Map Projections / Skills by Dan Snipes

APPENDIX MAP SCALE AND PROJECTIONS

Phillip C. Muercke

naided , our huma n sen ses pro vide a Li mited view of ou r sur ro un ding-s oTo overcome th ose limi tations, human kind has dev eloped pow erful vehicles of thought and commu nication , such as lan gu age, mathema tics, an d gra phics. Ea ch of t hose 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 u nders tand their rul es, biases, and disto rtion s. T he same is tr ue for the special form of grap hics we cal l ma ps: we mus t master th e logic beh ind th e mapping pro cess befo re we can use maps effectively. A funda mental issue in cartography, the science and art of ma king maps , is th e vast differ enc e between the size and geom etry of wh at is being mapped-the real wo rld, we will call it- and th at of the map itself. Scale and pro jection are the basic cartogra ph ic concep ts that help us understand th at difference and its effects.

Word Statement "One inch equals tell miles"

Graph ic Scale

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Representative Fraction

Map Scale Ou r sens es are dwarfed by th e immensity of our planet; we can sen se directly onl y 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 th e en tire stree t where we live. Ca rtography helps us expand what we can see at one tim e by let tin g us view the scene from some distan t vantage point. T he gre ater th e imagin ary distance between that po sition and the object of our obs ervation, the larger the area the map can cover, but the sma ller th e featu res will appe ar on the map . That redu ction is define d by the map scale, the rati o of th e distan ce on the map to th e dis tan ce on th e earth. M ap users need to kno w about ma p scale for two reasons: so tha t they can convert measurements on a map into mean ingful real -world measur es and so that they can know how abstract th e cartographic representation is.

REAL-WORLD MEASURES . A m ap 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 ca n co m pu te th e ac tu al size o f its features (le ng th, area, and vo lume) . Suc h ca lculations are helped by three express ions o f a m ap 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 u ni ts." For ex ample, th e express ion" I inc h to I 0 miles" means th at I inch on th e ma p rep rese 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 nu mber. Both u nits are exp resse d in mean ing ful terms, such as inches o r centimeters and miles or kilo me ters. Word statements are not intende d fo r pr ecise calcul at ion s but give the map user a rough idea of size an d dista nce. A graphicscale, such as a bar graph, is concrete an d th erefore over com es th e ne ed to visual ize inch es and miles tha t is assoc iated with a wo rd state ment of scale (see Figu re A- I) . A gra phic scale permi ts dir ect visual co mpar ison of featu re sizes and the dis tances be tween featu res. N o rule r is req uired; any m easuring aid will do . I t needs only be compared with the scaled bar; if the len gth of 1 to othpick is eq ual to 2 mil es o n th e g rou nd and the ma p dista n ce eq ual s the len gt h o f 4 too thpicks, th en the gro und distan ce is 4 tim es 2, or 8 miles. Gra phic

502

633,600

FIGURE A-l Co mmon expressions of map scale.

scales are esp ecially co nvenient in this age of cop yin g mach in es, whe n we are more likely to be wo rkin g with a copy than wit h 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 for m of a map scale is the representativefraction (RF) . An RF defines the ratio between the distance on the ma p and the distance on the earth in fraction al terms, such as 11633,600 (also written 1:633,600). The nume rator o f the fraction always refers to th e distanc e on the map, and th e deno mina tor 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 sma ll relative to the size of the earth , it makes sense to use small un its, such as inch es or centime te rs. T hus th e RF 1:633,600 migh t be read as " I inch on th e map to 633,600 inches on the earth." H er ein lies a pro blem with th e RF. Mea n ingful map -distance units im ply a de nomi nator so large th at it is im possible to visualize. Th us, in pr actice, readin g th e ma p scale invo lves an addi tio nal ste p of convert in g th e de no minator to a m eanin gful gro und measur e, such as miles or kilometers. The unwieldy 633,600 becomes the more ma nageable 10 miles when divided by th e num ber of inche s in a mile (63,36 0). O n th e plus side, the RF is good for calc ulat io ns. In part icul ar, th e g ro und dist an ce be twee n poin ts can be eas ily deter min ed fr om a map wi th an R F. 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 m ap with an RF of 1/126, 720 woul d signify a gr ound dis ta nce of 5 X 126,72 0, w hic 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 ta nce is

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 impor tant. 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

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

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 relation ship between map scale and map geometry brings us to the topic of map projections.

Map Projections

1:1,000.000

1:100 ,000

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

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 math ematical minds for centuries. Yet no one has ever found a perfect solu tion; 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 dis tance, 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 cre ated 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 computer driven 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

Planar

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

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o n the proj ecti on is symmetri cal about , and increases with distance from, that poin t o r line. That con ditio n is illustrat ed for th e cylindrical and planar classes of proje cti on s in Figure A-7 . If th e point or line of contact is chang ed to some oth er position on th e globe, th e distor tion pattern will be recen tered on the new position but will retain th e same sym metrica l form . Thus center ing a projection on the area of in terest on the ear th's surface can minimize the effects of projection disto rti on . And recogniz足 ing the general projection shape, associating it with a pers pec tive model , and recall ing the characteristic distortion pattern will pr ovide th e in fo r足 mat ion necessary to compensate for projection distortion ,

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

are often illustrated by using strat eg ically 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 approa ch dep end s o n findin g a pr o足 jection surface that is flat or that can be flatt en ed with out distorti on. 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 ova l 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 projecti on sur face and th e mode l at the point or line of contact is criti cal because disto rtio n o f spa tial properties

PRESERVED PROPERTIES. For a map projection to truthfully depic t the geometry of the earth's surface, it would have to pr eserve the spa足 tial attributes of distance, direction, area, shape, and proximity, That task can be readily accomplished on a globe, but it is not pos sibl e 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

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

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

Appendix: bo th dir ection and distanc e fro m a point, area must be dis tor ted. Simi larly, to pre serv e ar ea as well as direc tion from a point, distance has to be distorted. Because the earth's surface is continuous in all dir ections from every poi nt, d iscont inu it ies tha t violate proximity rela tio nshi ps must occur on all map projectio ns . The tr ick is to place those disconti nu ities where th ey will have th e least impact on the spatial rela tionships in which the ma p user is interes ted . We must be carefu l when we use spa tial terms, because the proper ties they refer to can be confusing . The geometry of the familiar plane is very different fro m that of a sphere; yet when we r efer to a flap map, we are in fact ma king reference to the spherical earth tha t was mapped . A shape-preserving pro jection, for examp le, is truthfu l to local shapes such as the right-angle crossing of latitude and longitude lines- but does no t preserve shapes at con tin ental or global levels. A distance-preserving project ion can preserve that prop erty from one point on the ma p in all directions or from a number of points in severa l direc tion s, bu t 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 simultan eously. Thus a shap e-, distance-, or direction-preserving projection is truthfu l to th ose prop erties onl y in part. P artia l tru ths are no t th e only conse quenc e of t r ansfo rm in g a sphere in to a flat sur face. Some projections exp loi t that tr ansfor ma tio n by expressing trai ts th at ar e of con sidera ble value for sp ecific applications. One of th ose is the famous sha pe-preservi n g M ercator projection (F igu re A-B). That cy lin dr ical pro jection was derived ma thema tically in th e 1500s so that a compass bea ring (called r hum b lines) betwee n any two po ints on the earth would plot as stra ight lines on the map. That trait let na vigat ors plan , plo t, an d foll ow courses between orig in an d de stina tion, but it was achieved at the expense of extreme areal distortion toward th e margins of th e projec tio n (see An tarcti ca in Figure A-B) . Although the Mercator projection is ad mirably suited for its inten ded pur pose, its wid esp read but in ap propriate use for nonn aviga tional purposes has dr awn a gre at deal of cri ticism .

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Map Scale and Projections

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FIGURE A-9 A gnomonic projection (A) and a Mercator projection (B), bo th o f value to long-dist anc e n avigators.

T he gnomonic projection is also usefu l for naviga tion. It is a plan ar projection with the valuable ch aracteristic of showing the sho rtest (or great circle) rou te between any two poin ts on the earth as straight lines. Long-distance 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 th e stra igh t 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 co nstru ct an approxim ation of th at course on the Me rca tor projection . Navigating the shortest course between origin and destination th en invol ves following the stra igh t segmen ts of the course and making directi ona l corrections between segments. L ike 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 tha n navigation or com municati ons. PROJECTIONS USED IN TEXTBOOKS. Although a map projection cannot be free of distortion, it can represent on e 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, belon gs to th e oval category an d show s are a accurately, alth ough 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 placin g th e in terruptions in the con tinen ts. Obviously, tha t type of interrup ted pro jection severely distorts pro ximity relation ships. Conse qu entl y, in differen t locatio ns th e properties of dista nce, direction, and shape are also distorted to varying degrees. T he distor tion pattern mimics tha t of cylindrical proj ections, with the equatori al zon e the most faithfully represented (Figu re A-II ). An alternative to special-p roper ty projections such as the equ al area Goode's homolosine is the compromise projection. In that case no spec ial pr oper ty is achi eved at th e expense of oth ers, and dis tortion is ra ther even ly distri buted among the var ious 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 th at of Good e's ho molosine . But the Robinson projection shows th e No rth Pole and the South Pole as lines th at are sligh tly mo re th an half th e 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 th e hig h latitudes (near th e 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:

Map Scale and Projections

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

all lat itude and lon gitude lin es int ersec t at right angles, as the y do on th e ea rt h, so we kno w that th e Ro binson projecti on does not pr eserve directi on o r shape either. However, it has fewer interruption s than the Good e's hom olosin e does, so it pre serves proximi ty better. Overall, th e Rob inson projecti on does a goo d job of representing spatial rela tionships, especially in th e low to middle latitud es and along the central meridian.

FIGURE A-ll The distortion pattern of th e interrup ted Goo de's homolosine projection, which mimics that of cylindrical projections. (C07l11esy of ACSM)

Scale and Projections in Modern Geography Com puters have drastically changed the way in which maps are made and used . In the pree lectron ic ag e, m aps wer e so lab ori ou s, time co nsumi ng , and expens ive to make th at relat ively few were created. Fru strated , geog raphe rs and o ther scientists o fte n found th em selves trying to use maps for purposes no t intende d by th e map design ers. But to day anyone wit h access to com puter ma pping facilities can crea te projections in a flash. Thus project ion s will be increasingly tai lore d to specific needs, and mor e and m or 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 con stru ction of projections, of course. Once maps and related geographical

data are entered into computers, many types of analyses can be carr ied out involving map scales and proj ections. 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 pr ocesses is geographical information system, or GIS (Figure A-l3). Within a GIS, projections provide th e mech anism 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 responsi bility because the map user is also the map ma ker. N ow more than ever, effective geography depends on knowledge of the close association be tween scale and projection.

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FIGURE A-12 T he com promise Robinson pro jection, which avoids th e interruptions of G oode's homol osine but preserves no special properties. (Courtesy of A CSM)

COM POSIT E OVER LAY

FIGURE A-13 Within a GIS , environ ment al 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.