map projections - welcome to cuny - the city university of new york

3Map projections

A map projection is a geometrical transformation of the earth's spherical or ellipsoidal surface onto aflat map surface. Much has been written about map projections, yet people still find this subject to beone of the most bewildering aspects of map use. Many people readily admit that they don't under-stand map projections. This shortcoming can have unfortunate consequences. For one thing, it hinderstheir ability to understand international relations in our global society. It also makes them easy preyfor politicians, special interest groups, advertisers, and others who, through lack of understanding orby design, use map projections in potentially deceptive ways.

Potentially, there are an infinite number of map projections, each of which is better suited for someuses than for others. How, then, do we go about distinguishing one projection from another andchoosing among them? One way is to organize the wide variety of projections into a limited numberof groups or families on the basis of shared attributes. Two approaches commonly used include clas-sifying the projections into families based on their geometrical distortion properties (relating todistance, shape, direction, and area) and examining the nature of the surface used to construct theprojection (a plane, a cone, or a cylinder), which helps us understand the pattern of spatial distor-tion over the map surface. The two approaches go hand in hand because the map user is concerned,first, with what spatial properties are preserved (or lost) and, second, with the pattern and extent ofdistortion. In this chapter, we will look closely at the distortion properties of map projections and thesurfaces used in the creation of projections.

Before we discuss map projection properties and families, we can help to clarify the issue of mapprojections with a discussion of the logic behind them. Why are projections necessary? We'll begin ourdiscussion with globes, a form of map you have probably looked at since childhood. Your familiaritywith globes makes it easy for us to compare them with flat maps.

36 Chapter 3 MAP PROJECTIONS

GLOBES VERSUS FLAT MAPS

Of all maps, globes give us the most realistic picture ofthe earth as a whole. Basic geometric properties such asdistance, direction, shape, and area are preserved becausethe globe is the same scale everywhere (figure 3.1).Globes have a number of disadvantages, however. Theydon't let you view all parts of the earth's surface simulta-neously—the most you can see is a hemisphere (half ofthe earth). Nor are globes useful to see the kind of detailyou might find on the road map in your glove compart-ment or the topographic map you carry when hiking.

Globes also are bulky and don't lend themselves toconvenient handling and storage. You wouldn't havethese handling and storage problems if you used abaseball-sized globe. But such a tiny globe would beof little practical value for map reading and analysis,since it would have a scale reduction of approximately1:125,000,000. Even a globe 2 feet (60 centimeters) indiameter (the size of a large desk model) still represents a1:20,000,000 scale reduction. It would take a globe about40-50 feet (12-15 meters) in diameter—the height of afour-story building!—to provide a map of the scale usedfor state highway maps. A globe nearly 1,800 feet (about550 meters) in diameter—the length of six U.S. footballfields!—would be required to provide a map of the samescale as the standard 1:24,000-scale topographic mapseries in the United States.

Another problem with globes is that the instrumentsand techniques that are suited for measuring distance,direction, and area on spherical surfaces are relativelydifficult to use. Computations on a sphere are far more

Figure 3.1 A typical world globe.

complex than those on a plane surface. (For a demon-stration of the relative difference in difficulty betweenmaking distance computations from plane and sphericalcoordinates, see chapter 11.)

Finally, globe construction is laborious and costly.High-speed printing presses have kept the cost of flatmap reproduction to manageable levels but have not yetbeen developed to work with curved media. Therefore,globe construction is not suited to the volume of mapproduction required for modern mapping needs.

It would be ideal if the earth's surface could be mappedundistorted onto a flat medium, such as a sheet of paperor computer screen. Unfortunately, the spherical earth isnot what is known as a developable surface, defined bymathematicians as a surface that can be flattened onto aplane without geometrical distortion. The only develop-able surfaces in our three-dimensional world are cylin-ders, cones, and planes, so all flat maps necessarily distortthe earth's surface geometrically. To transform a spheri-cal surface that curves away in every direction from everypoint into a plane surface that doesn't exhibit curvaturein any direction from any point means that the earth'ssurface must be distorted on the flat map. Map projectionaffects basic properties of the representation of the ,sphereon a flat surface, such as scale, continuity, and complete-ness, as well as geometrical properties relating to direc-tion, distance, area, and shape. What we would like todo is minimize these distortions or preserve a particulargeometrical property at the expense of others. This is themap projection problem.

THE MAP PROJECTION PROCESS

The concept of map projections is somewhat moreinvolved than is implied in the previous discussion.Not one but a series of geometrical transformations isrequired. The irregular topography of the earth's sur-face is first defined relative to a much simpler three-dimensional surface, and then the three-dimensionalsurface is projected onto a plane. This progressiveflattening of the earth's surface is illustrated in figure 3.2.

The first step is to define the earth's irregular surfacetopography as elevations above or sea depths below amore regular surface known as the geoid, as discussedin chapter 1. This is the surface that would result if theaverage level of the world's oceans (mean sea level) wereextended under the continents. It serves as the datum, orstarting reference surface, for elevation data on our maps(see chapter 1 for more on datums).

THE MAP PROJECTION PROCESS 37

4000- Topography0

2000

Geoid Surface a

50

100

150

200

250miles

Geoid SurfaceWGS 84 Ellipsoid Surface

100

.1

00

50

100

150

200

250miles

WGS 84Sphere

Ellipsoid

Figure 3.2 The map projection process involves a progressive flattening of the earth's irregular surface. The land elevation or

sea depth of every point on the earth's surface is defined relative to the geoid surface. The vertical differences between the

geoid and ellipsoid are so small that horizontal positions on the geoid surface are the same as on an oblate ellipsoid surface

such as VVGS 84. The geodetic latitude and longitude of the point is now defined. Geodetic latitudes and longitudes may be

converted into spherical coordinates for small-scale maps. Finally, the geodetic or spherical coordinates are transformed into

planar (x,y) map projection surface coordinates by manipulating the data on a computer.


The second step is to project the slightly undulatinggeoid onto the more regular oblate ellipsoid surface (seechapter 1 for further information on oblate ellipsoids).This new surface serves as the basis for the geodetic con-trol points determined by surveyors and as the datum forthe geodetic latitude and longitude coordinates found onmaps. Fortunately, the vertical differences between thegeoid and oblate ellipsoid surface are so small (less than100 meters on land) that latitude and longitude on theellipsoid is used as the horizontal position on the geoidas well. An additional step taken in making a small-scaleflat map or globe is to mathematically transform geo-detic coordinates into geocentric coordinates on a sphere,usually equal in surface area to the ellipsoid.

The third step involves projecting the ellipsoidal orspherical surface onto a plane through the use of mapprojection equations that transform geographic or spher-ical coordinates into planar (x,y) map coordinates. Thegreatest distortion of the earth's surface geometry occursin this step.

MAP PROJECTION PROPERTIES

Now that you understand the map projection process,you can see that it will lead to some distortion on the flatmap. Let's look now at how the properties of scale, com-pleteness, correspondence relations, and continuity willbe affected by the map projection process.

ScaleBecause of the stretching and shrinking that occurs in theprocess of transforming the spherical or ellipsoidal earthsurface to a plane, the stated map scale (see chapter 2for more information on scale) is true only at selectedpoints or along particular lines called points and linesof tangency—we will talk about these more later inthis chapter. Everywhere else the scale of the flat map isactually smaller or larger than the stated scale.

To grasp the idea of scale variation on map projections,you first must realize that there are in fact two map scales.One is the actual scale—this is the scale that you mea-sure at any point on the map; it will differ from one loca-tion!to another. Variation in actual scale is a consequenceof the geometrical distortion that results from flatteningthe earth.

The second is the principal scale of the map. This isthe scale of the generating globe—a globe reduced tothe scale of the desired flat map. This globe is then trans-formed into a flat map (figure 3.3). The constant scale of

the generating globe is the scale stated on the flat map,which is correct only at the points or lines where theglobe touches the projection surface. Cartographers callthis constant scale the principal scale of the map.

To understand the relation between the actual andprincipal scale at different places on the map, we com-pute a ratio called the scale factor (SF), which is definedas the following:

SF = Actual Scale Principal Scale

We use the representative fractions of the actual andprincipal scales to compute the SF. An actual scale of1:50,000,000 and a principal scale of 1:100,000,000would thus give an SF of the following:

1

SF = 50,000,0001

100,000,000

SF= 100,000,000 50,000,000

SF= 2.0

If the actual and principal scales are identical, then theSF is 1.0. But, as we saw earlier, because the actual scalevaries from place to place, so does the SF. An SF of 2.0 ona small-scale map means that the actual scale is twice aslarge as the principal scale (figure 3.3). An SF of 1.15 on asmall-scale map means that the actual scale is 15 percentlarger than the principal scale. On large-scale maps, theSF should vary only slightly from 1.0 (also called unity),following the general rule that the smaller the area beingmapped, the less the scale distortion.

CompletenessCompleteness refers to the ability of map projectionsto show the entire earth. You'll find the most obviousdistortion of the globe on "world maps" that don't actu-ally show the whole world. Such incomplete maps occurwhen the equations used for a map projection can't beapplied to the entire range of latitude and longitude. TheMercator world map (figure 3.19) is a classic example.Here the y-coordinate for the north pole is infinity,' sothe map usually extends to only the 80th parallel northand south. Omitting these high latitudes may be accept-able for maps showing political boundaries, cities, roads,and other cultural features, but maps showing physicalphenomena such as average temperatures, ocean currents,or landforms normally are made on globally complete

GeneratingGlobe

Actual = 1.50,000,000Scale

_

30°N-------Actual

= 186,600,000Scale

Actual = 1 . 100 000 000Scale ' '

Mercator Projection

Principal Scale1:100,000,000(everywhere)

MAP PROJECTION FAMILIES 39

Figure 3.3 Scale factors (SFs) greater than 1.0 indicate that the actual scales are larger than the principal scale of thegenerating globe.

projections. The gnomonic projection (figure 3.15) is amore extreme example, since it is limited mathematicallyto covering less than a hemisphere (the SF is infinitelylarge 90 degrees away from the projection center point).

Correspondence relationsYou might expect that each point on the earth would betransformed to a corresponding point on the map pro-jection. Such a point-to-point correspondence wouldlet you shift attention with equal facility from a featureon the earth to the same feature on the map, and viceversa. Unfortunately, this desirable property can't bemaintained for all points on many world map projections.As figure 3.4 shows, one or more points on the earthmay be transformed into straight lines or circles on theboundary of the map projection, most often at the northand south poles. You may notice that the SF in the east—west direction must be infinitely large at the poles on thisprojection, vet it is 1.0 in the north—south direction.

ContinuityTo represent an entire spherical surface on a plane, thecontinuous spherical surface must be interrupted at some

*point or along some line. These breaks in continuity formthe map border on a world projection. Where the map-maker places the discontinuity is a matter of choice. Onsome maps, for example, opposite edges of the map arethe same meridian (figure 3.4). Since this means featuresnext to each other on the ground are found at oppositesides of the map, this is a blatant violation of proximity

relations and a source of confusion for map users. Simi-larly, a map may show the north and south poles withlines as long as the equator. This means features adjacentto each other, but on opposite sides of the meridian usedto define the edge of the map, will be far apart along thetop or bottom edge of the map, while on the earth's sur-face they are at almost exactly the same location. Mapsof individual continents (except Antarctica) or nationsalmost always show these areas without breaks in con-tinuity. To do otherwise would needlessly complicatereading, analyzing, and interpreting the map.

MAP PROJECTION FAMILIES

As we mentioned previously, it is sometimes convenientto group map projections into families based on commonproperties so that we can distinguish and choose amongthem. The two most common approaches to groupingmap projections into families are based on geometricdistortion and the projection surface.

Projection families based on geometricdistortionYou can gain an idea of the types of geometric distortionsthat occur on map projections by comparing the grati-cule (latitude and longitude lines) on the projected sur-face with the same lines on a globe (see chapter 1 for moreon the graticule). Figure 3.4 shows how you might make


-7j-- ,_---.,

,31

-,;.„,

90°

Figure 3.4 The point-to-point correspondence between the globe and map projection may become point-to-line at somelocations. There also may be a loss of continuity, in which the same line on the globe forms two edges of the map.

this comparison (in figure 3.4, the drawing of a globe onthe left is actually a map projection of a hemisphere.) Askyourself several questions: To what degree do meridiansconverge? Do meridians and parallels intersect at rightangles? Do parallels shorten with increasing latitude?Are the areas of quadrilaterals on the projection the sameas on the globe?

We can use these observations to understand betterhow cartographers assign the types of geometric distor-tion you see on the map projection into the categoriesof distance, shape, direction, and area. Let's look at eachtype of distortion, beginning with variations in distance.

Equidistant map projections are used to show thecorrect distance between a selected location and anyother point on the projection. While the correct distancewill be shown between the select point and all others, dis-tances between all other points will be distorted. Notethat no flat map can preserve both distance and area.

DistanceThe preservation of spherical great circle distance on amap projection, called equidistance, is at best a partialachievement (see chapter 1 for more on great circles). Fora map projection to be truly distance-preserving (equi-distant), the scale would have to be equal in all directionsfrom every point. This is impossible on a .flat map. Sincethe scale varies continuously from location to locationon a map projection, the great circle distance betweenthe two locations on the globe must be distorted on theprojection. For some equidistant projections such asthe azimuthal equidistant projection that we will learnmore about later in this chapter, cartographers make theSF a constant 1.0 radially outward from a single pointlike the north pole (figure 3.14). Great circle distancesare eorrect along the lines that radiate outward fromthat point. Long-distance route planning is based uponknowing the great circle distance between beginningand ending points, and equidistant maps are excellenttools for determining these distances on the earth.

ShapeWhen angles on the globe are preserved on the map (thuspreserving shape), the projection is called conformal,meaning "correct form or shape." Unlike the property ofequidistance, conformality can be achieved at all pointson conformal projections. To attain the propert y of con-formalitT, the map scale must be the same in all directionsfrom a point. A circle on the globe will thus appear as acircle on the map. But to achieve conformality, it is nec-essary either to enlarge or reduce the scale by a differentamount at each location on the map (figure 3.5). Thismeans, of course, that the map area around each locationmust also vary. Tiny circles on the earth will always mapas circles, but their sizes will differ on the map. Since tinycircles on the earth are projected as circles, all directionsfrom the center of the circles are correct on the projectedcircles. The circles must be tiny because the constantchange in scale across the map means that they will be pro-jected as ovals if they were hundreds of miles in diameteron the earth. You can also see the distortion in the appear-ance of the graticule—while the parallels and meridianswill intersect at right angles, the distance between paral-lels will vary. Typically there will be a smooth increase ordecrease in scale across the map. Conformality appliesonly to directions or angles at—or in the immediatevicinity of—points, thus shape is preserved only in smallareas. It does not apply to areas of any great extent; theshape of large regions can be greatly distorted.

Point to Line

90

Point toPoint

co

ContinuityLoss

90°

MercatorConformalProjection

GeneratingGlobe

Skewing andshearing of

quadrilaterals

Tiny circles areequal in areabut different in

shape

Mollweide Map Projection


Figure 3.5 On a map using a conformal projection, identical

tiny circles (greatly magnified here) on the generating

globe are projected to the flat map as different sized circles

according to the local SF. The appearance of the graticule in

all the projected circles is the same as on the globe.

Conformal maps are best suited for tasks that involveplotting, guiding, or analyzing the motion of objects overthe earth's surface. Thus, conformal projections are usedfor aeronautical and nautical charts, topographic quad-rangles, and meteorological maps, as well as when theshape of environmental features is a matter of concern.

Direction

While conformal projections preserve angles locally,it's sometimes important to preserve directions glob-ally. Projections that preserve global directions are calledtrue direction projections. You will also sometimes seethese referred to as azimuthal projections. As you willsee in the next section, this term is also sometimes usedinstead of "planar" to describe projections. To avoid con-fusion, we will use the terms "true direction" and "planar"in this book. Unfortunately, no projection can representcorrectly all directions from all points on the earth asstraight lines on a flat map. But scale across the map canbe arranged so that certain types of direction lines arestraight, such as in figure 3.13 where all meridians radiat-ing outward from the pole are correct directionally. Alltrue direction projections correctly show the azimuth ordirection from a reference point, usually the center of themap (see chapter 12 for more on azimuths). Hence, greatcircle directions on the ground from the reference pointcan be measured on the map.

Generat ng Globe

\zz,

Figure 3.6 Tiny circles on the generating globe (greatly

enlarged here) are projected as ellipses of the same area

but different shape. This causes the directions of the major

and minor axes for each ellipse to be skewed. Similarly,

quadrilaterals are skewed and sheared on an equal area

projection.

True direction projections are used to create maps thatshow the great circle routes from a selected point to adesired destination. Special true direction projections formaps like this are used for long-distance route planningin air and sea navigation.

AreaWhen the relative size of regions on the earth ispreserved everywhere on the map, the projection is saidto be equal area and to have the property of equivalence.The demands of achieving equivalence are such thatthe SF can only be the same in all directions along one ortwo lines, or from at most two points. Since the SF andhence angles around all other points will be deformed,the scale requirements for equivalence and conformal-ity are mutually exclusive. No projection can be bothconformal and equal area—only a globe can be.

Adjusting the scale along meridians and parallels sothat shrinkage in one direction from a point is compen-sated for by exaggeration in another direction creates an

Planar

Cylindrical


equal area map projection. For example, a small circleon the globe with an SF of 1.0 in all directions may beprojected as an ellipse with a north–south SF of 2.0 andeast–west SF of 0.5 so that the area of the ellipse is thesame as the circle. Equal area world maps, therefore, com-pact, elongate, shear, or skew circles on the globe as wellas the quadrilaterals of the graticule (figure 3.6). The dis-tortion of shape, distance, and direction is usually mostpronounced toward the map's margins.

Despite the distortion of shapes inherent in equal areaprojections, they are the best choice for tasks that callfor area or density comparisons from region to region.Examples of geographic phenomena best shown on equalarea map projections include world maps of populationdensity, per capita income, literacy, poverty, and otherhuman-oriented statistical data.

Projection families based on map projectionsurfaceSo far we've categorized map projections into familiesbased on the types of geometric distortions that occuron them, realizing that particular geometric and otherproperties can be preserved under special circumstances.We can also categorize map projections based on thedifferent surfaces used in constructing the projections.

As a child, you probably played the game of castinghand shadows on the wall. The surface your shadow wasprojected onto was your projection surface. You discov-ered that the distance and direction of the light sourcerelative to the position of your hand influenced the shapeof the shadow you created. But the surface you projectedyour shadow onto also had a great deal to do with it. Ashadow cast on a corner of the room or on a curved sur-face like a beach ball or lampshade was quite differentfrom one thrown upon the flat wall.

You can think of the globe as your hand and the mapprojection as the shadow on the wall. To visualize this,imagine a transparent generating globe with the graticuleand continent outlines drawn on it in black. Then sup-pose that you place this globe at various positions rela-tive to a source of light and a surface that you project itsshadow onto (figure 3.7). In the case of maps, the flat sur-face that the earth's features are projected onto is calledthe developable surface. There are three basic projec-tiati families based on the developable surface—planar,conic, and cylindrical. Depending on which type ofprojection surface you use and where your light sourceis placed, you will end up with different map projectionscast by the globe. All projections that you can create inthis light-casting way are called true perspective.

Strictly speaking, very few projections actually involvecasting light onto planes, cones, or cylinders in a physi-cal sense. Most are not purely geometric and all use a setof mathematical equations in map projection softwarethat transform latitude and longitude coordinates on theearth into x,y coordinates on the projection surface. Thex,y coordinates for a coastline, for example, are the end-points for a sequence of short straight lines drawn by aplotter or displayed on a computer monitor. You see thesequence of short lines as a coastline. Projections thatdon't use developable surfaces are more common becausethey can be designed to serve any desired purpose, can bemade conformal, equal area, or equidistant, and can bereadily produced with the aid of computers. Yet even non-developable surface projections usually can be thought ofas variations of one of the three basic projection surfaces.

Figure 3.7 Planes, cones, and cylinders are used as true

perspective map projection surfaces.

Gnomonic

Orthographic

Stereographic


Figure 3.8 The three trueperspective planar projections(orthographic, gnomonic, andstereographic) can be thought of asbeing constructed by changing thelight-source location relative to thegenerating globe (they are actuallyconstructed by mathematicalequations in mapping software).

PlanarPlanar projections (also called azimuthal projections)can be thought of as being made by projecting onto aflat plane that touches the generating globe at a point orslices through the generating globe. If you again consideryour childhood shadow-casting game, you will recallthat the projection surface was only one of the factorsinfluencing the shape of the shadow cast upon the wall.The other influence was the distance of the light sourcefrom the wall. Thus, there will be basic differences withinthe planar projection family, depending on where theimaginary light source is located.

This family includes three commonly used trueperspective planar projections, which can be con-ceived of as projecting onto a plane tangent to thegenerating globe at a point of tangency (figure 3.8):orthographic (light source at infinity), stereographic(light source on the surface of the generating globeopposite the point of tangency), and gnomonic (lightsource at the center of the generating globe). Otherprojections, such as the azimuthal equidistant and

1Lambert azimuthal equal area (described later in thischapter), are mathematical constructs that cannot becreated geometrically and must be built from a set ofmathematical equations.

CylindricalCylindrical projections can be thought of as beingmade by projecting onto a cylinder that touches the gen-erating globe along any small circle or slices through thegenerating globe. As with planar projections, cylindricalprojections can also be distinguished by the location ofthe light source relative to the projection surface. Trueperspective cylindrical projections can be thought ofas projecting onto a cylinder tangent to the generatingglobe along a great circle line of tangency (usually theequator). This family includes two commonly used trueperspective projections (figure 3.9): central cylindrical(light source at the center of the generating globe) andcylindrical equal area (linear light source akin to a fluo-rescent light with parallel rays along the polar axis). Theseare described in more detail later in this chapter. Thewhole world can't be projected onto the central cylindri-cal projection because the polar rays will never intersectthe cylinder. For this reason, it is often cut off at somespecified latitude north and south.

Conic

The last family of map projections based on the surfaceused to construct the projection is conic projections,which can be thought of as projecting onto a cone withthe line of tangency on the generating globe along anysmall circle or a cone that slices through the generatingglobe. We usually select a mid-latitude parallel as a line

Cylindrical equalarea projection

3 .6

Central cylindrical projectionProjection

plane

Generating'globe

Figure 3.9 True perspective central cylindrical and

cylindrical equal area projections.

a result, the central part of the proj ection has a slightlysmaller SF than the stated scale and the edges don't haveSFs as large as with the tangent case, thus providing agreater area of minimum distortion.

Obviously, there is minimal distortion around thepoint or line of tangency. This explains why earth curva-ture may often be ignored without serious consequencewhen using flat maps for a local area, but only if theline—or lines—of tangency are set to fall within thearea being mapped. But as the distance from the pointor line of tangency increases, so does the degree of scaledistortion. By the time a projection has been extended toinclude the entire earth, scale distortion may have greatlyaffected the earth's appearance on the map.


of tangency. This projection family includes one trueperspective conic projection, which can be conceivedof as projecting onto a cone tangent to the generatingglobe along a small circle line of tangency, usually in themid-latitudes. The one commonly used true perspectiveprojection in this family is the central conic projection(light source at the center of the generating globe).

To fully understand map projection families based onthe projection surface, it is also necessary to understandvarious projection parameters, including case and aspect,among others.

MAP PROJECTION PARAMETERS

Tangent and secant caseThe projection surface may have either a tangent or secantrelationship to the globe. A tangent case projectionsurface will touch the generating globe at either a point(called a point of tangency) for planar projections oralong a line (called a line of tangency) for conic or cylin-drical projections (figure 3.10). The scale factor (SF) is 1.0at the point of tangency for planar projections or alongthe line of tangency for cylindrical and conic projections.The SF increases outward from the point of tangency orperpendicularly away from the line of tangency.

A secant case planar projection surface intersects thegenerating globe along a small circle line of tangency(figure 3.10). Secant case conic projections have two smallcircle lines of tangency, or standard parallels, usually atthe mid-latitudes. Secant case cylindrical projectionshave two small circle lines of tangency that are equidis-tant from the parallel where the projection is centered.For example, a secant case cylindrical projection centeredat the equator might have the 10°N and 10°S parallels aslines of tangency. All secant case projections have an SFof 1.0 along the lines of tangency. Between the lines oftangency, the SF decreases from 1.0 to a minimum valuehalfway between the two lines, while outside the lines, theSF increases from 1.0 to a maximum value at the edge ofthe map. In other words, the SF is slightly smaller in themiddle part of the map and slightly larger than the statedscale at the edges. For planar projections, the SF increasesoutward from the circle of tangency and decreases inwardto a minimum value at the center of the circle of tangency.

The advantage of the secant case is that it minimizesthe overall scale distortion on the map. This is becausethe SF is 1.0 along a circle instead of at a single point (fora secant case planar projection) or two lines instead ofone (for secant case conic or cylindrical projections). As

Point of tangency

SECANT CASELine of tangency

Largest „.--"--41:7-'tI-i

scale I I I, , ,, .t._

SF = 1 atLine of

tangency I

Largestscale 4

SECANT CASE

Largest,-- 'scale

,

aSF = 1 at/;Lines of

/tangency\

Largestscale

biact

mallastscal

TANGENT CASE SECANT CASE

SF = 1 atLine of -

tangency

Largest scale\-+n

Largest scale

SF = 1 atLines Of , TtangencTT Y

Largest scaleft

Largest scale

MAP PROJECTION PARAMETERS 45

PLANAR PROJECTIONS CYLINDRICAL PROJECTIONSTANGENT CASE

TANGENT CASE

CONIC PROJECTIONS

Figure 3.10 Tangent and secant cases of the three basic projection surfaces.

Secant case projections are common for all threeprojections families—planar, conic, and cylindrical. Youwill also find tangent case planar and cylindrical, buttangent case conic projections are little used in mapping,as their relatively small area of minimal scale distortionlimits their practical value.

Although secant case planar projections have lessoverall distortion, the tangent case is often used insteadfor equidistant projections. Secant case cylindrical

projections are best suited for world maps since they haveless overall distortion than tangent case projection—thisdistortion is minimized in the mid-latitudes, which isalso where the majority of the earth's populatiOn is found.Secant case conic projections are best suited for maps ofmid-latitude regions, especially those elongated in aneast—west direction. The United States and Australia,for example, meet these qualifications and are frequentlymapped using secant case conic projections.

Planar Projection AspectsOblique PolarEquatorial


AspectMap projection aspect refers to the location of thepoint or line(s) of tangency on the projection surface(figure 3.11). A projection's point or line(s) of tangencycan in theory touch or intersect anywhere on the devel-opable surface. When a tangent case projection point orline of tangency is at or along the equator, the resultingprojection is said to be in equatorial aspect. When thepoint or line of tangency is at or encircles either pole, theprojection is said to be in polar aspect. With cylindri-cal projections, the term transverse aspect is also used.Transverse aspect occurs when the line of tangency forthe projection is shifted 90 degrees so that it followsa pair of meridians (figure 3.12). Any other alignment ofthe point or line(s) of tangency to the globe is a projec-tion in oblique aspect.

Aspects of secant case projections are defined in amanner similar to tangent case aspects. For example, apolar aspect, secant case planar projection has the northor south pole as the center of its circle of tangency. Atransverse aspect, secant case cylindrical projection hastwo lines of tangency—equally spaced from the meridian

—that would be its line of tangency in its tangent case.The aspect that has been used the most historically for

each of the three projection families based on the devel-opable surface (planar, conic, and cylindrical) is referredto as the normal aspect. The normal aspect of planar pro-jections is polar; the normal aspect of conic projections is

the apex of the cone above a pole; and the normal aspectof cylindrical projections is equatorial.

The normal aspect of planar projections is polar—theparallels of the graticule are concentric around the centerand the meridians radiate from the center to the edgesof the projection. In the normal aspect for conic projec-tions, parallels are projected as concentric arcs of circles,and meridians are projected as straight lines radiatingat uniform angular intervals from the apex of the cone(figure 3.10).

The graticule appears entirely different on normalaspect and transverse aspect cylindrical projections. Youcan recognize the normal aspect of cylindrical projec-tions by horizontal parallels of equal length, verticalmeridians of equal length that are also equally spaced,and right angle intersections of meridians and parallels.Transverse aspect cylindrical projections look quite dif-ferent. The straight line parallels and meridians on thenormal aspect projection become curves in the transverseaspect. These curves are centered on the vertical line oftangency in the tangent case or halfway between the twovertical lines of tangency in the secant case (figure 3.12).

As we have seen, the choice of projection surfaceaspect leads to quite different appearances of the earth'sland masses and the graticule. Yet the distortion proper-ties of a given projection surface remain unaltered whenthe aspect is changed from polar to oblique or equato-rial. The map on the left in figure 3.12 is a good example.

Figure 3.11 Tangent caseequatorial, oblique, and polaraspects of the planar projectionsurface dramatically affect theappearance of the graticuleand land areas. Scale distortionincreases radially away from thepoint of tangency, no matterwhere the point is located onthe globe.

Transverse

CYLINDRICAL PROJECTION ASPECTS

Equatorial(normal)

MAP PROJECTION PARAMETERS 47

• In its normal aspect tangent case, the SF is 1.0 at theequator and increases north and south at right angles tothe equator. In its transverse aspect tangent case, the SFis 1.0 along the pair of meridians that form the line oftangency (that is, a selected meridian and its antipodalmeridian). The SF again increases perpendicularly to theline of tangency, which is due east and west at the inter-section of each parallel and the pair of meridians. The SFwill be exactly the same distance above and below theequator on the normal aspect and exactly the same dis-tance again to the left and right of the meridian pair onthe transverse aspect projection.

Other map projection parametersThere are a few other map projection parameters that areuseful to know about as they can have an impact on theappearance and appropriate use of a map projection. Thecentral meridian (also called the longitude of origin orless commonly the longitude of center) defines the originof the longitudinal x-coordinates. This is usually modi-fied to so that it is in the center of the area being mapped.The central parallel (also called the latitude of origin orless commonly the latitude of center) defines the origin ofthe y-coordinates—an appropriately defined map projec-tion will be modified so that this parameter is definedrelative to the mapped area, although this parameter maynot be located at the center of the projection. In particu-lar, conic projections use this parameter to set the originof the y-coordinates below the area of interest so that ally-coordinates are positive.

Figure 3.12 Tangent case

equatorial and transverse

aspects of the cylindrical

projection surface dramatically

affect the appearance of

• the graticule and land areas.

Scale distortion increases

perpendicularly away from the

line of tangency, no matter

where the line is located on

the globe.


COMMONLY USED MAPPROJECTIONS

As stated above, one way that map projections arecommonly grouped into families is based on the projec-tion surface—a plane, cone, or cylinder. We'll use thesesurfaces to structure our discussion of commonly usedmap projections.

Planar projections

OrthographicThe orthographic projection is how the earth wouldappear if viewedfrom a distant planet (figure 3.13). Sincethe light source is at an infinite distance from the gener-ating globe, all rays are parallel. This projection appearsto have been first used by astronomers in ancient Egypt,but it came into widespread use during World War IIwith the advent of the global perspective provided by theair age. It is even more popular in today's space age, oftenused to show land-cover and topography data obtainedfrom remote sensing devices. The generating globe andhalf-globe illustrations in this book are orthographicprojections, as is the map on the front cover of the book.The main drawback of the orthographic projection is thatonly a single hemisphere can be projected. Showing theentire earth requires two hemispherical maps. Northernand southern hemisphere maps are commonly made, butyou may also see western and eastern hemisphere maps.

StereographicProjecting a light source from the antipodal point onthe generating globe to the point of tangency creates thestereographic projection (figure 3.14). This is a confor-mal projection, so shape is preserved in small areas. TheGreek scholar Hipparchus is credited with inventingthis projection in the second century BC. It is now mostcommonly used in its polar aspect and secant case formaps of polar areas. It is the projection surface used forthe Universal Polar Stereographic grid system for polarareas, as we will see in the next chapter. A disadvantageof the stereographic: conformal projection is that it is gen-erally restricted to one hemisphere. If it is not restrictedto eine hemisphere, then the distortion near the edgesincreases to such a degree that the geographic featuresin these areas are basically unrecognizable. In past cen-turies, it was used for atlas maps of the western or easternhemisphere.

Figure 3.13 The orthographic projection best shows the

spherical shape of the earth.

Figure 3.14 Polar stereographic projection of the northern

hemisphere. Since this is a conformal projection, tiny circles

on the generating globe are projected as circles of the same

size at the point of tangency to four times as large at the

equator.

COMMONLY USED MAP PROJECTIONS 49

GnomonicProjecting with a light source at the center of thegenerating globe to a tangent plane produces the gno-monic projection. One of the earliest map projections,the gnomonic projection was first used by the Greekscholar Males of Miletus in the sixth century BC forshowing different constellations on star charts, whichare used to plot planetary positions throughout the year.The position of constellations in the sky over the yearwas used as a calendar, telling farmers when to plant andharvest crops, and when floods would occur. Horoscopesand astrology also began with the ancient Greeks overtwo thousand years ago. Many believed that the positionof the sun and the planets had an effect on a person's lifeand that future events in their lives could be predictedbased upon the location of celestial bodies in the sky.

The gnomonic projection is the only projection withthe useful property that all great circles on the globe areshown as straight lines on the map (figure 3.15). Sincea great circle route is the shortest distance between twopoints on the earth's surface, the gnomonic pro jection isespecially valuable as an aid to navigation (see great circledirections in chapter 12). The gnomonic projection is alsoused for plotting the global dispersal of seismic and radiowaves. Its major disadvantages are increasing distortionof shape and area outward from the center point and theinability to project a complete hemisphere.

Figure 3.15 Polar gnomonic projection of the northernhemisphere from 15°N to the pole. All straight lines on theprojection surface are great circle routes. Note the severeshape distortion of this projection compared to the polarstereographic projection shown in figure 3.13.

Azimuthal equidistantThe azimuthal equidistant projection in its polar aspecthas the distinctive appearance of a dart board—equallyspaced parallels and straight-line meridians radiatingoutward from the pole (figure 3.16). This arrangement ofparallels and meridians results in all straight lines drawnfrom the point of tangency being great circle routes.Equally spaced parallels mean that great circle distancesare correct along these straight lines. The ancient Egyp-tians apparently first used this projection for star charts,but during the air age it also became popular for useby pilots planning long-distance air routes. In the daysbefore electronic navigation, the flight planning roomin major airports had a wall map of the world that usedan oblique aspect azimuthal equidistant projection cen-tered on the airport. You will also find them in the publicareas of some airports. All straight lines drawn from theairport are correctly scaled great circle routes. This is oneof the few planar projections that can show the entiresurface of the earth.

Figure 3.16 Polar aspect azimuthal equidistant worldmap projection tangent at the pole. Great circle distancesare correct along straight lines outward from the pointof tangency at the north pole since the north-south SFis always 1. The east-west SF increases to a maximum ofinfinity at the south pole, where there is a point-to-linecorrespondence.


Lambert azimuthal equal areaIn 1772 the mathematician and cartographer JohannHeinrich Lambert published equations for the tan-gent case planar Lambert azimuthal equal area pro-jection, which, along with other projections he devised,carries his name. This planar equal area pro jection isusually restricted to a hemisphere, with polar and equa-torial aspects used most often in commercial atlases(figure 3.17). More recently, this projection has beenused for statistical maps of continents and countries thatare basically circular in overall extent, such as Australia,North America, and Africa. You will also see the oceansshown on maps that use the equatorial or oblique aspectsof this projection. The Lambert azimuthal equal area pro-jection is particularly well suited for maps of the PacificOcean, which is almost hemispheric in extent.

Cylindrical projections

EquirectangularThe equirectangular projection is also called theequidistant cylindrical or geographic projection. Thissimple map projection, nearly 2,000 years old, is attrib-uted to Marinus of Tyre, who is thought to have con-structed the projection about in 100 AD. Parallels andmeridians are mapped as a grid of equally spaced horizon-tal and vertical lines twice as wide as high (figure 3.18).The equal spacing of parallels means that the projection isequidistant in the north–south direction with a constantSF of 1.0. In the east–west direction the SF increasessteadily from a value of 1.0 at the equator to infinity ateach pole, which is projected as a straight line.

You may see world maps showing elevation data orsatellite imagery on this projection. This choice of pro-jection is not based on any geometrical advantages, butrather on the simplicity of creating flat world maps whenthey had to be done by hand.

Figure 3.17 The Lambert azimuthal equal area projection isoften used for maps of continents that have approximatelyequal east-west and north-south extents. This map ofNorth America in the box is part of an oblique aspect of theprojection centered at 45°N, 100°W.

-. , 111110

—-2

1

SF0

411104W

..,.:

..,

0

..

,

u)

1/Ilk

Q0

411

•.• •

IimIlIln

5------EP----r

VW VW

2 0..... ...

,

Figure 3.18 Equirectangular world map projection.

Figure 3.19 Mercator

projection shown with

rhumb lines between

selected major world

cities.


Mercator

Constructed by Gerhardus Mercator in 1569, theMercator projection is a tangent case cylindrical con-formal projection. As with all conformal projections,shape is preserved in small areas. This projection offers aclassic example of how a single projection can be used bothpoorly and well. Looking at the projection (figure 3.19),we can imagine Mercator started the construction of hisprojection with a horizontal line to represent the equatorand then added equally spaced vertical lines to representthe meridians. Mercator knew that meridians on theglobe converge toward the poles, so the meridians he haddrawn as parallel vertical lines must become progressivelymore widely spaced toward the poles than they would beon the generating globe. He progressively increased thespacing of parallels away from the equator so that theincrease matched the increased spacing between themeridians. As a result of this extreme distortion towardthe poles, he cut his projection off at 80°N and S. Thisnot only produced a conformal map projection, but alsothe only projection on which all lines of constant com-pass direction, called rhumb lines, are straight lines onthe map.

Navigators who used a magnetic compass immediatelysaw the advantage of plotting courses on the Mercatorprojection, since any straight line they drew would be aline of constant bearing. This meant they could plot a

course on the map and simply maintain the associatedbearing during passage to arrive at the plotted location.You can see how navigators would prefer a map on whichcompass bearings would appear as straight rhumb lines(see chapter 13 for more on rhumb line plotting). TheMercator projection has been used ever since for nauticalcharts, such as small-scale piloting charts of the oceans.The large-scale nautical charts used for coastal navigationcan be thought of as small rectangles cut out of a worldmap that uses the Mercator projection.

Of course, using these maps for navigation is not reallyas simple as this. Recall that the gnomonic projection isthe only projection on which all great circles on the gen-erating globe are shown as straight lines on the flat map.Lines drawn on a Mercator projection show constantcompass bearing, but they are not the same as the greatcircle route, which is the shortest distance between twopoints on a globe. Therefore, the Mercator projection isoften used in conjunction with the gnomonic projectionto plot navigational routes. The gnomonic projection isused first to determine the great circle route between twopoints, and then the route is projected to transform thegreat circle to a curve on a Mercator projection. Finally,the curve is translated into a series of shorter straightline segments representing portions of the route, eachwith constant compass direction (see figure 12.28 inchapter 12 for an illustration of this process).


The use of the Mercator projection in navigation is anexample of a projection used for its best purpose. A pooruse of the Mercator projection is for wall maps of theworld. We saw earlier that this projection cannot coverthe entire earth, and is often cut off. at 80°N and S. Cut-ting off part of the world does create a rectangular projec-tion surface with a height-to-width ratio that fits wallsvery well. The problem, of course, is the extreme scaleenlargement and consequent area distortion at higher lat-itudes. The area exaggeration of North America, Europe,and Russia gives many people an erroneous impression ofthe size of the land masses.

Gall-PetersThe Gall-Peters projection is a variation of thecylindrical equal area projection. Its equations werepublished in 1885 by Scottish clergyman James Gallas a secant case of the cylindrical equal area projectionthat lessens shape distortion in higher latitudes by plac-ing lines of tangency at 45°N and 45°S. Arno Peters, aGerman historian and journalist, devised a map basedon Gall's projection in 1967 and presented it in 1973 asa "new invention" superior to the Mercator world projec-tion (figure 3.20).

The projection generated intense debate becauseof Peters's assertion that this was the only "nonracist"world map. Peters claimed that his map showed devel-oping countries more fairly than the Mercator projec-tion, which distorts and dramatically enlarges the size ofEurasian and North American countries. His assertionwas a bit of a straw dog, since the Mercator projection wasdesigned and admirably suited for navigation and neverintended for comparing country sizes.

The first English version of the Gall-Peters projectionmap was published in 1983, and it continues to havepassionate fans as well as staunch critics.

Although the relative areas of land masses aremaintained, their shapes are distorted. According toprominent cartographer Arthur Robinson, the GallPeters map is "somewhat reminiscent of wet, ragged longwinter underwear hung out to dry on the Arctic Circle."Although several international organizations haveadopted the Gall-Peters projection, there are other equalarea world projections, such as the Mollweide projection(figure 3.24), that distort the shapes or land masses farless. Maps based on the Gall-Peters iprojection continueto be published and are readily available, though fewmajor map publishers use the projection today.

Figure 3.20 Gall-Peters projection world map.


Transverse Mercator

We saw earlier that Lambert constructed his azimuthalequal area projection in 1772. That same year, he alsoconstructed the transverse Mercator projection, alongwith the Lambert azimuthal equal area projection andanother described later in this chapter. In Europe thisprojection is called the Gauss-Kruger, in honor of themathematicians Carl Gauss and Johann Kruger wholater worked out formulas describing its geometric dis-tortion and equations for making it on the ellipsoid.Lambert's idea for the transverse Mercator projectionwas to rotate the Mercator projection by 90 0 so that theline of tangency became a pair of meridians—that is, anyselected meridian and its antipodal meridian (figure 3.12,bottom right). The resulting projection is conformal, asis the Mercator projection, but rhumb lines no longer arestraight lines. Along the central meridian of the projec-tion (the vertical meridian that defines the y-axis of theprojection), the SF is 1.0, and the scale increases perpen-dicularly away from the central meridian. Thus, narrownorth–south strips of the earth are projected with nolocal shape distortion and little distortion of area.

You're likely to see the transverse Mercator projec-tion used to map north–south strips of the earth calledgores (figure 3.21), which are used in the constructionof globes. Because printing the earth's surface directlyonto a round surface is very difficult, instead, a map ofthe earth is printed in fiat elongated sections and thenattached to a spherical object. The narrow, 6°-widezones of the universal transverse Mercator grid system(described in chapter 4) are' based on a secant case trans-verse Mercator projection. North–south trending zonesof the U.S. state plane coordinate system (also explainedin chapter 4) are also based on secant cases of the projec-tion. Most 1:24,000-scale U.S. Geological Survey (US GS)topographic maps are projected on these state planecoordinate system zones.

Figure 3.21 Gores of the globe on transverse Mercatorprojections 30° wide at the equator centered at 90°W and120°W. To make a world globe, the highlighted portion ofeach map would be cut out and pasted onto the globe andother gores would be used to cover the remaining area.

Figure 3.22 Lambertconformal conic

projection used for amap of the United States.

Straight lines are veryclose to great circle

routes, particularly onnorth-south paths.


Conic map projections

Lambert conformal conicThe Lambert conformal conic projection is another ofthe widely used map projections constructed by Lambertin 1772. It is a secant case normal aspect conic projectionwith its two standard parallels placed so as to minimizethe map's overall scale distortion. The standard parallelsfor maps of the conterminous United States are placed at33°N and 45°N to keep scale distortion at the map's edgesto less than 3 percent (figure 3.22). Just as the transverseMercator projection is used as the basis for the state planecoordinate system zones in north–south trending states,the Lambert conformal conic projection is used as thebasis for system zones in east–west trending states inthe United States like Oregon and Wisconsin (see chap-ter 4 for more on the state plane coordinate system usingexamples from these two states). As noted earlier, theseprojections are in turn used for the 1:24,000-scale USGStopographic maps within the state.

One major use of the Lambert conformal conicprojection is for aeronautical charts. All U.S. 1:500,000-scale sectional charts can be thought of as smaller rectan-gles of the 'national map described in the above paragraph.Recall that navigators (like aviators) prefer navigationalcharts that use conformal projections, which preserveshapes and directions locally. Equally important is thefact that straight lines drawn on the 1:500,000-scalecharts are almost great circle routes on the earth's surface(figure 3.22).

Albers equal area conicMathematically devised in 1805 by the Germanmathematician Heinrich C. Albers, the Albers equalarea conic projection was first used in 1817 for a map ofEurope. This is probably the projection used most oftenfor statistical maps of the conterminous United Statesand other mid-latitude east–west trending regions. Forexample, you will see this projection used for U.S. sta-tistical maps created by the Census Bureau and otherfederal agencies. The secant case version that has beenused for nearly 100 years for the conterminous UnitedStates has standard parallels placed at 29.5°N and 45.5°N(figure 3.23). This placement reduces the scale distortionto less than 1 percent at the 37th parallel in the middle ofthe map, and to 1.25 percent at the northern and south-ern edges of the country. USGS products, such as thenational tectonic and geologic maps also use the Albersprojection, as do recent reference maps and satellite imagemosaics of the country at a scale of around 1:3,000,000.

The reason for the widespread use of the Albersprojection is simple—people looking at maps usingthis projection can assume that the areas of states andcounties on the map are true to their areas on the earth.

Pseudocylindrical and other projectionsWe have seen that map projections can be classed intofamilies based on the nature of the surface used to con-struct the projection. This classification results in planar,conic, and cylindrical families. But these three fami-lies constitute only a small portion of the vast array of

Figure 3.23 Albers equal areaconic projection of the UnitedStates.


projections that have been constructed by cartographers.We also noted that map projections could be classifiedbased on their geometric distortion properties. Usingthis classification, we have shown that some preserveareas or shapes, while others have no special propertyaside from holding overall scale distortion to a minimumor presenting a pleasing visual image. For many purposes,a projection that "looks right" is more important than aprojection that rigidly provides area, distance, shape, ordirection fidelity.

Let's look at a last set of projections that can bedescribed using one or both of the classifications above.These include pseudocylindrical and other projectionsthat are of special interest. Pseudocylindrical map pro-jections can be conceived of as juxtaposing a number ofpartial cylindrical maps. They are similar to cylindri-cal projections in that parallels are horizontal lines andmeridians are equally spaced. The difference is that allmeridians except the vertical-line central meridian arecurved instead of straight.

MollweideYou've probably seen world maps in the shape of an ellipse

# twice as wide as it is high. Most likely you were looking atthe Mollweide projection, constructed in 1805 by theGerman mathematician Carl B. Mollweide. This ellipti-cal equal area projection most commonly uses the equa-tor as the standard parallel and the prime meridian as thecentral meridian of the projection (figure 3.24). Parallelsare horizontal lines, but they are not equally spaced as on

the generating globe. Instead, they reduce in distance asthe poles are approached. The elliptical shape of this pro-jection makes it look more "earth-like," and the overalldistortion in shape is less than on other equal area worldprojections such as the Gall-Peters.

You'll find the Mollweide projection used for mapsthat show a wide range of global phenomena, from popu-lation to land cover to major diseases. Cartographers havedevised other orientations of the projection by adjust-ing the central meridian to better show the oceans or tocenter attention on a particular continent.

Figure 3.24 Mollweide projection used for a world map.


SinusoidalAccording to some sources, Jean Cossin of Dieppe,France, appears to be the originator of the sinusoidalprojection, which he used to create a world map in1570. Others suggest that Mercator devised it, since itwas included in later editions of his atlases. This easilyconstructed equal area projection (figure 3.25) wasused by Nicholas Sanson (ca. 1650) of France for atlasmaps of the world and separate continents, and by JohnFlamsteed (1729) of England for star maps. Hence youmay also see it called the Sanson-Flamsteed projection.

In addition to correctly portraying the relative areasof continents and countries, the sinusoidal world projec-tion has an SF of 1.0 along the central meridian, and theeast—west SF also is 1.0 anywhere on the map. This meansthat the projection is equidistant in the east—west direc-tion and in the north—south direction along the centralmeridian, but only in these directions. Note the severeshape distortion at the edges of the map (figure 3.25).

HomolosineYou may have seen world map projections made bycompositing different projections along certain parallelsor meridians. The uninterrupted homolosine projec-tion (figure 3.26, top), constructed in 1923 by Americangeography professor J. Paul Goode, is a composite oftwo pseudocylindrical projections. The sinusoidal pro-jection is used for the area from 40°N to 40°S latitude,while Mollweide projections cover the area from 40°Nto 40°S to the respective poles. Since the componentprojections are equal area, the homolosine projection isas well. Notice in figures 3.25 and 3.26 that the shapesof the continents look less distorted on the homolosineprojection than on the sinusoidal projection, particularlyin polar areas.

An interrupted projection is one in which thegenerating globe is segmented in order to minimize the

Figure 3.25 Sinusoidal equal area world map projection.

distortion within any lobe (section) of the projection.Shape distortion at the edges of the map can be lessenedconsiderably by interrupting the composite projectioninto separate lobes that are pieced together along a centralline, usually the equator. With interruption, the betterparts of the projection are repeated within each lobe.

The Goode interrupted homolosine projection(figure 3.26, bottom), created in 1923 by Goode from theuninterrupted homolosine projection, is an interruptedpseudocylindrical equal area composite map projectionused for world maps as an alternative to portraying theearth on the Mercator world map projection (figure 3.19).The projection is a composite of twelve segments thatform six interrupted lobes. The six lobes at the top andbottom are Mollweide projections from 40°N or 40°Sto the pole, eaeh with a different central meridian. Thesix interior regions from the equator to 40°N or 40°Sare sinusoidal projections, each with a different centralmeridian. If you look carefully along the edges of thelobes, you can see a subtle discontinuity at the 40th par-allels. The two northern sections are usually shown withsome land areas repeated in both regions to show theGreenland land mass without interruption.

You will find the Goode homolosine projection usedfor maps in commercial world atlases to show a variety ofglobal information. It is a popular projection for showingphysical information about the entire earth, such as landelevations and ocean depths, land-cover and vegetationtypes, and satellite image composites. When viewingthese maps, remember that the continuity of the earth'ssurface has been lost completely by interrupting the pro-jection. This loss of continuity may not be apparent if thegraticule is left off the projection, as may be the case withsome global distribution maps found in world atlases.

RobinsonIn 1963, the American academic cartographer Arthur H.Robinson constructed a pseudocvlindrical projectionthat is neither equal area nor conformal but makes thecontinents "look right." A "right-appearing" projectionis called orthophanic. For the Robinson projection(figure 3.27), Robinson visually adjusted the horizontalline parallels and curving meridians until they appearedsuitable for a world map projection to be used in atlasesand for wall maps. To do this, Robinson represented thepoles as horizontal lines a little over half the length ofthe equator. You may have seen this projection used forworld maps created by the National Geographic Society.It has also been used for wall maps of the world that showthe shape and area of continents with far less distortionthan wall maps that use the Mercator projection.

_--Lobes

IV16,11weide 40°N ON

40°6, 1

MollWeid6I

Lobes - Lobes ,

—40°S

Figure 3.27 Robinson

pseudocylindrical world map

projection.


Figure 3.26 Uninterrupted

homolosine and Goode

interrupted homolosine equal

area world projections.

Uninterrupted Homosoline Projection

Goode Homosoline Projection

Figure 3.28 Aitoff projection used to create a world map.


AitoffOther map projections can be constructed by mathe-matically modifying widely used projections developedcenturies earlier. In 1889, the Russian cartographerDavid Aitoff published a modification of the equato-rial aspect azimuthal equidistant projection that todaycarries his name (figure 3.28). For the Aitoff projec-tion, Aitoff simply doubled the horizontal scale of theazimuthal equidistant projection, creating an ellipticalprojection with the same two-to-one width-to-heightratio as the Mollweide projection (figure 3.24). Unlikethe Mollweide projection, parallels are not straighthorizontal lines, and the map is neither equal area norequidistant. The Aitoff projection is an interesting com-promise between shape and area distortion, suggestingthe earth's shape with less polar shearing than on mapsthat use the Mollweide projection.

The Winkel tripel projection was not used widely until1998 when the National Geographic Society announcedthat it was adopting the Winkel tripe! projection as itsstandard for maps of the entire world. As a result, use ofthe Winkel tripel projection has increased dramaticallyover the last few years.

Figure 3.29 Winkel tripel projection used to create aworld map.

Winkel tripelNew map projections may also be constructed asmathematical combinations of two existing projections.Perhaps the best known example is the Winkel tripelprojection constructed in 1921 by the German cartog-rapher Oswald Winkel. The term tripel is not someone'sname, but rather a German word meaning a combina-tion of three elements. Winkel used the term to. empha-size that he had constructed a compromise projectionthat was neither equal area, conformal, nor equidistant,but rather minimized all three forms of geometric dis-tortion. He accomplished this by averaging the x- andyloordinates computed for the equirectangular andAitoff world projections at the same map scale. The result-ing projection is similar to the Robinson projection, butif you look closely you will see that parallels are not thestraight horizontal lines characteristic of pseudocylindri-cal projections. Rather, they are slightly curving, nonpar-allel lines (figure 3.29).

PROJECTIONS ON THE SPHEREAND ELLIPSOID

Mapmakers have a general rule that small-scale mapscan be projected from a sphere, but large-scale mapsalways must be projected from an ellipsoidal surfacesuch as the WGS 84 ellipsoid. We saw in chapter 1 thatsmall-scale world or continental maps such as globes andworld atlas sheets normally use coordinates based onan authalic or other auxiliary sphere. This was becauseprior to using digital computers to make these types ofmaps numerically, it was much easier to construct themfrom spherical geocentric coordinates. Equally impor-tant, the differences in the plotted positions of sphericaland corresponding geodetic coordinates are negligibleon small-scale maps.

Large-scale maps must be projected from anellipsoidal surface because, as we saw in chapter 1, thespacing of parallels decreases slightly but significantlyfrom the pole to the equator. We noted that on theWGS 84 ellipsoid the distance between two points onedegree apart in latitude (between 00 and 1°) at the equa-tor is 68.703 miles (110.567 kilometers), shorter than the69.407 mile (111.699 kilometer) distance between twopoints at 89°N and 90°N.

PROJECTIONS ON THE SPHERE AND ELLIPSOID 59

Let's see the differences in length when we projectone degree of geodetic latitude along a meridian at thepole, at the 45th parallel, and at the equator using atransverse Mercator projection at a scale of 1:1,000,000(figure 3.30). At this scale, the length of a degree of lati-tude at the pole is projected as slightly over I millime-ter longer than a degree of latitude at the equator. Thisdifference on these two projections may seem minimal,but on the ground it represents a distance of nearly akilometer! At larger map scales the difference in lengthbecomes more noticeable—for instance, slightly over1 centimeter on polar and equatorial maps at a scale of1:100,000. Topographic and other maps at this scale andlarger are projected from an ellipsoid so that accurate dis-tance and area measurements can be made on them. Thesame holds true for large-scale equidistant projectionssuch as the polar aspect azimuthal equidistant projection(figure 3.16), since the spacing of parallels on the projec-tion must be slightly lessened from the pole outward toreflect their actual spacing on the earth.

We saw in chapter 1 and in table C.2 in appendix Cthat the only place on the earth where the 69.05 mile(111.12 kilometer) per degree spacing of parallels on theauthalic or rectifying sphere is essentially the same as onthe WGS 84 ellipsoid is in the mid-latitudes close to the45th parallel. You will find that the transverse Mercatoror another map projection based on the sphere or theWGS 84 ellipsoid will have the same spacing of parallelsin locations that straddle the 45th parallel.

Every map projection has its own virtues andlimitations. You can evaluate a projection only in light ofthe purpose for which a map is to be used. You shouldn'texpect that the best projection for one situation will bethe most appropriate for another. Fortunately, the mapprojection problem effectively vanishes if the cartogra-pher has done a good job of considering projection prop-erties and if you are careful to take projection distortioninto consideration in the course of map use.

At the same time, however, there is the very realpossibility that the mapmaker through ignorance or lackof attention can choose an unsuitable projection or pro-jection parameters. Therefore is it of utmost importanceto map users to understand the projection concepts dis-cussed in this chapter to be assured that the maps we use

• have been made with careful consideration of all the deci-sions required to make an appropriate map projection.

Since most map use takes place at the local level, whereearth curvature isn't a big problem, global map projec-tions aren't a great concern for many users. With regionsas small as those covered by topographic map quadran-gles, your main projection-related problem is that, while

UTM Projections of One Degreeof Geodetic Latitude along a Meridian

90° 45° 1°

— E-I- CDif. CD

CD

= CD-

89°

Figure 3.30 Transverse Mercator map projections of onedegree of geodetic latitude on the WGS 84 ellipsoid alonga meridian at the pole, midway between pole and equator,and at the equator.

the individual sheets match in a north–south direction,they don't fit together in an east-west direction. Yet eventhis difficulty won't be a serious handicap unless you tryto create a large map mosaic.

The age of computers has changed how we think aboutmap projections. You no longer need to "make do" withinappropriate projections. Computer-generated projec-tions are available for almost any use. Furthermore, itis now practical for you to sit down at a computer andconstruct your own projections. Most important, per-haps, you can manipulate projections on the computerin search of the ideal projection base for a given applica-tion. All these benefits can only be realized, of course, ifyou know enough about projections to take advantage ofthese opportunities that computers provide.

Being map-savvy also helps you evaluate projectionspublished in magazines and newspapers. World mapsin the popular media often lack latitude-longitude grids.

44° 777


This masks the extreme spatial distortion inherent insuch maps and gives the impression that the continentsare truly represented. Thus, the map user is well advisedto reconstruct the latitude-longitude grid mentally as afirst step in map reading.

Understanding the concepts above will go a long waytoward helping you make the right decisions, but thereare also books and Web sites dedicated to map projec-tions. National mapping agencies such as the USGS andswisstopo, the Swiss national mapping agency, also offeruseful map projection information on their Web sitesand in their publications.

NOTES

1. The equation for the sphere isy = in ('tan (x/4 +where p is the latitude in radians (there are 2 ,7 radians ina 360 0 circle, so 1 radian is approximately 57.295 degrees).The value of this equation at 90° (7r/2 radians) is infinity.

map projections - welcome to cuny - the city university of new york

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