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 distortion arereviewed, with each method described and illustrated in graphical form. The symbolization methods are collected underten 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 approximately 2,000 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 flawed insome way. The somewhat flawed and flattened geo-spatial products that result take forms such as mapsprinted 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 (NCGIA 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 Department ofGeography at East Carolina University, Greenville, NC 27858.Email:<mulcahyk@mail.ecu.edu>. Keith Clarke is Professor andChair of the Geography Department, University of California, SantaBarbara, 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, toretain 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 globe tothe 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 methodswere devised by Steinwand et al. (1995) toconvey the special problems of transforming rasterdata.

Maps that show projection 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 of Map

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

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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 variety ofmethods because this review is intended as aresource for a broad community of users with awide range of needs and cartographic back-ground.

Ten Distortion MethodsFamiliar ShapesThe use of a familiar shape to visualize distortion isthe simplest and often the most engaging ofmethods for symbolizing distortion. A face or regularshape 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 listed byTobler (1964), who included work by Reeves (1910), Gedymin (1946), and others. Reeves (1910) employeda single profile of a face that stretched nearly frompole to pole in drawings for an educational lecture.An illustration of part of his work is included inFigure 1. His objective was to "give an idea at aglance of the exaggerations and distortions of a fewof the best-known projections." He mentionedhaving created these displays specifically "for alecture to boys" but noted that there are "those ofus who can more readily appreciate what the shapeof the average man's head is than they can thegeneral form of the land masses of the earth."Robinson et al. (1995) used the profile in the samefashion as Reeves, with a single head covering boththe northern and southern hemispheres.

Gedymin's (1946) use of the human face tosymbolize map projection distortion differed from theapproaches of Reeves (1910) and Robinson et al. (1995). Gedymin inscribed the face twice, once in thesouthern hemisphere and once in the northern. Theeffect on a Mercator projection is a classic alienhead with small facial features and an enormouscranium in the Northern Hemisphere and anormal cranium and ballooning neck in theSouthern Hemisphere. This two-headed version

Figure 1. Profile of the human head is an example of the useof familiar shapes for depicting distortion. The profile is shownover a graticule in the globular projection at top and is shown inthe orthographic projection in the bottom view (after Reeves1910).

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 face created from cir-cles, a triangle, and a line. The reader is instructed tothink of the facial features as continents. The face-asimple plan of circle eyes, triangle nose, and astraight-line mouth-is transformed into fourprojections. The cartoon face method seems morefriendly, when compared to the sometimesgrotesquely distorted human heads, yet it is hard tosay 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

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Figure 2. Spherical "squares" is another example of the use offamiliar shapes for depicting distortion (after Chamberlin 1947).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

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

The use of familiar shapes is the most accessible ofall of the visualization methods. Familiar shapes arelimited, however, in that the amount andcharacteristics of distortion are not communicated

explicitly 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 may or 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 Distortion

The use of a simple geometric shape is the basisfor displaying map projection distortion in aninteractive software package created by Brainerdand Pang (1998). 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 and isreferred 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 as asimple outline or a continuous raster fill thatdepicts either area or angular distortion. The user isable to interactively manipulate the ring on aglobe-like view on one screen and see the resultantdistortions on a second screen. The strength of thisapproach is the interactivity of the software combinedwith 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 view with acone centered over the North Pole. The coneexplicitly refers to the classification system for mapprojections that relates each projection type to asimple planar geometric shape, cone, or cylinder.A ring is shown floating on the surface of the globein Southern Africa. On the right side of the graphicis the map that results from the application of aconic projection. The floating ring over SouthernAfrica now appears distorted in both shape andarea.

The software requires further development, andthere 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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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.

ComparisonsThe mental act of comparison isinherent in nearly all of thesymbolization methods describedin this paper. For example, thefamiliar shapes method previouslydescribed functions by allowing theviewer to compare the originalshape to the transformed shapes.Of particular interest, however, isthe comparison of sets of isolines,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 projection issuperimposed 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 of low distortion on both pro-jections. A comparison of the grat-icules and the shorelines in anydirection from the origin indicates anincreasing 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 projection.The Lambert azimuthal equal-area projection isless dramatic in exposing the angular distortion thatresults from maintaining the equal-areaproperty. Still, land masses and countriestowards the outer edge of the maps, such asAntarctica

and Greenland, have a highly altered appearance.Another example is found in the online materialsof Peter Dana (1995) which effectively compare 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 close juxtaposition 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

Figure 3. Spherical and plane triangles on the Winkle Tripel projection is a visualizationsimilar to Fisher and Miller (1944). Familiar shapes such as the triangles can alsoprovide distortion measurements over large areas.

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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. The leftmap view also shows the size and orientation of the floating ring on the globe. The right map view shows the resultant conicprojection 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 characteristics for a GIS termed: Coordinated Information onthe European Environment (CORINE). 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 of theMercator, transverse Mercator, and the stereographicprojection for a conformal map of Latin America.

Overall, comparisons are one of the most flexibleand accessible methods for serving both educationaland analytical purposes. Having the graticules,shorelines or isolines overlapping on the samemap makes feature comparison very easy for mapusers. Comparisons are beneficial from a mapmaker'sperspective because they are very simple to create. Themapmaker can employ existing software packages tolayer vector maps for an effective cartographicvisualization. For example, the ArcView 3.2 softwarepackage was used to create Figure 5.

Isolines of Projection Distortion

Although isolines are mentioned in the previoussection on comparisons, they are an importantsymbolization method in their own right. The use ofisolines for the display of map projection distortionis effective for depicting the magnitude and thedistribution of any distortion measurement.Isolines are lines that represent equality withrespect to a given variable such as contourlines of equal elevation and isobars of atmospheric

pressure. Isolines can be used to map any variablethat 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 extensively byMaling (1992), by Bugayevskiy and Snyder (1995),and almost exclusively by Canters and Decleir (1989) in their directory of world map projections.The isoline approach also is a very commondistortion method in reference books on projections.

The two world maps in Figure 6 depict mapprojection distortion on the Winkle Tripel pro-jection. This projection was recently adopted by theNational Geographic Society after using the verysimilar Robinson projection from 1988 to 1998because the "Winkel Tripel 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 scale distortion. This projection isneither equivalent nor conformal, but instead seeksto portray the world in a balanced fashion suitable forworld reference maps.

Isoline mapping is a quantitative symbolizationmethod and provides absolute values of distortion.In addition to its effectiveness at portraying theamount and distribution of distortion, the ability todetermine absolute values is its greatest strength asa symbolization method.

Perspective Surface

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

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be employed because it provides a very intuitive cartographic display. The surfacecan represent a wide variety ofdifferent distortion measurements, such as the magnitude of angular or areadistortion. Clarke and Teague(1998) used this method tocreate a series of small multiple 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 as apixel. Each time a raster dataset 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 100

increments over the map. The map is presentedhere as a perspective surface with dots indicatingthe location of data points. The Mollweideprojection shows minimum distortion atapproximately 40 degrees north and south of theequator, where the projection is true to scale. InFigure 7, these locations appear as "valleys" along theparallels at approximately 40 degrees north andsouth. Beyond 40 degrees north and south,distortion increases rapidly towards the poles.

There are minor difficulties associated with cre-ating and viewing world map projection distortion asa continuous surface. A general principle is that mapprojection distortion is at a minimum at particularparallels or central points and increases towards amaximum at the edges of the map. A world map isgenerally located such that minimal distortion is atthe 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

to depict the magnitude and spatial distribution ofdistortion, but this symbolization method may havethe greatest impact when used along with aprimary 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 assigninga numerical measure to each pixel of a rasterdataset. To illustrate, Figure 8 shows the 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 beviewed at 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 was used

Figure 5. Comparison of the Lambert azimuthal equal-area and stereographic projectionsusing graticules, shorelines, and country boundaries. Point of origin is 0°N, 72.5°W.

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

by Clarke and Teague (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 methodsby Shuey and Tessar (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 effectively portrayed. Color methods alsohave the advantage of being able to display morethan one variable at the same time. These methodsmay be particularly useful to researchers accustomedto viewing global data sets in continuous colordisplays. As with perspective surfaces, colormethods can be effectively used as supplements toillustrate the consequences of projection selection.

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, theindicatrix is based upon Tissot's (1881)theorem of map projection distortion.Imagine an infinitely small circle onthe globe. This circle is thentransformed by a map projection and,in the process, it changes in size and/or shape due to scale change. The circlesmay remain circular but more oftenbecome ellipses with the semimajorand semi-minor axis reflecting theamount, type, and direction of theprojection distortion. Tissot's indi-catrix has been used extensively to

enhance understanding of projections inspecialized cartographic literature (Snyder and Voxland 1989). The indi-catrix is also used as an educational toolin 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 Tissot'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, thesinusoidal projection has the geometric property ofequal area, and so any area on the map accuratelyreflects the same relative area on the globe. UsingTissot's indicatrix, the ellipses indicate the equalarea property by retaining the same areathroughout the map. The amount and direction ofmaximum 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 property ofconformality, meaning all local angles remaintrue. In this case, ellipses near the equator are theonly ones exhibiting near-true equal area properties.With increased distance from the equator, north orsouth, the amount of area distortion increasesdramatically.

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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 variousprojections. A secondproblem is that the dis-tortion distribution shownby Tissot's indicatrix onlydescribes the infinitesimalareas near the center of theellipses, which is not thesame as the area of theellipses indicated visually.This may lead to misinter-pretations of the location ofdistortion. A third prob-lem, raised by Brainerd andPang (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 evaluation ofmany 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 of thegraticule 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 viewedorthogonally (looking straight down), which isthe way we look at a flat map.

• Meridians converge toward the poles anddiverge toward the equator.

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

spacing decreases from the equator to thepole.

• When both 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 two meridians (a given distance apart) is thesame anywhere between the same parallels.

The globe-like map appearing in Figure 10 isan 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 characteristics ofthe graticule on a globe can be illustrated using thisglobe-like projection. To evaluate another projection,comparisons can be made between the globe-likegraticule and the cylindrical projection above or thepseudocylindrical projection below.

The significance of the appearance of the graticulewas highlighted by Canters and Decleir (1989). Theyclassified projections into a system' based uponMaling (1973) and Tobler (1962). Canters

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 increasingdistortion.

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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 into athree-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 represented by parallel straight lines."

The other axis consists of the description of theparallels as being equally spaced, having decreasingparallel 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

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. Steinwand etal. (1995) were the first to address, and publish on,the problem of selecting the "best" projection forthis 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 shown inFigure 11A were formed on a Plate Carree pro-jection and then transformed to the Lambert azi-muthal equal area projection as seen in Figure11B.

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, thetype, 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 advantage ofusing the grid squares over Tissot's indicatrix is thatthis method is not limited to the globe-to-map caseand can be used for examining changes in distortionbetween different projections.

Checkerboard MethodThe checkerboard method was the second method

developed by Steinwand et al. (1995) as a raster-based analytical tool. A regular checkerboard patternmimics the state of the discrete raster structure,which does not respond to projection trans-formations like vector data. Vector data consistof locations 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 to theway cartographers have manually transformed mapfeatures for centuries. The transformation of 'araster data structure is a different matter. Raster dataconsist of a matrix of data values usually referred toas pixels. The transformation of a raster data setoccurs in two stages. First, a new geometric space forthe projected matrix is created. Then, for eachpixel in the new geometric space, the originalmatrix is searched for the appropriate attributevalue, and this attribute value is then transferred tothe new projected matrix. The result is often adramatic change in map features. Lines maybecome stair-stepped and pixels duplicated ordropped. 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 terms ofgeometrical alterations. The checkerboard projectedin Figure 12 is a fragment from a global vector-basedgrid data set by Shuey and Murphy (2000). It wasoriginally created on a curved sphere andtransformed to a Plate Carree projection. It is usedhere to show the extent of distortion that occurs tothe individual pixels.

Figure 9. Tissot's indicatrix shown at 300 increments.Upper map shows the equal-area sinusoidal projection and thelower 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 basic concept of geometricchanges occurring during projection transfor-mation, the checkerboard illustrates the changesthat occur within a raster data structure. Users ofspatial 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 effectively addresses this issue.

DiscussionThis review has 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, given thatmap and GIS users often have an unrealistic levelof faith in their base map and little knowledge ofprojections. Nyerges and Jankowski (1989) went so faras to say, "few people, even few cartographers,commonly know which projection is good

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Figure 10. Graticule patterns for several projections including theorthographic projection as reference globe at the center, theMercator 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 (registration,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 spatial datahas brought the projection issue to the forefrontbecause important decisions are often based

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 problem free from the projection standpoint.

Compounding the problem of deficiencies inthe educational process, the support for map

projections in many software packages is limited.Often only a few projections are provided to theuser. In comparing the implementation of equalarea world map projections in commercial software,

Mulcahy (1999) found that ESRI's ArcInfo 7.04software had the greatest number at seven, while the

ERDAS Imagine 8.1 and Clark Lab's Idrisi softwarepackages had only one each. Furthermore, there isno access to distortion measurement or the display ofprojection distortion in popular GIS and remotesensing software.

Distortion displays should be closely linked withtheir 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 carriedalong with the map data during analysis and appliedas a reality check or accuracy filter when necessary.This might be called truth in labeling. If distortiondisplays were provided, the GIS user would beencouraged to pay considerably more attention toprojection issues throughout the entire mappingprocess. The inherent distortion could be includedas a parallel or inset map or as an overlay to theprimary map.

Shuey and Tessar (1999) hope to implement aWeb based on-demand service that would allow auser to submit a data set to a geographic server. Theserver would then calculate distortion levels andreturn 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 expressions ofthe analytical treatment of distortion such as usingthe 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.Finally, a number of methods seem inspired simply bythe need to communicate map projection distor

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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 include thetesting of the authors' opinions with varioususer groups, from children being introducedto map projections, to researchers studyingvarious aspects of global systems. Thesemethods can be presented in many settings,from textbooks and popular GIS and remotesensing software to on-demand Web-basedservers. The authors hope that increasedawareness of these methods of projectiondistortion symbolization will enhanceunderstanding of the principles and conse-quences of projection distortion.

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

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

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