28NAUTICAL CHARTSthesimpleconic projection isnot perspectivesince onlytheparallels areconcentric circles.Thedistance alonganyme-ridian between consecutiveparallels is in correctrelationtomeridiansareprojectedgeometrically,eachbecominganthe distance on the earth, and, therefore, can be derivedelementofthecone.Whenthisprojection isspreadoutflatmathematically.The pole isrepresented by a circle (Figureto forma map,themeridians appear as straight lines con-312b).The scaleis correct along anymeridian and alongverging attheapex of the cone.The standard parallel.the standard parallel. All other parallels are too great inwhere the cone is tangent to the earth, appears as the arc oflength, with the errorincreasingwith increased distancea circle with its center at the apex of the cone.The otherfromthestandard parallel.Sincethescale isnotthe same inall directions about every point, the projection is neither aconformal norequal-areaprojection.Its non-conformal na-ture is its principal disadvantagefor navigation.Since the scale is correct along the standard paralleland variesuniformlyon eachside,withcomparativelylittledistortion near the standard parallel, this projection is usefulfor mapping an area covering a large spread of longitudeandacomparativelynarrowbandoflatitude.Itwasdevel-oped by Claudius Ptolemy in the second century A.D. tomap just such an area: the Mediterranean Sea.313.LambertConformalProjectionTheuseful latituderangeofthe simpleconicprojectioncan be increased by using a secant cone intersecting theearth at twostandard parallels.SeeFigure313.The area be-tween thetwo standard parallels is compressed, and thatbeyond isexpanded.Suchaprojection is called either a se-cant conic or conic projection with two standardparallels.Figure312a.Asimpleconicprojection.90'E12060°sO'E150'e518090'W150W12OM20WFigure312b.AsimpleconicmapoftheNorthernHemisphere
28 NAUTICAL CHARTS the simple conic projection is not perspective since only the meridians are projected geometrically, each becoming an element of the cone. When this projection is spread out flat to form a map, the meridians appear as straight lines converging at the apex of the cone. The standard parallel, where the cone is tangent to the earth, appears as the arc of a circle with its center at the apex of the cone. The other parallels are concentric circles. The distance along any meridian between consecutive parallels is in correct relation to the distance on the earth, and, therefore, can be derived mathematically. The pole is represented by a circle (Figure 312b). The scale is correct along any meridian and along the standard parallel. All other parallels are too great in length, with the error increasing with increased distance from the standard parallel. Since the scale is not the same in all directions about every point, the projection is neither a conformal nor equal-area projection. Its non-conformal nature is its principal disadvantage for navigation. Since the scale is correct along the standard parallel and varies uniformly on each side, with comparatively little distortion near the standard parallel, this projection is useful for mapping an area covering a large spread of longitude and a comparatively narrow band of latitude. It was developed by Claudius Ptolemy in the second century A.D. to map just such an area: the Mediterranean Sea. 313. Lambert Conformal Projection The useful latitude range of the simple conic projection can be increased by using a secant cone intersecting the earth at two standard parallels. See Figure 313. The area between the two standard parallels is compressed, and that beyond is expanded. Such a projection is called either a secant conic or conic projection with two standard Figure 312a. A simple conic projection. parallels. Figure 312b. A simple conic map of the Northern Hemisphere
29NAUTICALCHARTSThepolyconicprojection iswidelyused in atlases,par-ticularly for areas of large range in latitude and reasonablylargerangein longitude,suchas continents.However,sinceit is not conformal, this projection is not customarily usedinnavigation.315.AzimuthalProjectionsIf points on theearthareprojecteddirectlytoaplane sur-face, a map is formed at once, without cutting and flattening,or"developing"This can be considered a special case of a conicprojection in whichthe cone has zero heightThe simplest case of the azimuthal projection is one inwhich the plane is tangent at one ofthepoles.The meridians arestraightlinesintersectingatthepole,andtheparallelsareconcentric circles with their common center at the pole.Theirspacingdepends uponthemethodusedtotransferpointsfromtheearthtotheplaneIf theplane istangent at somepoint other than apolestraight lines through the point oftangency are great circles,and concentriccircleswiththeircommon center atthepointof tangency connect points of equal distance from thatpoint. Distortion, which is zero at the point of tangency, in-Figure313.A secantcone for a conic projection with twocreases along anygreat circle through this point.Along anystandard parallels.circle whose center is the pointof tangency,the distortionis constant.The bearing ofany point from thepoint of tan-If in such a projection the spacing of the parallels is al-gencyiscorrectlyrepresented.It isforthisreasonthatthesetered, suchthatthedistortionisthesamealongthemasprojections are called azimuthal. They are also called ze-nithal Several of the common azimuthal projections arealongthemeridians,theprojectionbecomesconformalThis modification produces theLambert conformal pro-perspective.jection.If the chart is not carried far beyond the standardparallels, and if these are not a great distance apart, the dis-316.GnomonicProjectiontortion over the entire chart is small.A straight line on this projection so nearly approximates aIfaplaneis tangentto theearth,andpoints areprojectedgreat circlethat thetwo are nearlyidentical.Radiobeacon siggeometricallyfrom the center oftheearth, the result is agnonals travel great circles, thus, they can be plotted on thismonic projection. See Figure 316a. Since the projection isprojection without corection.Thisfeature,gained without sac-perspective, it canbe demonstratedbyplacing a light at therificing conformality,has madethis projectionpopularforcenter ofa transparent terrestrial globe and holding aaeronauticalcharts becauseaircraff make wide use ofradioaidsto navigation.Except in highlatitudes,where a slightlymodifiedform ofthis projection has been used for polar charts,it has notreplacedtheMercatorprojectionformarinenavigation.314.Polyconic ProjectionThe latitude limitations of the secant conic projection canbe minimized by using a series of cones. This results in a poly-conic projection.In this projection, each parallel is thebaseofatangentcone.Atthe edgesofthechart, theareabetweenparallels is expanded to eliminate gaps. The scale is correct alongany parallel and along the central meridian of the projectionAlong other meridians the scale increases with increased differ-ence of longitudefrom the central meridian.Parallels appear asnonconcentric circles; meridians appear as curved lines con-Figure316a.An obliquegnomonicprojection.verging toward thepoleand concavetothecentral meridian
NAUTICAL CHARTS 29 If in such a projection the spacing of the parallels is altered, such that the distortion is the same along them as along the meridians, the projection becomes conformal. This modification produces the Lambert conformal projection. If the chart is not carried far beyond the standard parallels, and if these are not a great distance apart, the distortion over the entire chart is small. A straight line on this projection so nearly approximates a great circle that the two are nearly identical. Radio beacon signals travel great circles; thus, they can be plotted on this projection without correction. This feature, gained without sacrificing conformality, has made this projection popular for aeronautical charts because aircraft make wide use of radio aids to navigation. Except in high latitudes, where a slightly modified form of this projection has been used for polar charts, it has not replaced the Mercator projection for marine navigation. 314. Polyconic Projection The latitude limitations of the secant conic projection can be minimized by using a series of cones. This results in a polyconic projection. In this projection, each parallel is the base of a tangent cone . At the edges of the chart, the area between parallels is expanded to eliminate gaps. The scale is correct along any parallel and along the central meridian of the projection. Along other meridians the scale increases with increased difference of longitude from the central meridian. Parallels appear as nonconcentric circles; meridians appear as curved lines converging toward the pole and concave to the central meridian. The polyconic projection is widely used in atlases, particularly for areas of large range in latitude and reasonably large range in longitude, such as continents. However, since it is not conformal, this projection is not customarily used in navigation. 315. Azimuthal Projections If points on the earth are projected directly to a plane surface, a map is formed at once, without cutting and flattening, or “developing.” This can be considered a special case of a conic projection in which the cone has zero height. The simplest case of the azimuthal projection is one in which the plane is tangent at one of the poles. The meridians are straight lines intersecting at the pole, and the parallels are concentric circles with their common center at the pole. Their spacing depends upon the method used to transfer points from the earth to the plane. If the plane is tangent at some point other than a pole, straight lines through the point of tangency are great circles, and concentric circles with their common center at the point of tangency connect points of equal distance from that point. Distortion, which is zero at the point of tangency, increases along any great circle through this point. Along any circle whose center is the point of tangency, the distortion is constant. The bearing of any point from the point of tangency is correctly represented. It is for this reason that these projections are called azimuthal. They are also called zenithal. Several of the common azimuthal projections are perspective. 316. Gnomonic Projection If a plane is tangent to the earth, and points are projected geometrically from the center of the earth, the result is a gnomonic projection. See Figure 316a. Since the projection is perspective, it can be demonstrated by placing a light at the center of a transparent terrestrial globe and holding a Figure 313. A secant cone for a conic projection with two standard parallels. Figure 316a. An oblique gnomonic projection
30NAUTICALCHARTSflat surfacetangenttothesphereIn an oblique gnomonic projection the meridians ap-pear as straight lines converging toward the nearer pole.The parallels, except the equator, appear as curves (Figure316b).Asinallazimuthalprojections,bearingsfromtheR8point of tangency are correctly represented.The distancescale, however, changes rapidly,Theprojection is neitherconformal nor equal area.Distortion is so great thatshapesas wellas distances and areas,areverypoorly representedexcept near the pointof tangencyFigure317a.Anequatorial stereographicprojectionFigure 316b.An oblique gnomonic map with point oftangency at latitude30°N, longitude90°W.The usefulness of this projection rests upon the factthat any great circle appears on the map as a straight line,giving charts made on this projection thecommon namegreat-circlechartsGnomonic charts aremostoftenusedforplanningthegreat-circle track between points.Points along thedeter-mined track are then transferred to a Mercator projection.Thegreat circle is thenfollowedbyfollowingtherhumblinesfromonepointtothenext.Computerprogramswhichautomatically calculate great circle routes betweenpointsand providelatitudeandlongitudeofcorresponding rhumblineendpoints arequicklymakingthis use ofthegnomonicchartobsolete.317.StereographicProjectionFigure 317b.A stereographic map of the WesternA stereographic projection results from projectingHemisphere.points on the surfaceoftheearth onto a tangent plane,fromapointonthesurfaceoftheearthoppositethepoint oftan-great circles through the point of tangency appear asgency (Figure 317a). This projection is also called anstraight lines.Other circles such as meridians and parallelsazimuthal orthomorphicprojectionappearas either circles or arcs of circles.The scale of the stereographic projection increasesThe principal navigational use of the stereographicwith distance from the point of tangency, but it increasesprojection is for charts of the polar regions and devices formore slowlythan in the gnomonic projection.The stereo-mechanical or graphical solutionof the navigational trian-graphic projection can showan entire hemispherewithoutgle. A Universal Polar Stereographic (UPS)gridexcessive distortion (Figure317b).As inother azimuthalmathematically adjusted to the graticule, is used as areferprojections,encesystem
30 NAUTICAL CHARTS flat surface tangent to the sphere. In an oblique gnomonic projection the meridians appear as straight lines converging toward the nearer pole. The parallels, except the equator, appear as curves (Figure 316b). As in all azimuthal projections, bearings from the point of tangency are correctly represented. The distance scale, however, changes rapidly. The projection is neither conformal nor equal area. Distortion is so great that shapes, as well as distances and areas, are very poorly represented, except near the point of tangency. The usefulness of this projection rests upon the fact that any great circle appears on the map as a straight line, giving charts made on this projection the common name great-circle charts. Gnomonic charts are most often used for planning the great-circle track between points. Points along the determined track are then transferred to a Mercator projection. The great circle is then followed by following the rhumb lines from one point to the next. Computer programs which automatically calculate great circle routes between points and provide latitude and longitude of corresponding rhumb line endpoints are quickly making this use of the gnomonic chart obsolete. 317. Stereographic Projection A stereographic projection results from projecting points on the surface of the earth onto a tangent plane, from a point on the surface of the earth opposite the point of tangency (Figure 317a). This projection is also called an azimuthal orthomorphic projection. The scale of the stereographic projection increases with distance from the point of tangency, but it increases more slowly than in the gnomonic projection. The stereographic projection can show an entire hemisphere without excessive distortion (Figure 317b). As in other azimuthal projections, great circles through the point of tangency appear as straight lines. Other circles such as meridians and parallels appear as either circles or arcs of circles. The principal navigational use of the stereographic projection is for charts of the polar regions and devices for mechanical or graphical solution of the navigational triangle. A Universal Polar Stereographic (UPS) grid, mathematically adjusted to the graticule, is used as a reference system. Figure 316b. An oblique gnomonic map with point of tangency at latitude 30°N, longitude 90°W. Figure 317a. An equatorial stereographic projection. Figure 317b. A stereographic map of the Western Hemisphere
31NAUTICALCHARTS318.OrthographicProjectionlines andtheparallels as equally spaced concentriccirclesIf theplane is tangentat somepoint other thana pole,theIfterrestrial points are projected geometrically from in-concentric circles representdistances from the point oftan-finity to a tangent plane, an orthographic projectiongency. In this case, meridians and parallels appear as curves.results(Figure318a).Thisprojection is not conformal; norThe projection can be used to portray the entire earth, thedoes it result in an equal area representation. Its principalpoint 180°from the point oftangency appearing as the largestuse is in navigational astronomy because it is useful for il-of the concentric circles.Theprojection is not conformal,lustrating and solving the navigational triangle.It is alsoequal area,orperspective.Nearthepointoftangencydistor-useful for illustrating celestial coordinates.If the plane istion is small, increasing with distance until shapes near thetangentatapoint on the equator,theparallels(including theopposite side oftheearthare unrecognizable (Figure319)equator)appearas straight lines.Themeridians would apThe projection is useful because it combines the threepear as ellipses, exceptthatthemeridianthroughthepointfeatures ofbeingazimuthal,havingaconstantdistancescaleoftangencywould appearasastraight lineand theone90°fromthepointoftangencyand permitting theentireearthtoaway would appear as a circle (Figure318b).be shownon onemap.Thus, ifan importantharbororairportis selected as thepoint of tangency,thegreat-circle course,distance,and track from thatpointtoany otherpoint on the319.AzimuthalEquidistantProjectionearth arequicklyand accuratelydetermined.For communi-cation work withthe station atthepointoftangency,thepathAn azimuthal equidistant projection is an azimuthalof an incoming signal is at once apparent if the direction ofprojection in which the distance scale along any great circlearrival has been determined and thedirectionto train a direc-through the point of tangency is constant. If a pole is thetional antenna can be determined easily.The projection ispoint of tangency, the meridians appear as straight radialalso usedforpolar charts andforthestar finder,No.2102DFigure318a.Anequatorialorthographicprojection.Figure318b.AnorthographicmapoftheWesternHemisphere
NAUTICAL CHARTS 31 318. Orthographic Projection If terrestrial points are projected geometrically from infinity to a tangent plane, an orthographic projection results (Figure 318a). This projection is not conformal; nor does it result in an equal area representation. Its principal use is in navigational astronomy because it is useful for illustrating and solving the navigational triangle. It is also useful for illustrating celestial coordinates. If the plane is tangent at a point on the equator, the parallels (including the equator) appear as straight lines. The meridians would appear as ellipses, except that the meridian through the point of tangency would appear as a straight line and the one 90° away would appear as a circle (Figure 318b). 319. Azimuthal Equidistant Projection An azimuthal equidistant projection is an azimuthal projection in which the distance scale along any great circle through the point of tangency is constant. If a pole is the point of tangency, the meridians appear as straight radial lines and the parallels as equally spaced concentric circles. If the plane is tangent at some point other than a pole, the concentric circles represent distances from the point of tangency. In this case, meridians and parallels appear as curves. The projection can be used to portray the entire earth, the point 180° from the point of tangency appearing as the largest of the concentric circles. The projection is not conformal, equal area, or perspective. Near the point of tangency distortion is small, increasing with distance until shapes near the opposite side of the earth are unrecognizable (Figure 319). The projection is useful because it combines the three features of being azimuthal, having a constant distance scale from the point of tangency, and permitting the entire earth to be shown on one map. Thus, if an important harbor or airport is selected as the point of tangency, the great-circle course, distance, and track from that point to any other point on the earth are quickly and accurately determined. For communication work with the station at the point of tangency, the path of an incoming signal is at once apparent if the direction of arrival has been determined and the direction to train a directional antenna can be determined easily. The projection is also used for polar charts and for the star finder, No. 2102D. Figure 318a. An equatorial orthographic projection. Figure 318b. An orthographic map of the Western Hemisphere
32NAUTICALCHARTS9091008OS0300003000C6060Figure319.An azimuthal equidistant mapof theworld with the point of tangency latitude 40°N, longitude 100°WPOLARCHARTS320.PolarProjectionsthrough a full 3600without stretching or resuming its formerconical shape.Theusefulness ofsuchprojections is also limitedSpecial consideration isgivento the selection of pro-bythefactthat thepole appears as anarc ofacircle instead ofajections forpolarcharts because thefamiliar projectionspoint. However, by using a parallel very near the pole as thebecome special cases with unique features.higher standard parallel,a conic projection with two standardparallels can be made.Thisrequires little stretching to completeIn the case ofcylindrical projections in which the axis ofthethe circles oftheparallels and eliminate that ofthepole.Such acylinder is parallel to the polar axis of the earth, distortion be-projection, called a modified Lambert conformal or Ney'scomesexcessiveandthescalechangesrapidly.Suchprojectionsprojection, is useful for polar charts. It is particularly familiar tocannotbecarried tothepoles.However,boththetransverseandoblique Mercator projections areused.thoseaccustomedtousingtheordinaryLambert conformalcharts in lower latitudes.Conic projections with their axes parallel to the earth's po-laraxisarelimitedintheirusefulnessforpolarchartsbecauseAzimuthal projections areintheir simplestform whenparallels of latitude extending through a full 360° of longitudetangent at a pole.This is because the meridians are straightappear as arcs of circles rather than full circles. This is because alines intersecting at the pole,andparallels are concentriccone,when cut along an elementand flattened, does not extendcircles withtheir commoncenteratthepole.Withinafew
32 NAUTICAL CHARTS POLAR CHARTS 320. Polar Projections Special consideration is given to the selection of projections for polar charts because the familiar projections become special cases with unique features. In the case of cylindrical projections in which the axis of the cylinder is parallel to the polar axis of the earth, distortion becomes excessive and the scale changes rapidly. Such projections cannot be carried to the poles. However, both the transverse and oblique Mercator projections are used. Conic projections with their axes parallel to the earth’s polar axis are limited in their usefulness for polar charts because parallels of latitude extending through a full 360° of longitude appear as arcs of circles rather than full circles. This is because a cone, when cut along an element and flattened, does not extend through a full 360° without stretching or resuming its former conical shape. The usefulness of such projections is also limited by the fact that the pole appears as an arc of a circle instead of a point. However, by using a parallel very near the pole as the higher standard parallel, a conic projection with two standard parallels can be made. This requires little stretching to complete the circles of the parallels and eliminate that of the pole. Such a projection, called a modified Lambert conformal or Ney’s projection, is useful for polar charts. It is particularly familiar to those accustomed to using the ordinary Lambert conformal charts in lower latitudes. Azimuthal projections are in their simplest form when tangent at a pole. This is because the meridians are straight lines intersecting at the pole, and parallels are concentric circles with their common center at the pole. Within a few Figure 319. An azimuthal equidistant map of the world with the point of tangency latitude 40°N, longitude 100°W