Symbolization of Map Projection Distortion:A Review

Karen A. Mulcahy and Keith C. ClarkeABSTRACT: One of the most fundamental steps in map creation is the transformation of information from thesurface of a globe onto a flat map. Mapmakers have developed and used hundreds of different map projections overthe past 2,000 years, yet there is no perfect choice because every map projection uniquely alters some aspect of spaceduring the transformation process. Detailed information about the type, amount, and distribution of distortion isessential for choosing the best projection for a particular map or data set. The distortion inherent in projections canbe measured and symbolized much like any other map variable. Methods for symbolizing map projection distortionare reviewed, with each method described and illustrated in graphical form. The symbolization methods are collectedunder ten separate headings organized from simple to more complex in terms of interpretation. Most of thesemethods are highly effective at communicating distortion, yet they are rarely used beyond textbooks and technicaldocumentation. Map projections and the distortions they carry need to be better understood by spatial data developers,distributors, and users. Map distortion should be carried along with map data as confidence layers, and the easilyaccessible distortion displays should be available to help in the selection of map projections. There is a suitably widearray of symbolization methods to match any need from basic education to research.

KEYWORDS: Map projection distortion, cartographic symbolization, visualization

Introduction

Cartographers have been portraying ourglobe on flat surfaces for approximately2,000 years (Snyder 1993).The results have

been far from perfect. It is impossible to depictrounded objects, such as the globe, on flat surfaceswith complete fidelity. Thus, all maps are flawedin some way.The somewhat flawed and flattenedgeo-spatial products that result take forms such asmaps printed on paper, virtual maps on computerscreens, or virtual maps in computer databases.These flat map products cannot be true represen-tations of our globe. Alternatives to flattening theglobe such as Dutton's global hierarchical coor-dinate system (Dutton 2000), or methods to per-form spatial analysis on the surface of the globe(NCGrA 1997), are being developed but, in themeantime, map projections must be used.

As the globe is flattened, areas may expand orcontract, distances may change, and angles may bebent out of shape. The amount and types of dis-tortion introduced may be controlled through theselection of appropriate map projections. Choosingthe best projection for a map or map product is atask that has haunted cartographers for ages. The

Karen Mulcahy is Assistant Professor in the Departmentof Geography at East Carolina University, Greenville, NC27858. Email:<mulcahyk@mail.ecu.edu>. Keith Clarke isProfessor and Chair of the Geography Department, Universityof California, Santa Barbara, Ellison Hall 361, Santa Barbara,CA 93106-4060. Email: <kclarke@geog.ucsb.edu>.

secret lies in choosing an appropriate projectionthat will allow the final product, in whatever form,to retain the most important properties for a par-ticular use. Cartographers have measured, catego-rized, and organized various distortion characteris-tics resulting from the transformation of the globeto the flattened map. The most common types ofdistortion that are measured include changes inscale, distance, area, and maximum angular defor-mation.

In addition to the inevitable distortions listedabove, the data structure used to store map coor-dinate and attribute information adds anotherdimension to the problem of map projection distor-tion. Steinwand et al. (1995) present an argumentfor specifically addressing the projection of rasterdata sets as opposed to vector data sets. When pro-jecting raster data sets, pixel attribute values maychange or pixels may multiply to fill space or maybe squeezed out entirely. Two visualization meth-ods were devised by Steinwand et al. (1995) toconvey the special problems of transforming rasterdata.

Maps that showprojection distortion have beenincluded as backdrops to a base map, as supple-mental insets, or as parallel maps. For example,Snyder and Voxland (1989) created parallel dis-tortion maps for most maps in their Album ofMap Projections. As another example, Canters andDecleir (1989) provided a display of isolines ofangular and area distortion on their maps inThe World in Perspective: A Directory of World MapProjections. Cartographers have shown remarkable

CartograPhy and GeograPhic Information Science, 101.28, No.3, 2001, pp.167-181

versatility in the display of distortion using famil-iar shapes, direct comparisons of isolines of dis-tortion or graticules, and interactive visualizationtools.

Map users and researchers have a wide rangeof possibilities for evaluating and symbolizing mapprojection distortion. In this paper, these sym-bolization methods are presented under 10 sep-arate headings. From familiar shapes to checker-boards, these cartographic symbolization methodsare arranged in approximate order of interpretivecomplexity. We have chosen to provide a varietyof methods because this review is intended asa resource for a broad community of users witha wide range of needs and cartographic back-ground.

Ten Distortion Methods

Familiar ShapesThe use of a familiar shape to visualize distortionis the simplest and often the most engaging ofmethods for symbolizing distortion. Aface or regu-lar shape is created on a base map and then trans-formed (Figure 1). The human head or face is apopular theme. A number of examples were listedby Tobler (1964), who included work by Reeves(1910), Gedymin (1946), and others. Reeves(1910)employed a single profile of a face that stretchednearly from pole to pole in drawings for an educa-tional lecture. An illustration of part of his workis included in Figure 1. His objective was to "givean idea at a glance of the exaggerations and distor-tions of a few of the best-known projections." Hementioned having created these displays specifi-cally "for a lecture to boys" but noted that thereare "those of us who can more readily appreciatewhat the shape of the average man's head is thanthey can the general form of the land masses ofthe earth." Robinson et al. (1995) used the profilein the same fashion as Reeves, with a single headcovering both the northern and southern hemi-spheres.

Gedymin's (1946) use of the human face tosymbolize map projection distortion differed fromthe approaches of Reeves (1910) and Robinson etal. (1995). Gedymin inscribed the face twice, oncein the southern hemisphere and once in the north-ern. The effect on a Mercator projection is a clas-sic alien head with small facial features and anenormous cranium in the Northern Hemisphereand a normal cranium and ballooning neck in theSouthern Hemisphere. This two-headed version

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Figure 1. Profile of the human head is an example of theuse of familiar shapes for depicting distortion. The profile isshown over a graticule in the globular projection at top andis shown in the orthographic projection in the bottom view(after Reeves 1910).

effectively conveys the concept of increased dis-tortion with increased distance from the equatornorth and south on the Mercator.

A rather different example is found in theHammond International Atlas of the World (Hammond,Inc. 1994)that uses a cartoon facecreated from cir-cles, a triangle, and a line. The reader is instructedto think of the facial features as continents. Theface-a simple plan of circle eyes, triangle nose,and a straight-line mouth-is transformed intofour projections. The cartoon face method seemsmore friendly, when compared to the sometimesgrotesquely distorted human heads, yet it is hardto say which method is most effective. The morecheerful cartoon face may be more inviting toexamination while the grotesquely distorted headmay provide a greater initial visual impact.

The cartoon face used in the Hammond Atlascombines simple geometric figures into a face, but

Cartography and GeograPhic Information Science

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Figure 2. Spherical "squares" is another example of the useof familiar shapes for depicting distortion (after Chamberlin1947), The projections illustrated are: A. Orthographic; B.Stereographic; and C. Gnomonic.

simple shapes are also used individually or in sets.For example, Figure 2 shows a set of three projec-tions, each with five "spherical" squares createdafter Chamberlin's (1947) model. A more continu-ous view of distortion is seen in Figure 3, an illus-tration created after Fisher and Miller (1944) whoemployed an unbroken net of equilateral spherical

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triangles and compared these to a set of equilat-eral planar triangles.

The use of familiar shapes is the most accessibleof all of the visualization methods. Familiar shapesare limited, however, in that the amount andcharacteristics of distortion are not communicatedexplicitly when using this method. Furthermore,the use of a familiar shape is implemented as alearning device rather than as an analytical tool.On the positive side, while a student mayor maynot grasp the notion of transforming between thesurface of the globe and a flat map, the essentialidea is that something has changed. Features nowappear in different shapes or sizes in an immedi-ately recognizable way. Familiar shapes are highlyrecommended as an introduction to the concept ofmap projection distortion.

Interactive Display of DistortionThe use of a simple geometric shape is the basisfor displaying map projection distortion in aninteractive software package created by Brainerdand Pang (l998). It is intended as a tool for mapusers in a wide range of applications and disci-plines. In the software, a ring floats freely uponthe surface of the globe like a contact lens andis referred to as the "Floating Ring" by the devel-opers. The ring can be scaled as well as movedover the surface of the globe. It can be shown asa simple outline or a continuous raster fill thatdepicts either area or angular distortion. The useris able to interactively manipulate the ring on aglobe-like view on one screen and see the resul-tant distortions on a second screen. The strengthof this approach is the interactivity of the softwarecombined with multiple visual cues.

Figure 4 shows the contents of the two mapviewing sections of the software screens for theFloating Ring. On the left is the globe-like viewwith a cone centered over the North Pole. Thecone explicitly refers to the classification systemfor map projections that relates each projectiontype to a simple planar geometric shape, cone,or cylinder. A ring is shown floating on the sur-face of the globe in Southern Africa. On the rightside of the graphic is the map that results fromthe application of a conic projection. The floatingring over Southern Africa now appears distortedin both shape and area.

The software requires further development,and there are theoretical issues that influence itsdesign and reduce its possible effectiveness. Theseissues are understandable given that this contri-bution to cartographic visualization comes fromcomputer science. During future development

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Figure 3. Spherical and plane triangles on the Winkle Tripel projection is a visual-ization similar to Fisher and Miller (1944). Familiar shapes such as the trianglescan also provide distortion measurements over large areas.

A.

and Greenland, have a highly altered appearance.Another example is found in the online materialsof Peter Dana (1995)whicheffectivelycompare theLambert conformal conic to the Albers equal-areaconic for North and Central America. The com-parison approach is excellent as an educationaldevice because it goes a step beyond the use offamiliar shapes by overlaying one map directlyupon the other. The closejuxtaposition of the mapfeatures encourages detailed examination of thedifferences due to the projections.

Although comparisons of graticules and shore-lines are often used for educational purposes, theyhave analytical applications as well. Maling (1992)provided an example of comparing isolines of dis-

ComparisonsThe mental act of comparisonis inherent in nearly all of thesymbolization methods describedin this paper. For example, thefamiliar shapes method previouslydescribed functions by allowingthe viewer to compare the originalshape to the transformed shapes.Of particular interest, however,is the comparison of sets of iso-lines, shorelines, and graticules.Comparisons, like familiar shapes,are frequently used in educationalcontexts, although they also havebeen employed in analytical stud-ies.

In Figure 5, the Lambertazimuthal equal-area projectionis superimposed on the stereo-graphic projection at the samescale and with the same origin at 0degrees latitude and 72.5 degreeswest longitude. Features overlapperfectly near the origin of themap. This is the correspondingarea oflow distortion on both pro-jections. A comparison of the grat-icules and the shorelines in anydirection from the origin indicatesan increasing offset with increaseddistance from the origin.

Although the scale, origin, andextent of these two maps are thesame, the two projections differ greatly from oneanother. Due to an increasing scale distortion thatincreases radially from the projection center, onlythe centers of the projections remain true to scale.The stereographic projection has the property ofconformality where local angles are retained inthe output map. To maintain conformality, scale,and hence, area enlargement increases dramati-cally with distance from the origin of the projec-tion. The Lambert azimuthal equal-area projec-tion is less dramatic in exposing the angular distor-tion that results from maintaining the equal-areaproperty. Still, land masses and countries towardsthe outer edge of the maps, such as Antarctica

of the floating ring software, theinclusion of a cartographer in thedevelopment team might assist inmaking this an even more effec-tive exploratory learning tool formap projections.

170 CartograPhy and GeograPhic InfOrmation Science

Figure 4. Contents from map views in the Floating Ring interactive software. The globe-like view on the left indicates theorientation and type of projection that is shown on the right: namely, a conic projection centered over the North Pole. Theleft map view also shows the size and orientation of the floating ring on the globe. The right map view shows the resultantconic projection and floating ring.

tortion for two projections over an approximateoutline of Europe. His goal was to select the projec-tion that would have minimal distortion character-istics for a GIS termed: Coordinated Informationon the European Environment (CaRINE). Maling(1992) compared the relative merit of the azi-muthal equal-area projection and Bonne's pro-jection by overlaying isolines of maximum angu-lar distortion on an approximate coastline of theNorth American continent, and he compared iso-lines of particular scales on the normal aspect ofthe Mercator, transverse Mercator, and the stereo-graphic projection for a conformal map of LatinAmerica.

Overall, comparisons are one of the most flex-ible and accessible methods for serving both edu-cational and analytical purposes. Having the grat-icules, shorelines or isolines overlapping on thesame map makes feature comparison very easy formap users. Comparisons are beneficial from a map-maker's perspective because they are very simple tocreate. The mapmaker can employ existing softwarepackages to layer vector maps for an effectivecarto-graphic visualization. For example, the ArcView3.2softwarepackage wasused to create Figure 5.

Isolines of Projection DistortionAlthough isolines are mentioned in the previoussection on comparisons, they are an importantsymbolization method in their own right. The useof isolines for the display of map projection dis-tortion is effective for depicting the magnitudeand the distribution of any distortion measure-ment. Isolines are lines that represent equalitywith respect to a given variable such as contourlines of equal elevation and isobars of atmospheric

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pressure. Isolines can be used to map any vari-able that is assumed continuous. When applied todistortion, isolines are used (often with shadingbetween the lines) to identify the distribution andmagnitude of area exaggeration or angular defor-mation. They can be used to depict any distortionmeasure, however. Isolines were used extensivelyby Maling (1992), by Bugayevskiy and Snyder(1995), and almost exclusively by Canters andDecleir (1989) in their directory of world mapprojections. The isoline approach also is a verycommon distortion method in reference books onprojections.

The two world maps in Figure 6 depict mapprojection distortion on the Winkle Tripel pro-jection. This projection was recently adopted bythe National Geographic Society after using thevery similar Robinson projection from 1988 to1998 because the '''WinkelTripel improves on theRobinson by distorting the polar areas less" (Miller1999). The upper map depicts degree measure-ments of angular distortion and the lower indi-cates percentage of scaledistortion. This projectionis neither equivalent nor conformal, but insteadseeks to portray the world in a balanced fashionsuitable for world reference maps.

!soline mapping is a quantitative symboliza-tion method and provides absolute values of dis-tortion. In addition to its effectiveness at portray-ing the amount and distribution of distortion, theability to determine absolute values is its greateststrength as a symbolization method.

Perspective SurfaceAlthough not frequently used for symbolization ofprojection distortion, a perspective surface may

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Figure 5. Comparison of the lambert azimuthal equal-area and stereographic pro-jections using graticules, shorelines, and country boundaries. Point of origin is QON,72.5°W.

be employed because it pro-vides a very intuitive carto-graphic display. The surfacecan represent a wide varietyof different distortion mea-surements, such as the mag-nitude of angular or areadistortion. Clarke and Teague(1998) used this method tocreate a series of small mul-tiple displays with distortionmapped separately for eastwest, north south, and theoverall magnitude of posi-tional error.

Figure 7 portrays resultsfrom evaluating the pixelchanges that occur during theprojection of a raster data setto a Mollweide projection. Araster data structure is essen-tially a matrix, with eachelement being described asa pixel. Each time a rasterdata set is transformed, scalechange may cause change inpixel attribute values or the number of pixels.Pixels might be deleted in areas of scale reduction,while pixels may be duplicated in areas of scaleexpansion. Pixel duplication is evaluated at 10°increments over the map. The map is presentedhere as a perspective surface with dots indicatingthe location of data points. The Mollweide projec-tion shows minimum distortion at approximately40 degrees north and south of the equator, wherethe projection is true to scale. In Figure 7, theselocations appear as "valleys" along the parallelsat approximately 40 degrees north and south.Beyond 40 degrees north and south, distortionincreases rapidly towards the poles.

There are minor difficulties associated with cre-ating and viewing world map projection distortionas a continuous surface. A general principle is thatmap projection distortion isat a minimum at partic-ular parallels or central points and increasestowardsa maximum at the edges of the map. A world mapis generally located such that minimal distortion isat the center of the map. In Figure 7, distortion isviewed as a saddle-like object where the extremes ofdistortion occur at the edge. Finding a suitable view-point for the surface so that the distortion surface iseasily seen can be challenging.

The perspective surface is highly effective atpresenting a continuous depiction of the distribu-tion and relative magnitude of a variety of distor-tion measurements. A surface can be used alone

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to depict the magnitude and spatial distributionof distortion, but this symbolization method mayhave the greatest impact when used along witha primary map or dataset to illustrate the conse-quences of projection selection. For example, smallinset maps of a perspective surface might be pro-vided to data users as an integral part of the datadocumentation.

Color MethodsSimilar to the perspective surface, color methodscan provide a continuous display of distortionacross a map. Distortion is symbolized by assign-ing a numerical measure to each pixel of a rasterdataset. To illustrate, Figure 8 showsthe same datashown as a perspective surface. Compare the dis-tribution and relative magnitude of distortion inFigures 7 and 8. A continuous color distortion dis-play may consist of a single variable or multiplevariables. The color method can be nearly as intui-tive as the perspective surface when a single vari-able is being mapped, and it has the advantageof allowing the entire distortion distribution to beviewedat one time.

Color methods by Clarke and Mulcahy (1995)present a multivariate display using hues. Theeast-west scale distortion along the parallels wasmapped to red, the north-south scale distortionalong the meridians to green, and the conver-gence angle to blue. This same approach wasused

CartograPhy and GeograPhic Information Science

Tissot IndicatrixTissot's century-old method is an ele-gant and effective means for depict-ing map projection distortion. Tissotused a display method called the indi-catrix, which is visually based on thetransformation of small circles. Unlikethe circle used as a familiar shape,the indicatrix is based upon Tissot's(1881) theorem of map projection dis-tortion. Imagine an infinitely smallcircle on the globe. This circle isthen transformed by a map projectionand, in the process, it changes in sizeand/or shape due to scale change. Thecircles may remain circular but moreoften become ellipses with the semi-major and semi-minor axis reflectingthe amount, type, and direction ofthe projection distortion. Tissot's indi-catrix has been used extensively toenhance understanding of projectionsin specialized cartographic literature(Snyder and Voxland 1989). The indi-catrix is also used as an educationaltool in eXplaining map projection dis-tortion in many contemporary text-books in cartography (e.g., Dent 1999;Jones 1997; Kraak and Ormeling1996; and Robinson et. al. 1995).

To illustrate Tissor's indicatrix,ellipses are placed at the intersectionsof 30° graticules in Figure 9. The grati-

cule in Figure 9A shows the changes in shape thatrelate to angular and area distortion. For example,the sinusoidal projection has the geometric prop-erty of equal area, and so any area on the map accu-rately reflects the same relative area on the globe.Using Tissot's indicatrix, the ellipses indicate theequal area property by retaining the same areathroughout the map. The amount and directionof maximum angular change are also indicated, asellipses are increasingly distorted with increaseddistance from the center of the projection.

In the second example in Figure 9B, theellipses have retained a circular shape. This timethe projection is the Mercator that has the propertyof conformality, meaning all local angles remaintrue. In this case, ellipses near the equator arethe only ones exhibiting near-true equal area prop-erties. With increased distance from the equator,north or south, the amount of area distortionincreases dramatically.

Figure 6. Distortion expressed as isolines on the Winkle Tripel projection. A.Degree of angular deformation; B. Scale change.

by Clarke and league (1998) to depict locationaldistortion for a comparison of two vector maps.They presented several versions employing the red-green-blue (RGB)and the hue-saturation-intensity(HSI) color models. Other examples of color meth-ods by Shuey and TessaI' (1999) and Shuey andMurphy (2000) depict distortion in area, angle andshape. Each distortion variable is assigned to araster data layer (or a band) and referred to as "dis-tortion channel." Each distortion channel is thensymbolized using an additive RGB color model.

Color methods provide a visually continuousrepresentation of the magnitude of distortion.Both the spatial distribution and degree of distor-tion are effectivelyportrayed. Color methods alsohave the advantage of being able to display morethan one variable at the same time. These meth-ods may be particularly useful to researchers accus-tomed to viewing global data sets in continuouscolor displays. As with perspective surfaces, colormethods can be effectivelyused as supplements toillustrate the consequences of projection selection.

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Figure 7. Surface representation of distortion where the height of the surface indicates theamount of pixel loss for a raster data set that has been transformed to the Mollweide pro-jection. Bands of increasing color intensity reinforce the use of height to illustrate increas-ing distortion.

Although Tissot'smethod is effective andwidely accepted there havebeen criticisms of it. Oneproblem is that the indi-catrix is frequently inter-preted to be limited to theforward transformationfrom the globe to a mapprojection only. In com-puter applications there ismore frequently a need tomove data between vari-ous projections. A secondproblem is that the dis-tortion distribution shownby Tissot's indicatrix onlydescribes the infinitesimalareas near the center ofthe ellipses, which is notthe same as the area of theellipses indicated visually.This may lead to misinter-pretations of the locationof distortion. A third prob-lem, raised by Brainerdand Pang (1998), is thatthe use of the indicatrix obscures underlying data.We feel that ellipses may be superimposed on athematic map without obscuring the primary mapdata by symbolizing the ellipses by an outline onlyand not filling the symbols.

The Map GraticuleMany projections are evaluated or classified basedupon the appearance of the latitude and longi-tude grid-known as the graticule-on a map.Robinson et al. (1995) argued that the evaluationof many projections could be accomplished effec-tively by a visual analysis whereby the characteris-tics of the graticule on a map are compared to thegraticule on a reference globe (Figure 10). Theylisted nine comparative visual characteristics ofthe graticule on a globe.• Parallels are parallel.• Parallels, when shown at a constant interval,

are spaced equally on meridians.• Meridians and great circles on a globe appear

as straight lines when viewed orthogonally(looking straight down), which is the way welook at a flat map.

• Meridians converge toward the poles anddiverge toward the equator.

• Meridians, when shown at a constant interval,are equally spaced on the parallels, but their

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spacmg decreases from the equator to thepole.

• 'Whenboth are shown with the same intervals,meridians and parallels are equally spaced ator near the equator.

• When both are shown with the same intervals,meridians at 60° latitude are half as far apartas parallels.

• Parallels and meridians always intersect atright angles.

• The surface area bounded by any two parallelsand twomeridians (a givendistance apart) is thesameanywherebetweenthe same parallels.The globe-like map appearing in Figure 10

is an orthographic projection. The orthographicprojection is the preferred choice when a globe-like map view is required. Robinson (1986) makesextensive use of the orthographic as a referenceglobe specifically for assessing world map projec-tions. Each of the previously listed characteristicsof the graticule on a globe can be illustrated usingthis globe-like projection. To evaluate another pro-jection, comparisons can be made between theglobe-like graticule and the cylindrical projectionabove or the pseudocylindrical projection below.

The significance of the appearance of the grati-cule washighlighted by Canters and Decleir (1989).They classified projections into a system basedupon Maling (1973) and Tobler (1962). Canters

CartograPhy and GeograPhic Information Science

Figure 8. Color method showing the same data as the surface in Figure 6, where the continuously increasing color intensityindicates the amount of pixel loss for a raster data set that has been transformed to the Mollweide projection.

and Decleir arranged 68 world projections intoa three-by-four matrix. Along both axes, variouscharacteristics of the graticule were described. Onone axis, the various classes of projections aregrouped as:• Polyconic, where both parallels and meridians

are curved;• Pseudocylindrical, which have "curved merid-

ians and straight parallels;"• Pseudoconic, which "represents meridians as

concurrent curves and parallels as concentriccircular arcs;" and

• Cylindrical, which "comprises projectionswhere both parallels and meridians are repre-sented by parallel straight lines."The other axis consists of the description of the

parallels as being equally spaced, having decreas-ing parallel spacing, or having increasing parallelspacing (Canters and Decleir 1989).

The appearance of the graticule is useful inthe construction and use of these classification sys-tems described above and for the identification ofunknown projections. The appearance of the grati-cule may be used to help during map projectionselection. A study of the characteristics of the grati-cule is also useful in an educational context to illus-trate some of the basic map projection distortionconcepts. The use of the graticule in this instanceis somewhat limited because the graticule does notdirectly symbolize scale, area, or angular change

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as clearly as Tissot's indicatrix, isolines, color meth-ods, or a perspective surface.

Grid SquaresThe last two methods presented here-grid squaresand checkerboard-were developed during asearch for the best projection for a global rasterdatabase to be distributed by the USGS. Steinwandet al. (1995) were the first to address, and publishon, the problem of selecting the "best" projectionfor this type of data. Both grid squares and check-erboard methods were developed primarily toexplore the difference between projecting rasterand vector data, but both methods are useful fordepicting projection distortion in general.

The grid squares method is visually similar toTissot's indicatrix, with squares replacing circles.The grid squares method consists of the creationof sets of three-by-three vector-structured gridslocated at the intersection of selected parallelsand meridians. The original grid squares shownin Figure llA were formed on a Plate Carn~e pro-jection and then transformed to the Lambert azi-muthal equal area projection as seen in FigurelIB.

Although this method can be thought of as araster-styled Tissot's indicatrix, there is no explicitformula for determining angular distortion. Areais measured over a discrete region rather than

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within an infinitesimally small area. Nevertheless,the type, amount, direction, and overall spatialdistribution of map projection distortion through-out the map are communicated effectively by thismethod. Grid squares also have the advantage ofidentification, with the raster data structure high-lighting the data changes that occur when select-ing different projections. The primary advantageof using the grid squares over Tissot's indicatrix isthat this method is not limited to the globe-to-mapcase and can be used for examining changes in dis-tortion between different projections.

Checkerboard MethodThe checkerboard method was the second methoddeveloped by Steinwand et al. (1995) as a ras-ter-based analytical tool. A regular checkerboardpattern mimics the state of the discrete raster struc-ture, which does not respond to projection trans-formations like vector data. Vector data consist oflocations or points that may stand alone to rep-resent a feature or may be combined to formlines and polygons. These vector data are trans-formed point-by-point in a manner very similar tothe way cartographers have manually transformedmap features for centuries. The transformationof a raster data structure is a different matter.Raster data consist of a matrix of data values usu-ally referred to as pixels. The transformation of araster data set occurs in two stages. First, a newgeo-metric space for the projected matrix is created.Then, for each pixel in the new geometric space,the original matrix is searched for the appropri-ate attribute value, and this attribute value is thentransferred to the new projected matrix. The resultis often a dramatic change in map features. Linesmay become stair-stepped and pixels duplicatedor dropped. Since raster data are not as flexibleconcerning geometric transformations as vectordata, the checkerboard assists in communicatingthe differences between the two structures.

In the checkerboard method, a data set isdeveloped with alternating colored squares and isthen transformed by map projection. The patternformed by the alternating hues or values exposesthe spatial pattern and distribution of distortionand shape alterations, which provides insight intothe limitations of the raster structure in termsof geometrical alterations. The checkerboard pro-jected in Figure 12 is a fragment from a globalvector-based grid data set by Shuey and Murphy(2000). It was originally created on a curved sphereand transformed to a Plate Carn~e projection. Itis used here to show the extent of distortion thatoccurs to the individual pixels.

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Figure 9. Tissot's indicatrix shown at 30° increments.Upper map shows the equal-area sinusoidal projection andthe lower map, the conformal Mercator projection.

The checkerboard method is useful for edu-cational purposes, resembling the use of familiarshapes, but it has an important advantage. In addi-tion to communicating the basicconcept of geomet-ric changes occurring during projection transfor-mation, the checkerboard illustrates the changesthat occur within a raster data structure. Usersof spatial information systems such as image pro-cessing or geographic information systems areunaware that there is any difference between thetransformations of vector versus raster data. Thecheckerboard effectivelyaddresses this issue.

DiscussionThis reviewhas outlined the principal methods ofcartographic display of map projection distortion.Clearly, many methods have been designed andimplemented, and most of them are highly effec-tive at communicating the spatial pattern of mapprojection distortion. Nevertheless, the use of dis-tortion display is negligible beyond cartographicand map projection textbooks, manuals, and tech-nical documentation. This is unfortunate, giventhat map and GIS users often have an unrealisticlevel of faith in their base map and little knowl-edge ofprojections. Nyerges and Jankowski (1989)went so far as to say,"fewpeople, even fewcartog-raphers, commonly knowwhich projection is good

CaTtogmphy and GeogmPhic Infonnation Science

Figure 10. Graticule patterns for several projections includingthe orthographic projection as reference globe at the center,the Mercator above, and the Hammer below.

for what purpose and the trade-offs involved." Forthe GIS user working at continental and globalscales, an understanding of projections is criticalduring the data capture, management (registra-tion, merging, overlay), and analysis stages.

Cartographic education has tended to treatmap projections as peripheral, too technical, sim-plistically, or not at all. This trend has continuedthroughout the more general area of GIScience.Democratization of the access and use of spatialdata has brought the projection issue to the fore-front because important decisions are often based

VOL 28, No.3

upon spatial information. Raster GIS and imageprocessing, with its massive amount of hiddenimage resampling, can lead to greater errors thanprovided by the vector systems,yet neither is prob-lem free from the projection standpoint.

Compounding the problem of deficiencies inthe educational process, the support for mapprojections in many software packages is limited.Often only a few projections are provided to theuser. In comparing the implementation of equal-area world map projections in commercial software,Mulcahy (1999) found that ESRI's ArcInfo 7.04software had the greatest number at seven, whilethe ERDASImagine 8.1 and Clark Lab's Idrisi soft-ware packages had only one each. Furthermore,there is no access to distortion measurement or thedisplay of projection distortion in popular GIS andremote sensing software.

Distortion displays should be closely linkedwith their respective data sets and even with mapproducts. Map projection distortion, and, indeed,any other spatial error, could be considered a con-fidence, or uncertainty overlay, that can be car-ried along with the map data during analysis andapplied as a reality check or accuracy filter whennecessary. This might be called truth in labeling.If distortion displays were provided, the GIS userwould be encouraged to pay considerably moreattention to projection issuesthroughout the entiremapping process. The inherent distortion couldbe included as a paralIel or inset map or as an over-lay to the primary map.

Shuey and Tessar (1999) hope to implementa Web based on-demand service that would allowa user to submit a data set to a geographic server.The server would then calculate distortion levelsand return an image depicting the combineddistortion in area, angles and shapes. Providinginformation in the form of distortion displays ondemand is an important step leading towards theoptimal use of map projections.

ConclusionMuch creative energy has been expended at symbol-izing map projection distortion. Some distortionsymbolization methods seem logical expressionsof the analytical treatment of distortion such asusing the graticule, the Tissot indicatrix, the colormethods, the grid squares and the checkerboard.Other symbolization methods are borrowed fromthe standard cartographic symbolization toolboxsuch as the isoline method or perspective surface.FinalIy,a number of methods seem inspired simplyby the need to communicate map projection distor-

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/www.colorado.edu/geography/gcraft/notes/mapproj!mapproj.html].

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Dutton, G. 2000. Universal geospatial data exchangevia global hierarchical coordinates. InternationalConference on Discrete Global Grids, Santa Barbara,California,26-28March. [http://www.ncgia.ucsb.edu/glo-balgrids/papers/du tton. pdf].

Fisher, I., and O. M. Miller. 1944. World maps and globes.NewYork, NY:J. J. Little & Ives.

....o

Figure 11. Grid squares portrayed before and after transformation bya projection. A. Set of grid squares located at graticule intersectionson the Plate Carree projection. B. The same set of grid squares aftera transformation to the Lambert azimuthal equal-area projection (afterSteinwand et al. 1995).

Brainerd, J., and A. Pang. 1998. Floating ring: Anew tool for visualizing distortion in map projec-tions. Proceedings, Computer Graphics International98. June 22-26.

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REFERENCES

NOTENumerous terms specific to the map projection litera-ture are described in a glossary prepared for this work.It is located at <http://www.geog,.ucsb.edu/-kclarke/Papers/Distortion.html> .

ACKNOWLEDGMENTSPartial support for this project is acknowl-edged from National Science FoundationProject Varenius (SBR-9600465).

tion to inexperienced users, such as famil-iar shapes, interactive display of distortion,or comparisons. There is a wide of array ofmethods to choose from to suit any need fordepicting map projection distortion.

The arrangement of symbolizationmethods and many of the comments by theauthors on the effectiveness of the methodsare subjective. Future studies may includethe testing of the authors' opinions withvari-ous user groups, from children being intro-duced to map projections, to researchersstudying various aspects of global systems.These methods can be presented in manysettings, from textbooks and popular GISand remote sensing software to on-demandWeb-based servers. The authors hope thatincreased awareness of these methods of pro-jection distortion symbolization willenhanceunderstanding of the principles and conse-quences of projection distortion.

180 CartograPhy and GeograPhic InfoT1nation Science

•

1 F 5'" I

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"." >-p'-.II f·-..

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~:;~'h1", J'l'c\, " ~Il O)l

n.,~ ,'-Jj( 'Y'" ;::0:

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, ( :(,_Ji'

- - - .-..r:f ....:::Figure 12. The checkboard image was created from a dataset developed by Shuey and Murphy (2000), which wasoriginally created on a curved sphere and then transformedto a Plate Carree projection. The variations in the size andshape of the individual squares indicate the degree of datachange.

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aps comple ed during 2001 are eligible.Awards will be given for:

Professiona CategoryBest of ShowBest of Category: Thematic· Book/Atlas; Reference; Series

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Deadline for submission is December 1, 2001.Submit three copies of entries in the Professional category and two

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182 CartograPhy and Geo~aphic Information Science