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Map Projections

Map Projections

with

TNTmips®

TNTedit™

TNTview®

Introduction toPROJECTIONS

Map Projections

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Before Getting Started

It may be difficult to identify the important points in some illustrations without acolor copy of this booklet. You can print or read this booklet in color fromMicroImages� web site. The web site is also your source for the newest GettingStarted booklets on other topics. You can download an installation guide, sampledata, and the latest version of TNTlite.

http://www.microimages.com

Positional information in a georeferenced spatial object must refer to a particulargeographic reference system. Most of the standard reference systems locate posi-tions in a two-dimensional (planar) coordinate system. Such a system must use amap projection to translate positions from the Earth�s nearly spherical surface toa hypothetical mapping plane. Although the projection procedure inevitably in-troduces systematic spatial distortions, particular types of distortion can be mini-mized to suit the geographic scope and intended use of the map data. This bookletprovides an introduction to map projections and geographic reference systemsand their use in the TNT products. Procedures for selecting projection and refer-ence system parameters are presented in a series of simple exercises.

Prerequisite Skills This booklet assumes that you have completed the exercisesin Getting Started: Displaying Geospatial Data and Getting Started: Navigating.Those exercises introduce essential skills and basic techniques that are not cov-ered again here. Please consult those booklets and the TNTmips reference manualfor any review you need.

Sample Data The exercises presented in this booklet use sample data that isdistributed with the TNT products. If you do not have access to a TNT productsCD, you can download the data from MicroImages� web site. In particular, thisbooklet uses sample files in the SF_DATA data collection.

More Documentation This booklet is intended only as an introduction to mapprojections and geographic reference systems. Consult the TNTmips referencemanual, which contains more than 25 pages on geographic reference systems, formore information.

TNTmips® and TNTlite® TNTmips comes in two versions: the professional ver-sion and the free TNTlite version. This booklet refers to both versions as�TNTmips.� If you did not purchase the professional version (which requires ahardware key), TNTmips operates in TNTlite mode, which limits object size anddoes not allow export. All of the exercises can be completed in TNTlite using thesample geodata provided.

Randall B. Smith, Ph.D., 12 September 2002

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Map Projections

Introduction to Map ProjectionsA core feature of the TNT products is the capabilityto relate information to specific geographic locations.Spatial data in any of the available forms (raster,vector, CAD, or TIN objects) can be georeferenced,or related to a standard geographic reference sys-tem. The locations of elements within ageoreferenced object are then expressed in geo-graphic coordinates, and meaningful distances andareas can be measured or calculated.

Although Earth images and map data that youuse are typically rendered onto flat surfaces(such as a computer screen or a sheet of pa-per), the Earth’s surface obviously is not flat.As a result of Earth’s surface curvature, pla-nar maps of all but the smallest areas containsignificant geometric distortions of shapes, ar-eas, distances, or angles. In order to producetwo-dimensional maps that preserve geographic re-lationships and minimize particular types ofdistortion, several steps are required. We mustchoose a geometric model that closely approximatesthe shape of the Earth, yet can be described in simplemathematical terms. We must also adopt a coordi-nate system for referencing geographic locations inthe mapping plane. Finally we must choose a methodof transferring (or projecting) locations from the ide-alized Earth model to the chosen planar coordinatesystem.

You can choose coordinate system and map projec-tion parameters in TNTmips when you establishgeoreference control for your project materials,when you import georeferenced data, or when youwarp or resample georeferenced objects to a newprojection. In addition, the Spatial Data Display pro-cess in all TNT products allows you to change thecoordinate system and map projection for the layersin a group, either to control the display geometry orto provide coordinate readouts.

STEPS! launch TNTmips! select Display / Spatial

Data from the mainmenu

Pages 4-8 introduce theinterface used to selectcoordinate systems andmap projections, anddiscuss the availablepredefined coordinatesystems. Conceptsconcerning the shape of theEarth and geodetic datumand ellipsoid selection arecovered on pages 9-11.Map projections areintroduced on pages 12-14,and pages 15-21 discusssome widely usedexamples. Page 22 coversuse of map projections ingroup display. Resourcesfor further study arepresented on page 23.

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Map Projections

Parameters WindowSTEPS! click the Open icon

button on the DisplaySpatial Data toolbar andchoose Open Layoutfrom the menu

! navigate to the LAYOUTS

Project File in theSF_DATA data collectionand select LAYOUT2

! from the Options menuon the View windowchoose Position Report /Projection

The Coordinate System / Projection Parameters win-dow is the standard interface you use to set upgeographic reference parameters in the relevant pro-cesses. You can use this window to choose acoordinate system and map projection, and to specifythe geometric Earth model.

Choose Projection under the Position Report option toset the geographic reference system for the PositionReport fields at the bottom of the View window. We willuse this example to introduce selection procedures andthe important background concepts.

In most other relevantprocesses you canopen the CoordinateSystem / ProjectionParameters windowby pressing aProjection... buttonon the appropriatecontrol window.

The Datum and Ellipsoid parameters specify the geometricEarth model used to generate the map projection. Theellipsoid is usually determined by your choice of datum.

Click [System...] to selecta coordinate system.

Some coordinate systemsrequire you to select ageographic zone.

The Projection button is active when User-Defined is selected as the coordinatesystem, allowing you to choose a specificmap projection. In this example theprojection is already defined by the choice ofcoordinate system, so the button is dimmedand inactive.

Keep the CoordinateSystem / ProjectionParameters windowopen and continue tothe next page.

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Map Projections

Choosing a Coordinate SystemSTEPS! click [System] on the

Coordinate System /Projection Parameterswindow

! examine the choices inthe Coordinate Systemwindow

! click [OK] to close theCoordinate Systemwindow and continue tothe next page

The Coordinate System window allows you tochoose from several predefined map coordinate sys-tems that automatically specify an associated mapprojection. If necessary, you can use the User De-fined option to choose a map projection andcoordinate system that may be more appropriate foryour data and project needs.

The United States State Plane and Universal Trans-verse Mercator coordinate systems are planarcoordinate systems with associated map projec-tions. These systems can be used to accuratelyrepresent positions, distances, and shapes of fea-tures in small geographic areas, such as a state orprovince. Both of these systems use a set of geo-graphic zones, with each zone having its owncoordinate grid. Grid coordinates are customarilyreferred to as easting (relative to a north-south axis)and northing (relative to an east-westaxis). In order to minimize scale distor-tion effects related to the map projection,one of the coordinate axes may be locatedwithin the geographic zone. A large posi-tive value, called a false easting or falsenorthing, is usually assigned to this axis.This procedure moves the origin (0,0point) of the coordinate system outside thezone, ensuring that all northing and eastingvalues are positive.

The Latitude/Longitude and Geographicoptions both refer to the familiar globalsystem of spherical coordinates. This sys-tem uses a grid of east-west latitude lines(parallels) and north-south longitude lines(meridians) to designate position.

The predefined coordinate system choices are ex-plored in more detail on the following pages.

Grid reference 0,0

East-West Axis

with false northing

North-South Axis

Easting

Northing

Zone

Planar Coordinate Systems

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Map Projections

Universal Transverse Mercator SystemSTEPS! click [Zone...] on the

Coordinate System /Projection Parameterswindow

! examine the list ofavailable UTM zones,then click [Cancel] onthe Select Zone window

! click [Cancel] on theCoordinate System /Projection Parameterswindow

! move the screen cursorto the location of theintersection shownbelow and check thevalues in the PositionReport fields at thebottom of the Viewwindow

The group in the view window is referenced to theUniversal Transverse Mercator (UTM) coordinatesystem. This global coordinate system is commonlyused in the United States on topographic maps andfor some digital cartographic data. The UTM sys-tem divides the world into uniform zones with a widthof 6 degrees of longitude. The zones are numberedfrom 1 to 60 eastward, beginning at 180 degrees.Easting is measured from a zone’s central meridian,which is assigned a false easting of 500,000 meters.Northing is measured relative to the equator, whichhas a value of 0 meters for coordinates in the north-ern hemisphere. Northing values in the southernhemisphere decrease southward from a false north-ing of 10,000,000 meters at the equator. (You mustchoose either northern or southern hemisphere co-ordinates when choosing a UTM zone.)

The UTM coordinate system uses the TransverseMercator map projection, which minimizes shapedistortions for small geographic features. The in-herent accuracy of distance measurements (relatedto scale variations) is one part in 2500.

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 of theZone 10 centralmeridian.

4,166,094 mnorth of theequator.

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Map Projections

State Plane Coordinate SystemSTEPS! from the Options menu

on the View windowchoose Position Report /Projection

! click [System...] on theCoordinate System /Projection Parameterswindow

! from the CoordinateSystem window selectUnited States StatePlane 1927 and click[OK]

! click [Zone...], thenselect California III andclick [OK] in the SelectZone window

! click [OK] on theCoordinate System /Projection Parameterswindow

! from the Options menuon the View windowchoose Position Report /Units / Feet

! move the screen cursorto the location of thehighway intersection andcheck the values in thePosition Report fields

! set the Position Report /Units option back toMeters

The United States State Plane Coordinate Systemhas been widely used as a grid system for land sur-veys. It was devised to provide each state withrectangular coordinates that could be tied to loca-tions in the national geodetic survey system. Theoriginal system, based on the North American Da-tum 1927, uses coordinates in feet. A more recentrevision which measures distances in meters is basedon the North American Datum 1983.

Most states are divided into two or more overlap-ping state plane zones, each with its own coordinatesystem and projection. A few smaller states use asingle zone. The Lambert Conformal Conic pro-jection is used for zones with a larger east-west thannorth-south extent. Zones that are more elongate inthe north-south direction are mapped using theTransverse Mercator projection. Scale variationsare minimized to provide an accuracy of one part in10,000 for distance measurements. State Plane Co-ordinate tick marks and zone information can befound on U.S. Geological Survey topographic maps.

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

The intersection coordinatesyou determine may differslightly from those shown

here because of roundingeffects related to your

window size andzoom level.

Map Projections

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Latitude / LongitudeSTEPS� from the Options menu

on the View windowchoose Position Report /Projection

� click [System...] on theCoordinate System /Projection Parameterswindow

� from the CoordinateSystem window selectLatitude/Longitude andclick [OK]

� click [OK] to close theCoordinate System /Projection Parameterswindow

� move the screen cursorto the location of thehighway intersection andcheck the values in thePosition Report fields

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). Because this is a spherical coordinatesystem, latitude and longitude values are expressedas angles relative to standard reference planes. Lati-tude is measured from 0 to 90 degrees north and southof the equator. Longitude values range from 0 to180 degrees east or west of the Prime Meridian,which by international convention passes through theRoyal Observatory at Greenwich, England. (Southlatitude and west longitude coordinates are treatedas negative values in the TNT products, but you canuse the standard directional notation and omit theminus sign when entering values in process dialogs.)

Because the Latitude/Longitude system referenceslocations to a spheroid rather than to a plane, it isnot associated with a map projection. Use of lati-

tude/longitude coordinates cancomplicate data display and spa-tial analysis. One degree oflatitude represents the samehorizontal distance anywhere onthe Earth’s surface. However,because lines of longitude arefarthest apart at the equator andconverge to single points at thepoles, the horizontal distanceequivalent to one degree of lon-gitude varies with latitude.Many TNTmips processes ad-

just distance and area calculations from objects withlatitude/longitude coordinates to compensate for thiseffect, but these approximations are less accurate thancalculations with data projected to a planar coordi-nate system. If you have data with Geographiccoordinates, you will achieve the best results bywarping or resampling the data to a planar coordi-nate system.

Equator

Meridian

Parallel

North Pole

PrimeMeridian

0°N3

0°N6

0°N

30

°E

30

°W

60°W

Latitude

Longitude

0°E

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Map Projections

The Shape of the EarthFrom our traditional human vantage point on theground, the Earth’s surface appears rough and ir-regular. But spacecraft images show that on aplanetary scale, the Earth has a regular geometricshape with a very smooth surface. Knowledge ofthis shape is a prerequisite if we are to accuratelytransform geographic coordinates through a mapprojection to a planar 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 atthe equator. As a result the polar radius isslightly shorter than the equatorial radius. Ifexpressed as a fraction of the equatorial ra-dius, the difference according to currentmeasurements is about 1/298.257, a value knownas the polar flattening. The earth thus appearsslightly elliptical in a cross section through the poles.Rotating this ellipse about the polar axis results in athree-dimensional shape known as an ellipsoid. Itis this geometric shape that cartographers use as thereference surface for creating large-scale maps (suchas topographic maps).

While cartographers need a simple geometric rep-resentation of the Earth’s shape, geodesists areinterested in defining a level surface to provide abasis for land surveys. A level surface at any pointis the plane perpendicular to the local direction ofgravity (the direction in which the surveyor’s plumbbob points). Because of local topography and theirregular distribution of mass within the Earth, thelocal direction of gravity may not be exactly per-pendicular to the ideal ellipsoidal surface. Hencethe geoid (the level surface on which gravity is ev-erywhere equal to its strength at mean sea level) isnot perfectly ellipsoidal in shape. Instead ithas smooth, irregular undulations that de-part from the ideal ellipsoid by as muchas 100 meters.

Geodesy is the branch ofscience concerned withmeasuring the size andshape of the Earth.

Equatorial Radius

Pol

ar R

adiu

sSouth Pole

North Pole

The spherical latitude/longitude coordinate systemhas been adaptedmathematically to accountfor Earth’s ellipsoidal shape,yielding geodetic latitudeand longitude values.

Horizontal survey positionsare adjusted to conform toan ideal ellipsoidal surface.The elevations shown ontopographic maps, however,are expressed relative to themean sea level geoid.

Geoid

Ellipsoid

The Earth Ellipsoid(flattening exaggerated)

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Map Projections

Geodetic DatumsSTEPS! from the Options menu

on the View windowchoose Position Report /Projection

! reset the coordinatesystem to UniversalTransverse Mercator

! click [Datum...] on theCoordinate System /Projection Parameterswindow

! examine the list ofavailable geodeticdatums, then click[Cancel] on the SelectDatum window

! click [Cancel] on theCoordinate System /Projection Parameterswindow

Many different reference ellipsoids have been usedby cartographers over the years. Estimates of theellipsoid dimensions that best fit the overall shapeof the Earth have changed as new technologies havepermitted increasingly refined measurements of theplanet. In addition, any global best-fit ellipsoid doesnot fit all parts of the surface equally well becauseof the irregular undulations of the geoid. For thisreason many additional reference ellipsoids havebeen defined for surveying and mapping in differ-ent countries. Each regionally-defined ellipsoid hasbeen chosen to conform as closely as possible to thegeoid over that specific region. The resulting ellip-soids differ in their dimensions, the location of theircenters, and the orientation of their polar axes.

A geodetic datum is an ellipsoidal surface used asa reference for mapping horizontal positions in aparticular country or region. Each datum specifiesthe dimensions, location, and orientation of the ref-erence ellipsoid. The TNT products includespecifications for a great many geodetic datums.When you select a datum, the name of the associ-ated ellipsoid is automatically shown in the Ellipsoidfield of the Coordinate System / Projection Param-eters window.

Planar coordinates used to express positions forgeoreferenced data always refer to a specific geo-detic datum (usually that of the map from which thedata were abstracted). When you import orgeoreference data, be sure to check the datum speci-fication. (You may need to consult the

documentation or metadata that accompaniesthe data). Referencing map coordinates tothe wrong datum can lead to positioning er-rors of tens to hundreds of meters!

The lower part of the SelectDatum window identifies theappropriate region for theselected datum.

Because a geodetic datum specifies a particular ellipsoid, the Ellipsoid buttonis dimmed and inactive when you choose a predefined geodetic datum. (Thebutton is only active when Unspecified is selected from the datum list.)

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Map Projections

North American DatumsThe position of a given setof geodetic latitude andlongitude coordinates inNorth America can shift upto 300 meters with thechange from NAD27 toNAD83. The amount of theshift varies from place toplace. The illustrationshows position N 37° 43'31.84", W 122° 05' 01.03"in the Hayward, Californiaquadrangle plotted withrespect to both datums.

There 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 and orien-tation are specified relative to the Meades Ranchsurvey station, with the result that the ellipsoid isnot 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 thatthe accuracy of the survey network associated withthis datum is not sufficient for many modern needs.Surveying errors, destruction of survey monuments,and horizontal movements of the Earth’s crust haveled to horizontal errors in control point positions aslarge as 1 part in 15,000. The creation of satellite-based positioning systems also now requires the useof a global best-fit ellipsoid centered on the Earth’scenter of mass.

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 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 Projections

Map ProjectionsA 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. Threetypes of developable surfaces form the basis of mostcommon map projections: a cylinder, a cone, or theplane itself.

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.

CylindricalProjection

ConicProjection

AzimuthalProjection

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Map Projections

Map DistortionsProjecting 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. Exact distance measurements from mapfeatures would require constant scale throughoutthe map, but no map projection can achieve this. Inmost projections scale remains constant along oneor more standard lines, and careful positioning ofthese lines can minimize scale variations elsewherein the map. Specialized equidistant map projec-tions maintain constant scale in all directions fromone 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. A map projection with this prop-erty is an equal-area projection.

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 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 projectionsshow all great circles passing through the projec-tion center as straight lines.

Map scale is themathematical relationshipbetween distances on themap and correspondingdistances on the surface.

Equal-Area Projection

Conformal Projection

Shapes are distorted, but allmap features are shownwith the correct relativeareas.

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

A = 1

A = 1

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Map Projections

Choosing a Map ProjectionSTEPS! from the Options menu

on the View windowchoose Position Report /Projection

! click [System...] on theCoordinate System /Projection Parameterswindow

! choose User Definedfrom the CoordinateSystem window andclick [OK]

! click [Projection...] onthe Coordinate System /Projection Parameterswindow

! examine the list ofavailable projections inthe Select Projectionwindow, then click [OK]

! click [Cancel] on theCoordinate System /Projection Parameterswindow

If data you are importing or georeferencing requiresthe use of a map projection other than those pro-vided by the standard predefined coordinate systemoptions, choose User-defined as the system option.The Projection option button on the Coordinate Sys-tem / Projection Parameters window is then enabled.This button opens the Map Projection Selection win-dow, which lists a number of commonly usedprojections. Some representative examples are dis-cussed on the pages that follow, and illustrated usingmaps of North America with a superimposed lati-tude-longitude grid.

If necessary, georeferenced data can be transformedto a new map projection and coordinate system moreappropriate to the location, size, and shape of theproject area, and to the anticipated use of the data.A particular projection can be centered on the projectarea by choosing appropriate projection parameters.(The parameters vary depending on the projectiontype; the principal parameters for cylindrical, conic,

and azimuthal projections are discussed withthe examples that follow.) Because of vary-ing patterns of distortion, some projectionsare better for areas elongate in an east-westdirection, and others for areas elongate north-south. Large-scale maps used to determineor plot directions, such as navigational chartsand topographic maps, should use a confor-mal map projection. Equal-area projectionsare appropriate for smaller-scale thematic ordistribution maps.

When you select User-Defined for the coordinatesystem, the Projection Parameters panel opensautomatically. The parameters vary depending on theprojection you select. In general, they determine thegeographic positioning of the projection. All projectionsallow you to specify a False Northing and a FalseEasting value for the origin. When you import data in aspecific projection, be sure to consult the accompany-ing metadata for the relevant projection parameters.

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Map Projections

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. Thespacing between parallels increases away from the Equator to produce a confor-mal projection. The scale is true along the equator for a tangent Mercatorprojection (specified by the default setting for the True Scale Latitude param-eter). Assigning a different true scale latitude produces an intersecting cylindricalprojection with two standard parallels (with true scale) equidistant from the equator.

The poleward increase in spacing of parallels produces great distortions of areain high-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.

Projection Parameters:

• Central Meridian

• True-Scale Latitude

page 16

Map Projections

Transverse Mercator Projection

Transverse Mercatorprojection of NorthAmerica with centralmeridian at 100° westlongitude.

The Transverse Mercator projection is a conformal cylindrical projection withthe cylinder rotated 90 degrees with respect to the regular Mercator projection.The cylinder is tangent to a central meridian of longitude around its entire cir-cumference. The central meridian and equator are straight lines, but all othermeridians and 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.It forms the basis for the Universal Transverse Mercator Coordinate System, andis primarily used for large-scale (1:24,000 to 1:250,000) quadrangle maps. Thecentral meridian can be mapped at true scale (Central Scale parameter = 1.0), orat a slightly reduced constant scale (for example, the value 0.9996 used in theUTM system). In the latter case a pair of meridians bracketing the central onemaintain true scale, and the mean scale for the entire map is closer to the truescale.

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. In Europe it is sometimes called the GaussConformal or Gauss-Kruger projection.

Projection Parameters:

• Central Scale

• Central Meridian

• Origin Latitude

page 17

Map Projections

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.

Projection Parameters:• South Standard Parallel

• North Standard Parallel

• Central Meridian

• Origin Latitude

Lambert ConformalConic projection ofNorth America withcentral meridian at 100°west longitude, andstandard parallels at20° and 60° north

page 18

Map Projections

Albers Conic Equal-Area Projection

Albers Conic Equal-Area projection 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 latitude29.5º and 45.5º north as standard parallels. Mid-latitude distortion is minor formost normal conic projections, so that differences between them become obvi-ous only when the region mapped extends to higher or lower latitudes.

Projection Parameters:• South Standard Parallel

• North Standard Parallel

• Central Meridian

• Origin Latitude

page 19

Map Projections

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.Each parallel is the trace of a unique cone tangent to the globe at that latitude.The name 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 perpendicu-lar to 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.

Projection Parameters:• Central Meridian

• Origin Latitude

page 20

Map Projections

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 Center Longitude and Center Latitudeparameters. In general, the projection center should coincide with the center ofthe area to be mapped. Scale is true only at the center point, but deviation fromtrue scale for other points on the map is less than for other forms of azimuthalprojection.

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.

Projection Parameters:• Center Longitude

• Center Latitude

page 21

Map Projections

Stereographic Projection

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

The Stereographic projection is a conformal azimuthal projection. When usedfor large areas, so that the spherical Earth model can be used, it is also a trueperspective projection, unlike most map projections. Surface locations are pro-jected to a tangent plane using a single projection point on the surface of thesphere exactly opposite the center of the projection. When used to map smallerareas, so that the ellipsoidal Earth model is used, the projection is not perspec-tive, and, in order to maintain 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. (This Polar Sterographic formis provided as a separate selection in the Map Projection Selection window). Inthis form, the map shows meridians as straight radii of concentric circles repre-senting parallels. In an oblique Stereographic projection, such as the illustrationbelow, only the central meridian is straight. All other meridians and parallels arecircular arcs intersecting at right angles. Scale increases away from the centralpoint, which normally has true scale (Central Scale parameter = 1.0). Reducingthe Central Scale value produces an intersecting rather than tangential plane.Scale is then true along an ellipse centered on the projection center, and the meanscale for the entire map is closer to the true scale.

Projection Parameters:• Center Longitude

• Center Latitude

• Central Scale

page 22

Map Projections

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 to aspecific map coordinate system.

The Auto-Match option determines whetherobject coordinates are used or not, and, formultiple layers, which object provides thebasis for the group projection. To orient thegroup to a map projection, choose [None]

(or [First Raster or None] if there is no raster layerin the group). The Projection button then becomesactive, allowin you to open the Coordinate System /Projection Parameters window described earlier.

A vector, CAD, or TIN object in the display groupis projected precisely to the selected map coordi-nates, subject to the inherent accuracy of the elementcoordinates. Reorientation of a raster object is gov-erned by the Positional Accuracy option in the Viewwindow’s Option menu. The Exact option provides

an exact resampling to the projection, but can beslow. The quicker Affine Approximate optionchanges the orientation and scale in a uniform man-ner to approximately match the output projection.The Auto-Select option automatically chooses theappropriate method for the current zoom level.

As you gather spatial data from different sources for a project, you will probablyend up with data in a number of different coordinate systems and projections.Although the Display process can overlay layers with different map projectionsand coordinate systems with reasonable registration, redisplay times can belong. For maximum efficiency, materials that will be used together routinelyshould be reprojected to a common map projection and coordinate system.For raster images, choose Process / Raster / Resample / Automatic from theTNTmips main menu. For vector objects, choose Process / Vector / Warp.

In 2D Group or 3D GroupDisplay, open the GroupSettings window bychoosing Projection /Clipping from the Groupmenu. If you are viewing aLayout, click theGroup Settings iconbutton for the group.

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.

Map Projections

page 23

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.

NIMA Geospatial Sciences Publications:

http://www.nima.mil/GandG/pubs.html

This page at the U.S. National Imagery and Mapping 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).

page 24

Map ProjectionsAdvanced Software for Geospatial Analysis PROJECTIONS

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IndexAuto-match option....................................22coordinate system.............................3-5,8,14

geographic........................................5,8latitude/longitude............................5,8,9United States State Plane...................5,7Universal Transverse Mercator..........5,6

datum...............................................4,7,10,11GRS1980...........................................11North American 1927 (NAD27).........11North American 1983 (NAD83).........11World Geodetic System 1984..............11

developable surface...................................12distortion..................................................3,13easting......................................................5,6ellipsoid...............................................4,9,10geoid...........................................................9georeferencing.............................................3meridian

central.......................................6,7,15-19prime....................................................8

northing....................................................5,6origin latitude........................................16,17parallel, standard.....................................17,18projections

Albers conic equal area......................18azimuthal......................................12,13conformal...........................................13conic..................................................12equal-area...........................................13equidistant.........................................13Lambert azimuthal equal-area............20Lambert conformal conic...................7,17Mercator..........................................6,15polyconic...........................................19stereographic......................................21transverse Mercator..........................6,16

scale, map..................................................13true-scale latitude......................................15zone, geographic......................................5-7