projections that are interesting and beautiful, but not intended

for navigation between places or to visualize data. Examples of these artistic projections include

the heart-shaped Stabius-Werner projection (Figure 3o) and Waterman’s “Butterfly” projection

(Figure 3p).

Figure 3o. Stabius-Werner Projection

(Pseudoconic equal area) 8

Figure 3p. Waterman’s “Butterfly” projection 9

3.6 Conclusion

In this section, we have looked at some of the most frequently used projections. There are

hundreds of projections, each which distorts the world in a slightly different way. Keep in mind

that all maps have a scale and there a few important ways to indicate this scale. All maps also use

a projection that can be formed from a developable surface and can preserve one or two

properties at most.

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Resources:

 http://giscommons.org/earth-and-map-preprocessing/

 How to choose between different projections: http://www.geo.hunter.cuny.edu/~jochen/gtech201/lectures/lec6concepts/map%20coordi

nate%20systems/how%20to%20choose%20a%20projection.htm

 Comprehensive guides to projections: o http://www.csiss.org/map-projections/index.html o http://www.progonos.com/furuti/MapProj/Normal/ProjTbl/projTbl.html

1 Parts of this section are adapted from Campbell and Shin (2011). Essentials of Geographic Information Systems.

http://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=67 2 Penn State Geog 482 “The Nature of Geographic Information” Ch.2.5 3 http://krygier.owu.edu/krygier_html/geog_222/geog_222_lo/geog_222_lo13.html 4 http://www.geog.ucsb.edu/~dylan/mtpe/geosphere/topics/map/map1.html#proj 5 http://geokov.com/education/map-projection.aspx but © USGS 6 © 2013 Carlos A Furuti, http://www.progonos.com/furuti/MapProj/Normal/ProjCon/ProjConNP/projConNP.html 7 http://kartograph.org/showcase/projections/#robinson 8 © 1997 Carlos A Furuti, http://www.progonos.com/furuti/MapProj/Normal/ProjPCon/projPCon.html#Werner1 9 © 1997 Carlos A Furuti, http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly2.html#gnoct