Map Projections / Skills

APPENDIX MAP SCALE AND PROJECTIONS Phillip C. Muercke

Unaided, our huma n sen ses pro vide a Li mited view of ou r surro un ding-s oTo overcome th ose limi tations, humankind has developed powerful vehicles of thought and commu

nication , such as language, mathema tics, and graphics. Each of those tools is based on elabor ate rules; each has an informa tion bias, and each may distort its message, often in subtl e ways. Consequently, to use those aids effect ively, we mu st unders tand their rul es, biases, and disto rtion s. T he same is tr ue for the special form of graphics we cal l maps: we mus t master the logic behind the mapping pro cess before we can use maps effectively.

A funda mental issue in cartography, the science and art of making maps , is the vast differ enc e between the size and geom etry of what is being mapped-the real world, we will call it- and that of the map itself. Scale and pro jection are the basic cartographic concepts that help us understand th at difference and its effects.

Map Scale Ou r sens es are dwarfed by the immensity of our planet; we can sense directly only our local surroundings. T hus, we cannot possibly look at our who le state o r country at on e tim e, even though we may be able to see the en tire stree t where we live. Ca rtography helps us expand what we can see at one time by let ting us view the scene from some distant vantage point. T he gre ater the imagin ary distance between that position and the object of our obs ervation, the larger the area the map can cover, but the sma ller the featu res will appe ar on the map . That redu ction is defined by the mapscale, the rati o of the distance on the map to the distance on the earth. M ap users need to know about map scale for two reasons: so tha t they can convert measurements on a map into meaningful real -world measures and so that they can know how abstract the cartographic representation is.

REAL-WORLD MEASURES . A map can provide a useful sub stitute for the real wo rld for man y analytical purposes. With the scale of a ma p, for inst ance, we can co m pu te th e ac tu al size of its features (le ng th, area, and vo lume) . Suc h ca lculations are helped by three express ions of a map sca le: a wo rd sta tement, a graphic scale, and a representative fraction.

A word statement of a map scale com pares X un its on the map to Y un its on th e ea rt h, often abbr evia ted "X un it to Y uni ts." For example, th e express ion" I inc h to I 0 miles" means th at I inch on the ma p represe nts 10 mil es on the earth (Figur e A-I ). Because th e ma p is always sma ller than th e area that h as been map ped, the gr ound un it is alw ays th e larger number. Both u nits are exp resse d in meaning ful terms, such as inches or centimeters and miles or kilo me ters. Word statements are not intende d fo r precise calcul at ion s but give the map user a rough idea of size an d distance.

A graphicscale, such as a bar graph, is concrete and th erefore over com es th e need to visual ize inches and miles tha t is assoc iated with a word statement of scale (see Figu re A- I) . A gra phic scale permi ts direct visual compar ison of featu re sizes and the dis tances between featu res. N o ruler is req uired; any measuring aid will do . I t needs only be compared with the scaled bar; if the len gth of 1 to othpick is equal to 2 mil es on th e g round and the ma p distan ce eq ual s the len gt h of 4 too thpicks, then the ground distance is 4 times 2, or 8 miles. Gra phic

Word Statement

"One inch equals tell miles"

Graph ic Scale

10 Mlle.,

Representative Fraction

1

633,600

FIGURE A-l Co mmon expressions of map scale.

scales are especially convenient in this age of cop ying machines, whe n we are more likely to be wo rkin g with a copy than with the original map. If a map is reduced or enlarged as it is co pied, the gra phic scale will cha nge in prop ortion to th e change in the size of the map and th us will rema in accurate.

T he third form of a map scale is the representativefraction(RF) . An RF defines the ratio between the distance on the map and the distance on the earth in fraction al terms, such as 11633,600 (also written 1:633,600). The nume rator of the fraction always refers to the distance on the map, and the denominator always refers to the distan ce on the earth . N o units of measurement are given, but both numbers must be expressed in the same units. Because map distances are extremely small relative to the size of the earth , it makes sense to use small un its, such as inches or centime ters. T hus the RF 1:633,600 might be read as " I inch on th e map to 633,600 inches on the earth."

H erein lies a problem with the RF. Mea ningful map -distance units imply a denominator so large that it is im possible to visualize. Thus, in practice, readin g the map scale invo lves an addi tional step of converting the deno minator to a meanin gful ground measur e, such as miles or ki lometers. The unwieldy 633,600 becomes the more ma nageable 10 miles when divided by the num ber of inches in a mile (63,36 0).

O n th e plus side, the RF is good for calc ulat ions. In part icul ar, th e g ro und dist an ce be tween poin ts can be eas ily deter min ed from a map wi th an RF. O ne simply mul tiplies the di stance betw een th e points on the ma p by the den ominator of the RF. Thus a dis tance of 5 inc hes on a map with an RF of 1/126, 720 woul d signify a gr ound dis tance of 5 X 126,720, whic h equ als 633 ,60 0. Because all units are inc hes and th ere are 63 ,36 0 in ch es in a mile , the grou nd dis tance is

502

Appendix: Map Scale and Projections 503

633,600 -7 63,360, or 10 miles. Computation of area is equally straightforward with an RF. Computer manipulation and analysis of maps is based on the RF form of map scale.

GUIDES TO GENERALIZATION. Scales also help map users visualize the nature of the symbolic relation between the map and the real world. It is convenient here to think of maps as falling into three broad scale categories (Figure A-2). (Do not be confused by the use of the words large and small in this context; just remember that the larger the denominator, the smaller the scale ratio and the larger the area that is shown on the map.) Scale ratios greater than 1:100,000, such as the 1:24,000 scale of U.S. Geological Survey topographic quadrangles, are large-scale maps. Although those maps can cover only a local area, they can be drawn to rather rigid standards of accuracy. Thus they are useful for a wide range of applications that require detailed and accurate maps, including zoning, navigation, and construction.

At the other extreme are maps with scale ratios of less than 1:1,000,000, such as maps of the world that are found in atlases. Those are small-scale maps. Because they cover large areas, the symbols on them must be highly abstract. They are therefore best suited to general reference or planning, when detail is not important. Medium- or intermediate-scale maps have scales between 1:100,000 and 1:1,000,000. They are good for regional reference and planning purposes.

Another important aspect of map scale is to give us some notion of geometric accuracy; the greater the expanse of the real world shown on a map, the less accurate the geometry of that map is. Figure A-3 shows why. If a curve is represented by straight line segments, short segments (X) are more similar to the curve than are long segments (Y). Similarly, if a plane is placed in contact with a sphere, the difference between the two surfaces is slight where they touch (A) but grows rapidly with increasing distance from the point of contact (B). In view of the large

1:1,000.000

1:100 ,000

FIGURE A-2 The scale gradient can be divided into three broad categories.

FIGURE A-3 Relationships between surfaces on the round earth and a flat map.

Sphere

Ql c:: <tl

c:::

A

B

diameter and slight local curvature of the earth, distances will be well represented on large-scale maps (those with small denominators) but will be increasingly poorly represented at smaller scales. This close relationship between map scale and map geometry brings us to the topic of map projections.

Map Projections The spherical surface of the earth is shown on flat maps by means of map projections. The process of "flattening" the earth is essentially a problem in geometry that has captured the attention of the best mathematical minds for centuries. Yet no one has ever found a perfect solution; there is no known way to avoid spatial distortion of one kind or another. Many map projections have been devised, but only a few have become standard. Because a single flat map cannot preserve all aspects of the earth's surface geometry, a rnapmaker must be careful to match the projection with the task at hand. To map something that involves distance, for example, a projection should be used in which distance is not distorted. In addition, a map user should be able to recognize which aspects of a map's geometry are accurate and which are distortions caused by a particular projection process. Fortunately, that objective is not too difficult to achieve.

It is helpful to think of the creation of a projection as a two-step process (Figure A-4). First, the immense earth is reduced to a small globe with a scale equal to that of the desired flat map. All spatial properties on the globe are true to those on the earth. Second, the globe is flattened. Since that cannot be done without distortion, it is accomplished in such a way that the resulting map exhibits certain desirable spatial properties.

PERSPECTIVE MODELS. Early map projections were sometimes created with the aid of perspective methods, but that has changed. In the modern electronic age, projections are normally developed by strictly mathematical means and are plotted out or displayed on computerdriven graphics devices. The concept of perspective is still useful in visualizing what map projections do, however. Thus projection methods

504 Appendix: Map Scale and Projections

1 Reduce

(I Flatten

lat fv1a

FIGURE A-4 T he two-step processof creating a projection.

are often illustrated by using strat egically located light sourc es to cast shadows on a projection surface from a latitud e/ lon gitude net inscribed on a transparent globe.

The success of the perspective approach depend s on finding a projection surface that is flat or that can be flatt en ed with out distortion. The cone, cylinder, and plane possess those att ributes and serve as mod els for three general classes of map projections: conic, cylindrical, and planar (or azimuthal). Figure A-5 shows those three classes, as we ll as a fourth, a false cylindrical class with an oval shape. Although th e oval class is not of perspective origin, it appears to combine prop erties of th e cylindrical and planar classes (Figure A-6).

The re lationship between the projection sur face and the mode l at the point or line of contact is criti cal because distortio n of spa tial properties

zt OO M r:: c==J 0 -:>.~c=J <:>

Conic Cyl indri cal Planar Oval

FIGURE A-5 General classes of rnap projections. (Courtesyof A CSM)

Planar

Oval

FIGURE A-6 Th e visual properties of cylindrical and planar projections combined in oval projections. (Courtesyof ACSM)

on the proj ection is symmetri cal about , and increases with distance from, that point or line. That conditio n is illustrat ed for th e cylindrical and planar classes of proje ctions in Figure A-7 . If the point or line of contact is changed to some other position on the globe, the distor tion pattern will be recen tered on the new position but will retain the same symmetrica l form . Thus center ing a projection on the area of interest on the ear th's surface can minimize the effects of projection disto rti on . And recognizing the general projection shape, associating it with a pers pective model , and recall ing the characteristic distortion pattern will provide th e in format ion necessary to compensate for projection distortion ,

PRESERVED PROPERTIES. For a map projection to truthfully depic t the geometry of the earth's surface, it would have to preserve the spatial attributes of distance, direction, area, shape, and proximity, That task can be readily accomplished on a globe, but it is not possible on a flat ma p. To preserve area, for example, a mapmaker must stretch or she ar shapes; thu s area and shape cannot be preserved on the same ma p. To depict

Cy lindrical

FIGURE A-7 Characteristic patterns of distortion for two projection classes, Here, darker shading impliesgreater distortion. (Courtesy ofACSM)


both dir ection and distanc e fro m a point, area must be dis tor ted. Similarly, to pre serve area as well as direc tion from a point, distance has to be distorted. Because the earth's surface is continuous in all directions from every poi nt, d iscont inu it ies tha t violate proximity rela tio nships must occur on all map projections . The tr ick is to place those discontinu ities where they will have the least impact on the spatial rela tionships in which the map user is interes ted .

We must be carefu l when we use spa tial terms, because the properties they refer to can be confusing . The geometry of the familiar plane is very different from that of a sphere; yet when we refer to a flap map, we are in fact making reference to the spherical earth tha t was mapped . A shape-preserving pro jection, for examp le, is truthfu l to local shapessuch as the right-angle crossing of latitude and longitude lines- but does not preserve shapes at con tinental or global levels. A distance-preserving project ion can preserve that prop erty from one point on the map in all directions or from a number of points in severa l direc tion s, but distance cannot be pre served in the gen eral sense that area can be pres erved. Direction can also be generally preserve d from a single point or in several directio ns from a num ber of points but not from all poin ts simultaneously. Thus a shape-, distance-, or direction-preserving projection is truthfu l to those properties only in part.

P artia l tru ths are no t th e only consequence of t ransfo rm in g a sphere in to a flat sur face. Some projections exp loi t that tr ansformatio n by expressing trai ts th at ar e of con sidera ble value for specific applications. One of th ose is the famous sha pe-preserving M ercator projection (Figure A-B). That cylin dr ical pro jection was derived mathema tically in the 1500s so that a compass bea ring (called rhum b lines) between any two po ints on the earth would plot as stra ight lines on the map. That trait let navigators plan , plo t, an d foll ow courses between orig in an d destina tion, but it was achieved at the expense of extreme areal distortion toward th e margins of the projec tio n (see An tarcti ca in Figure A-B) . Although the Mercator projection is admirably suited for its intended pur pose, its wid espread but in appropriate use for nonnaviga tional purposes has dr awn a gre at deal of cri ticism.

~ "' ~ ~1 l f· ~ ~~(~ ~

;0. ~ h?'$'<> ~ ~

('--"- ~C:>< ~~~. ! \ ; k I\t (f"1r-ir> ~ ;,,~ I : :; ~"" J'I \ !r' , ,. . L

' H+~Ii' ~ " ~ : , ~ h fi 91 , .u , - mm~m

; I 1":-', ' m" a fRffi M, I ; ~ J~~r .--.-

1 ~ f"'" ~9 I lr ~r-Jr' (, " t'J

"~ ~ 11,. Ic' cI

\\ () I ~ v ir'

I 1"I'---, ~ ~

FIGURE A-8 The useful Mercator projection, showing extr eme area disto rtion in

the higher lati tudes. (Courtesy ofACSM)

FIGURE A-9 A gnomonic projection (A) and a Mercator projection (B), bo th of

value to long-distance n avigators.

T he gnomonic projection is also usefu l for naviga tion. It is a planar projection with the valuable characteristic of showing the sho rtest (or great circle) rou te between any two poin ts on the earth as straight lines. Long-dis tance navigators first plot the grea t circle course between origin and destin ation on a gnomonic projection (Figure A-9, top) . Next they trans fer the stra ight line to a M ercator pro ject ion, where it normally appears as a curve (Figure A-9 , botto m) . Finally, usin g stra igh t- lin e segment s, they constru ct an approximation of that course on the Me rca tor projection. Navigating the shortest course between origin and destination then involves following the stra ight segments of the course and making directi ona l corrections between segments. Like the Mercator projection, the specialized gnomonic projection dis to rts other spati al properties so severely that it should no t be used for any purpose other than navigation or communicati ons.

PROJECTIONS USED IN TEXTBOOKS. Although a map projection cannot be free of distortion, it can represent one or several spatial prop erties of the earth 's surfac e accurately if oth er properties are sacrificed. The two projections used for world maps th roughout this textbook illustrate that point well. Goode's homolosineprojection, shown in Figure A- I0, belongs to th e oval category an d shows are a accurately, although it gives the imp ression that the earth's surface has been torn , peeled, and flattened. T he interruptions in Figure A-IO have been placed in the major oceans, giving continuity to the land masses. Ocean areas could be featured instead by placing the in terruptions in the continents. Obviously, tha t type of interrup ted projection severely distorts pro ximity relationships. Consequentl y, in different locatio ns th e properties of distance, direction, and shape are also distorted to varying degrees. T he distor tion pattern mimics tha t of cylindrical projections, with the equatori al zone the most faithfully represented (Figu re A-II ).

An alternative to special-proper ty projections such as the equ alarea Goode's homolosine is the compromise projection. In that case no spec ial pr oper ty is achi eved at th e expense of others, and dis tortion is ra ther even ly distri buted among the various properties, instead of being focused on one or several pro perties . The Robinsonprojection, whic h is also used in this textbook, falls in to that category (Figure A- 12). Its oval projection has a glob al fee l, som ewhat like that of Good e's homolosine . But the Robinson projection shows the North Pole and the South Pole as lines that are sligh tly mo re than half the length of the equator, thus exagge rating dis tance s an d areas near the po les. Areas look larger than th ey reall y are in the hig h latitudes (near the po les) and sm aller than they rea lly are in the low latitude s (nea r the equa tor). In addi tion, not

506 Appendix: MapScale and Projections

FIGURE A-10 An inte rrupted Goode's hom olosine, an equal-area projection . (Courtesy ofA CSM)

all lat itude and longitude lines intersec t at r ight angles, as the y do on the eart h, so we kno w that the Robinson projection does not preserve direction or shape either. However, it has fewer interruptions than the Goode's hom olosin e does, so it pre serves proximi ty better. Overall, the Rob inson projection does a goo d job of representing spatial relationships, especially in the low to middle latitud es and along the central meridian.

Scale and Projections in Modern Geography Computers have drastically changed the way in which maps are made and used . In the pree lectron ic age, maps wer e so labori ou s, timeconsuming , and expens ive to make th at relat ively few were created. Fru strated , geographe rs and o ther scient ists o ften found them selves trying to use maps for purposes no t intended by th e map design ers. But to day anyone wit h access to com puter ma pping facil it ies can crea te projections in a flash. Thus project ion s will be increasingly tai lored to specific needs, and mor e and mor e scientists will do their own mapping rather than have som eon e else guess what they want in a map.

Computer mapping creates oppor tunities that go far beyond the construction of projections, of course. Once maps and related geographical

FIGURE A-ll The distortion pattern of the interrup ted Goo de's homolosine

projection, which mimics that of cylindrical projections. (C07l11esy of ACSM)

data are entered into computers, many types of analyses can be carr ied out involving map scales and projections. Distances, areas, and volumes can be computed; searches can be conducted; informatio n from different maps can be combined; optimal rout es can be selected ; facilities can be allocated to the most suitab le sites; and so on. The term used to describe such processes is geographical information system, or GIS (Figure A-l3). Within a GIS, projections provide the mechanism for linking data from different sour ces, and scale provides the basis for size calculations of all sorts. M astery of both projection and scale become s the user's responsibility because the map user is also the map maker. N ow more than ever, effective geography depends on knowledge of the close association between scale and projection.


---P '>: ~ . \ \ '\. ,, ~, <,<,~ ~~

~/ / 7. / ~$. U~ A '" \"""\ \~-\ ,\ " -,", ,',," ~

//7AI I Nx-J 1 .s> y \ .\ .\ \ \ "\ \ ~ ~"" 1111 k: l r7 1 I ~ -l: ~ ., .is \ \ \ ~ ~ \ \ \

/ . / / i » JI7 - - \ I ~ <; ~ ~ \ \ \,\ .

~ i"r'-> ' --, '7 I~ ~ A ~ h

\ i) \ 1\ ~ IS ' f?:~,

\ \ \ \ \ ~ f l I I (I ! ~ ! \ \ \ \\\\\1 y \ I l.---'

I / / / / IAIi Ii ""-.-,\ \ \ \ \L\ , \ 1/ / / / 1 / / / / /

~""''''~-,~ \ \ I· Hif-/ / //h//-,,--.: ,~ \ I / / / / //~ /./-'.-/-

FIGURE A-12 T he com promise Robinson projection, which avoids the interruptions of Goode's homol osine but preserves

no special properties. (Courtesy of A CSM)

COM POSITE OVERLAY

FIGURE A-13 Within a GIS , environ mental data attached to a com mon

terrestrial referen ce system, such as latitu dell ongitude, can be stacked in layers for

spatial comparison and analysis.

Map Projections / Skills

Documents

Transcript of Map Projections / Skills