Map Projections

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Map Projections

with

TNTmips®

TNTedit™

TNTview®

Introduction toPROJECTIONS

Map Projections

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Before Getting Started

You can print or read this booklet in color from MicroImages’ web site. Theweb site is also your source for the newest tutorial booklets on other topics.You can download an installation guide, sample data, and the latest versionof TNTmips.

http://www.microimages.com

Positions in a georeferenced spatial object must refer to a particular coordinatereference system. Many standard reference systems locate positions in a two-dimensional (planar) coordinate system. Such a system must use a map projec-tion to translate positions from the Earth’s nearly spherical surface to a hypotheticalmapping plane. Although the projection procedure inevitably introduces system-atic spatial distortions, particular types of distortion can be minimized to suit thegeographic scope and intended use of the map data. This booklet provides aconceptual introduction to map projections and geographic reference systems.

Prerequisite Skills This booklet assumes that you have completed the exercisesin the tutorial booklets Displaying Geospatial Data and Navigating. Those exer-cises introduce essential skills and basic techniques that are not covered againhere. Please consult those booklets for any review you need.

Sample Data This booklet does not use exercises with specific sample data todevelop the topics presented. You can, however, use the sample data that is dis-tributed with the TNT products to explore the ideas discussed on these pages. Ifyou do not have access to a TNT products DVD, you can download the data fromMicroImages’ web site. Make a read-write copy on your hard drive of the datasets you want to use so changes can be saved.

More Documentation This booklet is intended only as an introduction to con-cepts of map projections and spatial reference systems. Consult the tutorialbooklet entitled Coordinate Reference Systems for more information about howthese concepts are implemented in TNTmips.

TNTmips® Pro and TNTmips Free TNTmips (the Map and Image ProcessingSystem) comes in three versions: the professional version of TNTmips (TNTmipsPro), the low-cost TNTmips Basic version, and the TNTmips Free version. Allversions run exactly the same code from the TNT products DVD and have nearlythe same features. If you did not purchase the professional version (which re-quires a software license key) or TNTmips Basic, then TNTmips operates inTNTmips Free mode.

Randall B. Smith, Ph.D., 27 September 2011©MicroImages, Inc., 1998—2011

Map Projections

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Introduction to Map ProjectionsAlthough Earth images and map data that you useare typically rendered onto flat surfaces (such as yourcomputer screen or a sheet of paper), the Earth’s sur-face obviously is not flat. Because Earth has acurving, not-quite sphericial shape, planar maps ofall but the smallest areas contain significant geomet-ric distortions of shapes, areas, distances, or angles.In order to produce two-dimensional maps that pre-serve geographic relationships and minimizeparticular types of distortion, several steps arerequired. We must choose a geometric model(known as a geodetic datum) that closely ap-proximates the shape of the Earth, yet can bedescribed in simple mathematical terms. Wemust also adopt a coordinate system for refer-encing geographic locations in the mappingplane. Finally we must choose an appropriatemathematical method of transferring locations fromthe idealized Earth model to the chosen planar coor-dinate system: a map projection.

A coordinate system, datum, and map projection areall components of the coordinate reference systemfor a spatial object. You can choose coordinate ref-erence system parameters in TNTmips when youestablish georeference control for your project ma-terials, when you import georeferenced data, or whenyou warp or resample georeferenced objects to a newprojection. In addition, the Spatial Data Display pro-cess in all TNT products allows you to change thecoordinate reference system for the layers in a group,either to control the display geometry or to providecoordinate readouts.

The tutorial booklet entitled Coordinate ReferenceSystems introduces the mechanics of selecting coor-dinate reference system components. The presentbooklet provides a conceptual introduction to mapprojections, coordinate systems, and geodetic da-tums. We will begin with the latter concept.

The shape of the Earth andthe geodetic datum conceptare covered on pages 4-6.Page 7-9 discuss map scaleand the basics of coordinatesystems. Map projectionsare introduced on pages 10-12, and pages 13-19 presentsome widely used examples.Several common projectedcoordinate systems arediscussed on pages 20-21.Page 22 covers use of mapprojections in group display.Resources for further studyare presented on page 23.

Map Projections

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The Shape of the EarthGeodesy is the branch ofscience concerned withmeasuring the size andshape of the Earth.

The Earth Ellipsoid(flattening exaggerated)

Equatorial Radius

Pol

ar R

adiu

s

South Pole

North Pole

Geoid

Ellipsoid

The geographic positions ofsurvey benchmarks areeither measured or adjustedto conform to an idealellipsoidal surface. Theelevations shown ontopographic maps, however,are expressed relative to themean sea level geoid.

From our traditional human vantage point on theground, the Earth’s surface appears rough and irregu-lar. But spacecraft images show that on a planetaryscale, the Earth has a regular geometric shape with avery smooth surface. Knowledge of this shape is aprerequisite if we are to accurately transform geo-graphic coordinates through a map projection to aplanar coordinate system.

Sir Isaac Newton was the first to suggest that theEarth, because it rotates on its polar axis, is notquite spherical, but bulges outward slightly at theequator. The polar radius is thus slightly shorterthan the equatorial radius. If expressed as a frac-tion of the equatorial radius, the difference

according to current measurements is about 1/298.257, a value known as polar flattening. The

earth thus appears slightly elliptical in a cross sec-tion through the poles. Rotating this ellipse aboutthe polar axis results in a three-dimensional shapeknown as an ellipsoid. It is this geometric shapethat cartographers use as the reference surface forcreating large-scale maps (such as topographic maps).

While cartographers need a simple geo-metric representation of the Earth’sshape, geodesists are also interested in

defining a level surface to provide a basis for landsurveys. A level surface at any point is the planeperpendicular to the local direction of gravity (thedirection in which the surveyor’s plumb bob points).Because of local topography and the irregular distri-bution of mass within the Earth, the local directionof gravity may not be exactly perpendicular to theideal ellipsoidal surface. Hence the geoid (the levelsurface on which gravity is everywhere equal to itsstrength at mean sea level) is not perfectly ellipsoi-dal in shape. Instead it has smooth, irregularundulations that depart from the ideal ellipsoid by asmuch as 100 meters.

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Many different reference ellipsoids have been used by cartographers over the years.Estimates of the ellipsoid dimensions that best fit the overall shape of the Earthhave changed as new technologies have permitted increasingly refined measure-ments of the planet. In addition, any global best-fit ellipsoid does not fit all partsof the surface equally well because of the irregular undulations of the geoid. Forthis reason many additional reference ellipsoids have been defined for surveyingand mapping in different countries. Each regionally-defined ellipsoid has beenchosen to conform as closely as possible to the local geoid shape over that spe-cific region. The resulting ellipsoids differ in their dimensions, the location oftheir centers, and the orientation of their polar axes.

A horizontal geodetic datum specifies an ellipsoidal surface used as a referencefor mapping horizontal positions. The definition includes the dimensions andposition of the reference ellipsoid. Planar coordinates used to express positionsfor georeferenced data always refer to a specific geodetic datum (usually that ofthe map from which the data were abstracted).

The TNT products include specifications for a great many geodetic datums. Whenyou georeference new data, be sure to check the datum specification. (You mayneed to consult the documentation or metadata that accompanies the data). Refer-encing map coordinates to the wrong datum can lead to positioning errors of tensto hundreds of meters!

Geodetic Datums

Some of the many geodeticdatums available forselection in the TNT productsusing the CoordinateReference System window.

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North American DatumsThere are two geodetic datums in common use inNorth America. The North American Datum of 1927(NAD27) is an example of a regional datum, in whichthe ellipsoid (Clarke 1866) is tied to an initial pointof reference on the surface. NAD27 was developedin conjunction with the adjustment of a number ofindependent geodetic survey networks to form asingle integrated network originating at MeadesRanch, Kansas. The ellipsoid’s position is specifiedrelative to the Meades Ranch survey station, withthe result that the ellipsoid is not earth-centered.

The vast majority of U.S. Geological Survey topo-graphic maps have been produced using NAD27.Over the years, however, it has become clear that theaccuracy of the survey network associated with thisdatum is not sufficient for many modern needs. Sur-veying errors, destruction of survey monuments, andhorizontal movements of the Earth’s crust have ledto horizontal errors in control point positions as largeas 1 part in 15,000. The creation of satellite-basedpositioning systems also now requires the use of aglobal best-fit ellipsoid centered on the Earth’s cen-ter of mass (termed a geocentric ellipsoid).

As a result of these problems, the U.S. NationalGeodetic Survey has introduced a new datum, theNorth American Datum of 1983 (NAD83). The ref-erence ellipsoid used is that of the InternationalUnion of Geodesy and Geophysics Geographic Ref-erence System 1980 (GRS 1980), which isgeocentric, and thus has no single initial referencepoint on the surface. New latitude and longitudecoordinates were computed for geodetic controlpoints by least-squares adjustment. The calculationsincluded over 1.75 million positions obtained by tra-ditional survey and satellite observations, using sitesthroughout North America, Greenland, and the Car-ibbean.

The position of a given set ofgeodetic latitude andlongitude coordinates inNorth America can shift up to300 meters with the changefrom NAD27 to NAD83. Theamount of the shift variesfrom place to place. Theillustration shows position N37° 43' 31.84", W 122° 05'01.03" in the Hayward,California quadrangle plottedwith respect to both datums.

The U.S. Geological Surveyhas begun converting itsprimary series of topo-graphic maps (1:24,000scale 7.5-minute quad-rangles) to NAD83 as part ofthe periodic revisionprocess.

NAD27 NAD83

0 300 meters

The World Geodetic System1984 (WGS84) datum,developed by the U.S.Department of Defense, isalmost identical to NAD83.Positions in the two systemsagree to within about 0.1millimeter. WGS84 is thereference used for positionsdetermined from the GlobalPositioning Systemsatellites.

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Map Scale

A U S T R A L I A

Example of asmall-scale map

The Scale field at the bottomof TNT View windows showsthe map scale (at thecurrent zoom level) for thegeoreferenced data you areviewing.

Globes and traditional paper maps depictscaled-down versions of surface features.Map scale is defined mathematically as theratio between a unit distance on the map andthe actual distance (in the same units) that itrepresents in the real area depicted on themap. A map scale can then be representedas a fraction with 1 in the numerator (as in 1/ 250,000), but map scale is more commonlyprinted on maps using a colon as a separator(as in 1: 250,000).

Maps are can be roughly categorized by their scale.A small map that depicts a large area has a smallscale fraction (large number in the denominator), andcan be termed a small-scale map. Examples includethe small map of Australia shown on this page (scaleof 1: 50,000,000) and the page-size maps shown inprinted atlases. A map of the same size that showeda much smaller area of the world would have a largerscale fraction (smaller number in the denominator),and could be called a large-scale map. Topographicquadrangle maps (such as the excerpt on the pre-ceding page), which typically have map scalesbetween 1: 24,000 and 1: 100,000, are examples oflarge-scale maps. Large-scale maps can depict moremap features more accurately than can small-scalemaps.

Once map data have been converted to electronicform, they can be displayed at any scale. In the TNTproducts the current view scale is shown in the Scalefield at the bottom of View windows (see the bottomillustration). While exercising the freedom to viewyour map data at any scale, you should not lose sightof the fact that the map features were originally cre-ated at a particular map scale. The detail and levelof accuracy of the map features are commensuratewith that original map scale regardless of the currentscale at which you are viewing them.

Scale = 1 : 50,000,000

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The spherical latitude/longitude coordinate systemhas been adapted math-ematically to account forEarth’s ellipsoidal shape,yielding geodetic latitudeand longitude values.

Latitude / Longitude Coordinates

Latitude / Longitude is usedas the native coordinatesystem for several widelyavailable forms of spatialdata (including the U.S.Census Bureau’s TIGER /Line data, the Digital Chartof the World, and sometypes of USGS DigitalElevation Models andDigital Line Graphs).

The oldest global coordinate system is the Latitude/Longitude system (also referred to as Geographiccoordinates). It is the primary system used for de-termining positions in surveying and navigation. Agrid of east-west latitude lines (parallels) and north-south longitude lines (meridians) represent anglesrelative to standard reference planes. Latitude ismeasured from 0 to 90 degrees north and south of

the equator. Longitude values rangefrom 0 to 180 degrees east or west ofthe Prime Meridian, which by interna-tional convention passes through theRoyal Observatory at Greenwich, En-gland. (South latitude and westlongitude coordinates are treated asnegative values in the TNT products,but you can use the standard directionalnotation and omit the minus sign whenentering values in process dialogs.)

Because the Latitude/Longitude systemreferences locations to a spheroid rather

than to a plane, it is not associated with a map pro-jection. Use of latitude/longitude coordinates cancomplicate data display and spatial analysis. Onedegree of latitude represents the same horizontal dis-tance anywhere on the Earth’s surface. However,because lines of longitude are farthest apart at theequator and converge to single points at the poles,the horizontal distance equivalent to one degree oflongitude varies with latitude. Many TNTmips pro-cesses adjust distance and area calculations forobjects with latitude/longitude coordinates to com-pensate for this effect, but these approximations areless accurate than calculations with data projectedto a planar coordinate system. If you have data ofregional or smaller extent with Geographic coordi-nates, you will achieve better results by warping orresampling the data to a planar coordinate system.

Equator

Meridian

Parallel

North Pole

PrimeMeridian

0°N3

0°N6

0°N

30

°E

30

°W

60°W

Latitude

Longitude

0°E

Map Projections

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Planar Coordinate SystemsA large-scale map represents a small portion of the Earth’ssurface as a plane using a rectilinear grid coordinate sys-tem to designate location coordinates. A planar Cartesiancoordinate system has an origin set by the intersection oftwo perpendicular coordinate axes and tied to a knownlocation. The coordinate axes are normally oriented sothat the x-axis is east-west and the y-axis is north-south.Grid coordinates are customarily referred to as easting(distance from the north-south axis) and northing (dis-tance from the east-west axis). The definition of aCartesian coordinate system also includes the units usedto measure distances relative to these axes (such as metersor feet).

For very small areas (such as a building construction site)the curvature of Earth’s surface is so slight that groundlocations can be referenced directly to an arbitrary planar Cartesian coordinatesystem without introducing significant positional errors. For areas larger than afew square kilometers, however, the difference between a planar and curving sur-face becomes important when relating ground and map locations. A map projectionmust then be selected to relate surface and map coordinates to reduce undesirabledistortions in the map. In the TNT products a Cartesian coordinate system that isrelated to the Earth via a map projection is termed a projected coordinate system.

A number of projected coordinate systemshave been set up to represent large areas(states, countries, or larger areas) by subdi-viding the coverage area into geographiczones, each of which has its own origin. Tominimize variations in scale associated withthe map projection, one of the coordinateaxes (sometimes both) typically bisects thezone. In that case, in order to force all loca-tions within the zone to have positivecoordinate values, a large number is addedto one or both coordinates of the origin,termed false easting and false northing. Thisprocedure moves the 0,0 position of the co-ordinate system outside the zone to create afalse origin, as illustrated to the right.

Portion of a colororthophoto overlaidwith a map grid with200 meter spacingbetween the grid lines.

False origin(x = 0, y = 0)

North-South Axiswith false easting

Easting (x)

Northing (y)

Zone

Planar Coordinate Systemwith false easting

East-West Axis

True North-SouthAxis

True origin

Map Projections

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A map projection can be thought of as a process oras the output of the process. For example, a mapprojection can be described as a systematic repre-sentation of all or part of the Earth’s surface on aplane. But this representation is the result of a com-plex transformation process. The input for the mapprojection process is a set of horizontal positions onthe surface of a reference ellipsoid. The output is acorresponding set of positions in a reference planeat a reduced scale. In this sense, a map projection isa complex mathematical formula that produces thedesired coordinate transformations. However, amathematical approach is not required for under-standing the basic concepts surrounding the mapprojection process.

Transforming coordinates from the Earth ellipsoidto a map involves projection to a simple geometricsurface that can be flattened to a plane without fur-ther distortion (such as stretching or shearing). Sucha surface is called a developable surface. Three typesof developable surfaces form the basis of most com-mon map projections: a cylinder, a cone, or the planeitself.

Simple cylindrical projections are constructed us-ing a cylinder that has its entire circumference tangentto the Earth’s surface along a great circle, such asthe equator. Simple conic projections use a conethat is tangent to the surface along a small circle,such as a parallel of latitude. Projecting positionsdirectly to a plane tangent to the Earth’s surface cre-ates an azimuthal projection.

Regular cylindrical and conic projections orient theaxis of the cylinder or cone parallel to the Earth’saxis. If the axes are not parallel, the result is a trans-verse (perpendicular axes) or oblique projection.Additional variants involve the cylinder, cone, orplane cutting through the globe rather than beingmerely tangent to the surface.

Map Projections

CylindricalProjection

ConicProjection

AzimuthalProjection

Map Projections

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Projecting the Earth’s curving surface to a mappingplane cannot be done without distorting the surfacefeatures in some way. Therefore all maps includesome type of distortion. When selecting a map pro-jection, cartographers must decide whichcharacteristic (or combination thereof) should beshown accurately at the expense of the others. Themap properties that enter into this choice are scale,area, shape, and direction.

All maps are scaled representations of the Earth’ssurface. Measuring exact distances from any mapfeatures in any direction would require constant scalethroughout the map, but no map projection canachieve this. In most projections scale remains con-stant along one or more standard lines, and carefulpositioning of these lines can minimize scale varia-tions elsewhere in the map. Specialized equidistantmap projections maintain constant scale in all direc-tions from one or two standard points.

In many types of spatial analysis it is important tocompare the areas of different features. Such com-parisons require that surface features with equal areasare represented by the same map area regardless ofwhere they occur. An equal-area map projectionconserves area but distorts the shapes of features.

A map projection is conformal if the shapes of smallsurface features are shown without distortion. Thisproperty is the result of correctly representing localangles around each point, and maintaining constantlocal scale in all directions. Conformality is a localproperty; while small features are shown correctly,large shapes must be distorted. A map projectioncannot be both conformal and equal-area.

No map projection can represent all great circle di-rections as straight lines. Azimuthal projections showall great circles passing through the projection cen-ter as straight lines.

Equal-Area Projection

Conformal Projection

Shapes are distorted, but allmap features are shown withthe correct relative areas.

Small shapes maintaincorrect proportions, as localscale and angles areconstant around each point.Relative area of featuresvaries throughout the map.

A = 1

A = 1

Map DistortionsDue to the projection from acurving surface to a plane,map scale varies from placeto place and for differentdirections on maps. Thescale factor in any maplocation is the actual mapscale at that location dividedby the nominal map scale.For those special locationswhere the actual map scaleequales the nominal scale,the scale factor equals 1.0.

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Using Map ProjectionsThe TNT products provide built-in support for hundreds of projected coordinatesystems whose definitions include all parameters for the required map projection.So when you import or georeference new geodata, or reproject existing geodata,in most cases you do not need to select or set up a map projection directly. Youmerely select the predefined coordinate reference system or projected coordinatesystem that matches your data. (Several of the commonly-used projected coordi-nate systems are described later in this booklet.)

In some cases you may need to import or georeference spatial data referenced to aprojected coordinate system that is not among the predefined choices. Or youmay want to define your own projected coordinate system for a project. For thesecases, the Coordinate Reference System window in TNTmips allows you to setup and save a custom coordinate reference system with your choice of map pro-jection and projection parameters. The required procedures are introduced in thetutorial booklet entitled Coordinate Reference Systems.

All parameters for a custom map projectioncan be defined on the Projection panel ofthe Coordinate Reference System window.

A particular projection can be centered on the project area by choosing appropri-ate projection parameters. Because of varying patterns of distortion, someprojections are better for areas elongate in an east-west direction and others forareas elongate north-south. Large-scale maps used to determine or plot direc-tions, such as navigational charts or topographic maps, should use a conformalmap projection. Equal-area projections are appropriate for smaller-scale thematicmaps and maps that show spatial distributions. The following pages discuss someof the projections that are commonly used on paper maps and in the projectedcoordinate systems used with digital geodata.

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Mercator ProjectionOne of the best known map projections, the Mercator projection was devisedspecifically as an aid to navigation. A ship’s course can be plotted easily with theMercator projection because a course with constant azimuth (compass direction)is shown as a straight line.

The Mercator is a regular cylindrical projection (the cylinder axis passes throughthe north and south poles). Meridians of longitude are shown as equally spacedvertical lines, intersected at right angles by straight horizontal parallels. The spacingbetween parallels increases away from the Equator to produce a conformal pro-jection. The scale is true along the equator for a tangent Mercator projection,which has a natural origin at the equator. Assigning a different latitude of naturalorigin produces an intersecting cylindrical projection with two standard parallels(with true scale) equidistant from the equator.

The poleward increase in spacing of parallels produces great distortions of area inhigh-latitude regions. In fact, the y coordinate for the poles is infinity, so mapsusing the Mercator projection rarely extend poleward of 75 degrees latitude. TheMercator projection remains in common use on nautical charts. Because scaledistorion is minor near the equator, it also is a suitable conformal projection forequatorial regions.

Mercator projection ofNorth America centeredon 100° west longitude,with ten-degree latitude-longitude grid.

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Transverse Mercator Projection

Transverse Mercatorprojection of NorthAmerica with centralmeridian at 100° westlongitude.

The Transverse Mercator projection is a conformal cylindrical projection with thecylinder rotated 90 degrees with respect to the regular Mercator projection. Thecylinder is tangent to a central meridian of longitude around its entire circumfer-ence. The central meridian and equator are straight lines, but all other meridiansand parallels are complex curves.

Scale is constant along any meridian. Scale change along parallels is insignifi-cant near the central meridian, but increases rapidly away from it, so the TransverseMercator projection is useful only for narrow bands along the central meridian. Itforms the basis for the Universal Transverse Mercator Coordinate System, and isprimarily used for large-scale (1:24,000 to 1:250,000) quadrangle maps. The cen-tral meridian can be mapped at true scale (Central Scale parameter = 1.0), or at aslightly reduced constant scale (for example, the value 0.9996 used in the UTMsystem). In the latter case a pair of meridians bracketing the central one maintaintrue scale, and the mean scale for the entire map is closer to the true scale.

In the United States the Transverse Mercator projection is also used in the StatePlane Coordinate System for states (or individual state zones) which are moreelongate in the north-south direction. Gauss Conformal and Gauss-Kruger areEuropean names for the Transverse Mercator projection.

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Lambert Conformal Conic ProjectionThe Lambert Conformal Conic projection is normally constructed with a devel-opable surface that intersects the globe along two standard parallels. You mustspecify the latitude for each standard parallel when setting up the projection, aswell as the latitude to use as the origin for northing coordinates. Scale is truealong the standard parallels, smaller between them, and larger outside them. Areadistortion is also relatively small between and near the standard parallels. Thisprojection therefore is particularly useful for mid-latitude regions which are elon-gate in the east-west direction.

The parallels in the Lambert Conformal Conic projection are concentric circles,while the meridians are equally-spaced straight radii of these circles. The merid-ians intersect parallels at right angles (as expected in a conformal projection).Spacing of the parallels increases north and south from the band defined by thestandard parallels.

In the United States the Lambert Conformal Conic projection (also known asConical Orthomorphic) is used in the State Plane Coordinate System for statezones with greater east-west than north-south extent. It is also used for regionalworld aeronautical charts and for topographic maps in some countries.

Lambert ConformalConic projection ofNorth America withcentral meridian at 100°west longitude, andstandard parallels at 20°and 60° north latitude.

Map Projections

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Albers Conic Equal-Area Projection

Albers Conic Equal-Areaprojection of NorthAmerica with centralmeridian at 100° westlongitude, and standardparallels at 20° and 60°north latitude.

The Albers Conic Equal-Area Projection is commonly used to map large areas inthe mid-latitudes, such as the entire “lower 48” United States. In this normalapplication there are two standard parallels. Like other conic projections, theparallels are concentric circular arcs with equally-spaced meridians intersectingthem at right angles. The change in spacing between parallels is opposite fromthe Lambert Conformal Conic projection; parallels are more widely spaced be-tween the standard parallels, and more closely spaced outside them.

Each parallel has a constant scale, with true scale along the standard parallels,smaller scale between them, and larger scale outside them. To maintain equalarea, scale variations along the meridians show a reciprocal pattern; the increasein east-west scale outside the standard parallels is balanced by a decrease in north-south scale.

The Albers Conic Equal-Area projection has been used by the U.S. GeologicalSurvey for a number of small-scale maps of the United States, using latitude 29.5ºand 45.5º north as standard parallels. Mid-latitude distortion is minor for mostnormal conic projections, so that differences between them become obvious onlywhen the region mapped extends to higher or lower latitudes.

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Polyconic Projection

Polyconic projection ofNorth America withcentral meridian at100° west longitudeand the equator asorigin latitude.

The Polyconic projection was devised in the early days of U.S. government sur-veying and was used until the 1950’s for all large-scale U.S. Geological Surveyquadrangle maps (now superceded for most revised maps by Universal Trans-verse Mercator). The Polyconic projection produces extremely small distortionover small areas near the central meridian, despite being neither conformal norequal-area.

Parallels in the Polyconic projection are circular arcs, but are not concentric. Eachparallel is the trace of a unique cone tangent to the globe at that latitude. Thename thus refers to the fact that there are many cones involved in creating theprojection, rather than a single conic developable surface. When chosen as theorigin latitude, the equator is a straight line. The central meridian is also a straightline, but all other meridians are complex curves that are not exactly perpendicularto the parallels.

Scale is true along each parallel and along the central meridian. When the centralmeridian is chosen to lie within a large-scale map quadrangle, scale distortion isalmost neglible for the map. Because polyconic quadrangle maps are not pre-cisely rectangular, they cannot be mosaicked in both north-south and east-westdirections without gaps or overlaps.

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Lambert Azimuthal Equal-Area Projection

Lambert AzimuthalEqual-Area projectionof North Americacentered at 100° westlongitude and 50° northlatitude.

The Lambert Azimuthal Equal-Area projection transforms surface coordinatesdirectly to a plane tangent to the surface. The point of tangency forms the centerof the projection, and is specified by the longitude and latitude of natural origin.In general, the projection center should coincide with the center of the area to bemapped. Scale is true only at the center point, but deviation from true scale forother points on the map is less than for other forms of azimuthal projection.

Scale in the radial direction decreases away from the center. Scale perpendicularto a radius increases with distance from the center, as required to produce theequal-area property. Distortion is symmetric about the central point, so this pro-jection is appropriate for areas that have nearly equal north-south and east-westextents.

The pattern of meridians and parallels depends on the choice of central point. Ifthe projection is centered at a pole, meridians are straight radii and parallels areconcentric circles. In an oblique projection, such as the example illustrated be-low, only the central meridian is straight, and other meridians and parallels arecomplex curves. The Lambert Azimuthal Equal-Area projection has been usedcommonly for small-scale maps of the polar regions, ocean basins, and conti-nents.

Map Projections

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Stereographic Projection

Stereographicprojection of NorthAmerica centered at100° west longitudeand 50° north latitude.

The Stereographic projection is a conformal azimuthal projection. When used forlarge areas, so that the spherical Earth model can be used, it is also a true perspec-tive projection, unlike most map projections. Surface locations are projected to atangent plane using a single projection point on the surface of the sphere exactlyopposite the center of the projection. When used to map smaller areas, so that theellipsoidal Earth model is used, the projection is not perspective, and, in order tomaintain conformality, is not truly azimuthal.

The Stereographic projection is most commonly used to map polar regions, inwhich case the pole is chosen as the center point. (The Universal Polar Sterographiccoordinate system is an adjunct to the Universal Transverse Mercator system,extending this universal system to polar regions.) In this form, the map showsmeridians as straight radii of concentric circles representing parallels. In an ob-lique Stereographic projection, such as the illustration below, only the centralmeridian is straight. All other meridians and parallels are circular arcs intersect-ing at right angles. Scale increases away from the central point, which normallyhas true scale (scale factor at natural origin = 1.0). Reducing the scale valueproduces an intersecting rather than tangential plane. Scale is then true along anellipse centered on the projection center, and the mean scale for the entire map iscloser to the true scale.

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Universal Transverse Mercator SystemThe next two pages discuss several of the formally-defined projected coordinatesystems that are in widespread use in the United States and throughout the world.

The Universal Transverse Mercator (UTM) system is a global set of coordinatesystems commonly used in the United States on topographic maps and for large-scale digital cartographic data. The UTM system divides the world into uniformzones with a width of 6 degrees of longitude. The zones are numbered from 1 to60 eastward, beginning at 180 degrees. Easting is measured from a zone’s centralmeridian, which is assigned a false easting of 500,000 meters. Northing is mea-sured relative to the equator, which has a value of 0 meters for coordinates in thenorthern hemisphere. Northing values in the southern hemisphere decrease south-ward from a false northing of 10,000,000 meters at the equator. (You must chooseeither northern or southern hemisphere coordinates when choosing a UTM zone.)The scale factor at the central meridian is 0.9996. The illustration below shows asample location in UTM zone 10, a highway intersection in the Hayward Quad-rangle, California (USA).

The UTM coordinate system uses the Transverse Mercator map projection, whichminimizes shape distortions for small geographic features. The inherent accuracyof distance measurements (related to scale variations) is one part in 2500. TheGauss-Kruger zonation system is very similar to UTM, except for a scale factor of1.000 at the central meridian. Variants of the Gauss-Kruger zonation use either 6-degree or 3-degree geographic zones, and in some versions the zone number isadded to the beginning of the 500,000 meter false easting value at each centralmeridian (e.g. 4,500,000 meter false easting for Zone 4 central meridian).

Hayward Quadrangle Highway Intersection UTM CoordinatesEasting 580,349 m, Northing 4,166,094 m

Zone 10 (120° W to 126° W), North American Datum 1927

120° W123° W126° WCentral Meridian (500,000 m False Easting)

80,349 m east ofthe Zone 10central meridian.

4,166,094 mnorth of theequator.

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U.S. State Plane Coordinate SystemThe United States State Plane Coordinate System (SPCS) has been widely usedas a grid system for land surveys. It was devised to provide each state with rect-angular coordinates that could be tied to locations in the national geodetic surveysystem. The original system, based on the North American Datum 1927, usescoordinates in feet. Most states are divided into two or more overlapping stateplane zones, each with its own coordinate system and projection. A few smallerstates use a single zone. The Lambert Conformal Conic projection is used forzones with a larger east-west than north-south extent. Zones that are more elon-gate in the north-south direction are mapped using the Transverse Mercatorprojection. Scale variations are minimized to provide an accuracy of one part in10,000 for distance measurements. State Plane Coordinate tick marks and zoneinformation can be found on U.S. Geological Survey topographic maps.

With the development of the North American Datum 1983, a revised version ofthe State Plane Coordinate System based on that datum was also developed. Zoneswere redefined for some states, and northing and easting coordinates are nomi-nally in meters. However, some state and local governments require the use offeet as a measurement unit with SPCS83. To cover these variations, the TNTproducts provide two versions of each SPCS27 and SPCS83 zone, one in metersand one in US feet.

Hayward Quadrangle Highway IntersectionState Plane Coordinates

Easting 1,539,895 ft, Northing 419,137 ftCalifornia Zone III, North American Datum 1927

120° 30' W2,000,000 ft False Easting

36° 30' N0 ft

419,137 ftnorth of theorigin

460,105 ft west of the N-S axis;easting = 2,000,000 - 460,105 = 1,539,895 ft

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Using Map Projections in Group DisplayIf a geospatial object has been created in orreprojected to a specific map projection, the object’scoordinate system coincides with the map coordi-nate system. For many georeferenced objects,however, this is not the case. The Orientation / Pro-jection Options controls in the Group Settingswindow determine whether a display group is ori-

ented relative to object coordinates or toa specific map coordinate system.

The Auto-Match option determineswhether object coordinates are used ornot, and, for multiple layers, which objectprovides the basis for the group projec-tion. To orient the group to a map

projection, choose [None] (or [First Raster or None]if there is no raster layer in the group). The MatchLayer Projection button then becomes active; pressit and then the Redraw button to redraw the viewusing the Coordinate Reference System of the layerthat is currently active.

A vector, CAD, shape or TIN object in the displaygroup is projected precisely to the selected map co-ordinates, subject to the inherent accuracy of theelement coordinates. Reorientation of a raster ob-ject is governed by the Positional Accuracy option

in the View window’s Option menu. The Fastoption changes the orientation and scale in auniform manner to approximately match theoutput projection. The Exact option providesan exact resampling to the projection, but cantake slightly longer to redisplay the view.

As you gather spatial data from different sources for a project, you will probably end upwith data in a number of different coordinate systems and projections. The Displayprocess can overlay layers with different coordinate referenence systems withreasonable registration, but for maximum efficiency materials that will be used togetherroutinely should be reprojected to a common coordinate reference system. For rasterimages, choose Image / Resample and Reproject / Automatic from the TNTmips menu.For geometric objects (vector, CAD, or TIN), choose Geometric / Reproject.

The [First Raster or FirstLayer] and [First Layer]options always use theobject coordinates of theindicated layer to control thegroup orientation andprojection. “First” refers tothe lowest layer in the layerlist.

In Display, open the GroupSettings window for thedesired group by left-clickingon its Settings icon button inthe DisplayManagerwindow

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Looking FurtherReferences

Langley, Richard B. (February, 1992). Basic geodesy for GPS. GPS World, 3,44-49.

A valuable introduction to geodetic concepts, including the geoid, the el-lipsoid, and geodetic datums.

Maling, D. H., (1992). Coordinate Systems and Map Projections. Oxford:Pergamon Press. 255 pp.

Robinson, A. H., Morrison, J. L., Muehrcke, P. C., Kimerling, A. J., and Guptill,S. C. (1995). Elements of Cartography (6th ed.). New York: John Wiley& Sons, Inc. 674 pp.

Contains excellent chapters on Geodesy, Map Projections, and Referenceand Coordinate Systems.

Snyder, John P. (1987). Map Projections -- A Working Manual. U.S. GeologicalSurvey Professional Paper 1395. Washington, D.C.: U.S. GovernmentPrinting Office. 383 pp.

An exhaustive description and history of map projections and related con-cepts, including the mathematical details.

Snyder, John P., and Voxland, Philip M. (1989). An Album of Map Projections.U.S. Geological Survey Professional Paper 1453. Washington, D.C.: U.S.Government Printing Office. 249 pp.

Internet Resources

Map Projection Overview, Coordinate Systems Overview, and Geodetic DatumOverview:

http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj.html

These web pages by Peter H. Dana (part of The Geographer’s Craft Project)provide an illustrated discussion of each topic.

NGA Geospatial Sciences Publications:

http://earth-info.nga.mil/GandG/publications/index.html

This page at the U.S. National Geospatial Intelligence Agency websiteprovides links to a number of online documents on geodesy and mappingscience (in PDF or HTML format), including Geodesy for the Laymanand All You Ever Wanted to Know and Couldn’t Find Out About PrecisePositioning...(In Plain English).

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IndexAuto-match option....................................22coordinate system...........................3,8-9,20,21

geographic........................................8latitude/longitude............................8United States State Plane...................21Universal Transverse Mercator..........20

datum.................................................4-6North American 1927 (NAD27).........6North American 1983 (NAD83).........6World Geodetic System 1984..............6

developable surface...................................10distortion..................................................3,11easting...................................................9,20,21ellipsoid...............................................4-6geoid........................................................4-5georeferencing.............................................3meridian

central.......................................15-21prime....................................................8

northing...............................................9,20,21origin latitude........................................16,17parallel, standard.....................................15,16projections

Albers conic equal area......................16azimuthal..................................10,11,18conformal.......................11,13-15,17,19conic..........................................10,15-17equal-area....................................11,16-18equidistant.........................................11Lambert azimuthal equal-area............18Lambert conformal conic................15,21Mercator..........................................13polyconic...........................................17stereographic......................................19transverse Mercator..................14,20,21

scale, map...........................................6-7,11true-scale latitude......................................13zone, geographic..................................9,20,21

Advanced Software for Geospatial Analysis PROJECTIONS