6.4 MAP PROJECTIONS FOR LARGER-SCALE MAPPING* "ohn P. Snyder U.S. Geological Survey National Center, Stop 522 Reston, Virginia 22092 ABSTRACT After decaaes of using a single map projection, thc Polyconic, for its mp- i.ing program, the U.S. Geological Survey now uses several long-establjshcd ,-ojections for its published maps, both large and mal? scale. Tor naps cc 1:1,000,OCO-scale and larger, the most coimon projections are confowa:, such as the Transverse Mercator and Lambert Confornal Conic. projections for these scales should treat the Earth as an ellipsoid. In addition, the 'JSGS has conceived and designed some new projections, inc:udjng the Tpace Ob1 ique Vercator, the first nap projection designed to permit low-distortion {lapping of the Earth from satellite inagery, continucasly Collowing the graundtrack. The bSGS has programed nearly a1 1 yrti nent project ion equations for inverse and foniard calculations. These are used to plot naps or to transfom coordinates froill one projection to another. The CIS5 is also publishing its first comprehensive slap projxtion rcanual, describinq in detail and matheciatical ly all projections used by the agerlcy. The U.S. Geoloyical Scrvey wds created in 187c), and detailed large-scale napping 3f the country soon becane one of its primary objectives. Tt has relied heavily over the years 9n the foriner U.S. roast and Geodetic Survey (!JSC&GS, now the Yational Oceln Survey) &or guidance on nap projectiqns. Ati? the 1;te 1950's, only the Polyconic projection was used for the prirnary iJSGS napping product, i .e., large-scale quadrangle naps. The Polyconic projection was apparently 'nvented and certainly pronoted 5y f-erd-inand Rudolph Hassler, the first head of what was to hecoine known as the Coast and Geodetic Survey. In the 1959's, the USG5 quadrangle projection was changed to the Lanbert Co:iformal Conic and t.he Transve-w Mercator projec- tiocs, which had been adopted by the Coast and Geodetic Survey in the IQ3!7's for the State Plane Coordinate System. The developent of standardized zones bdsed upon the Universal Transverse Yereator and the polar Stereographic led to !JSGS use of these projections. Trl addition to these, the regii?ar Mercator, the Ob? ique Mercator, the Alhex Equal-Area hnic, and the .4zimith?l Equidistant have been used for other larger-scale nappinq by the IlSrS and other agencies. Although authors and organizations variously de <'inelarge- fYote: This paper is adaiited with sever31 ivodi'ications frmi one by the aiithor published it1 the ?roceedings of the 1381 ACSM Fall Technical Yeeting. and intermediate-scale mapping, for the purposes of this paper, the term "larger scale" will apply to scales larger than 1 to 2 million. When the space age added its impact to mapping, classical projections (Yercator, Lambert Conformal Conic, and Stereographic) were chosen fx the mapping of the Earth's Moon, three other planets, ana a nwber of other natural satellites. Some projections, especially the Space Oblique Mercator, originated within IlSGS to assist [napping from satellite imagery. TYPES OF PROJECTION Before describing the projections themselves, I 'd 1 ike to review Sriefly the different types. Equal-area or equivalent projections of the globe are used especially by geographers seeking to conpare land use, densities, and t.he like. On an equal-area projection, such as the Albers Equal-Area Con'c, a coin laid on one part of the map covers exactly the same area of the actual Earth as the same coin on any other part of the map. Shzpes, angles, and scale mst be distorted on most parts of such a map, but there are usually certain lines on an equal-area map as well as on other types of projections, along which there is no distortion Qfany kind. These so-called 'lst.dndard lines" nay be a merjdian, one or two parallels, lines which $re neither, or not a line but a point. More cononly used n larger-scale napping are conformal (orthmorphic) projections such as the Transverse Mercator and the Lambert Confornal Conic. The term nears that they are correct in shape, but, unlike the term "equal srea," :he conforma principle applies only to each infinitesimal elenent oc the nap. A.nqles at each point are correct, and consequently the local scale in every direction around any one point is constant, so the nap user can measure distance and direction between near points with a nininum oc difci- culty. Conformal maps mcy also be prepared by fitting together small pieces of other conformal nacs which have been enlarged or reduced; non-confornal projections rcqoire reshaping as well. When the region consists of more than a small element, distortioa in shape as well as area becomes apprecfable. This is especi&l!y serious with the most famous conformal projection -- the Mercator -- because cf its widespread use in classrooms, especiallj in the past. Recause tha-e is no angular distortion, all merid';;s intersect. parallels at right angles on a conformal pro<jection, just as the, 40 on the Earth. "Standard lines" may also be applied to a conformal ri,p to el irni- nate scale and area distortion along these lines and to t,iinimize distortion elsewhere. Sone nap projectims, such as the Azimuthal Equidistant, aye neither cqual- area nor confon,ial, but linear scale is correct along all lines radiating from the center, along mer'ldians, or following other special patterns. In additiotl, there are compromise projections, almost eqtirely restricted to srnall-scale mapping, which are used because they balance distortion in scale, area, and snape. 225 Projections are often classified by the type of surface onto which the Earth may be mapped. If a cylinder or cone that has been placed around a globe is unrolled, we have the concept gf cylindrical or conic projections, such as the Mercator or Lambert Confomal Conic, respectively. If the axis of the cone or cylinder coincides with the polar axis of the globe, the projection has equally spaced straight meridians, paral le1 on the cy1 indrical projections and converging on the conics. The paral le1s intersect the rwri- diaris at right dngles, being straight on the cylindricals and concentric circular arcs on the conics. The spacing of the parallels is seldorii projective. A plane laid tangent to the globe at the pole leads to polar azimlthal projec- tions, such as the polar Stereographic, with the parallels mapped as arcs of concentric circles and meridians as equally spaced rad'i of the circlt-1s. Scale reinains constant alony each parallel of latitude on a regular cylindrical, conic, or polar azimuthal projection, hut it changes one latitude to another. Directions of all points are correct as seen frog the center oc an azimuthal projection. If the cylinder or cone is secant instead of tangent to the glohe, the projec- tion conceptually has tw lines instead of one which are true to scale. Wrapping the cylinder about a meridian le,ids to tr3nsverse projections. Ry placiny a plane tangent to the Equator ixtead of a pole, equatorial aspects of azimuthal projections result. Tilting the cylinder, cone, or plane t,o reldte to another jioint on the Earth leads to an oblique projection, and the cieridians dnJ paral!els are 113 longer the straight lines or circular arcs they were in the norr:ial aspect. The lines of constant scale are correspond- inyly rotated. THE EAKTti h, IN ELLIPWID For i:iaps at scales si;ialler than 1:5,000,000, arld which cover regions larqer tha!i the Ilnited States , the distortions frwi i:iapping the spherical Earth' on flat p3per- are much greater than the slight additional corrections needed to coiqierisiite for the el 1 ipsoidal shape of the Farth. These corrections my then usually be ignored. The ellipsoid should be, and normdlly is, used for la;-ge-scale mapping of sm11 arms, or for loriq narroif strips. For such areas, the flattening of the round Earth (isual ly produces less distortion than the use of the sphere instead of the ellipsoid. A shift froin one ellipsoid to another has a negligible effect, ever1 on large- scale naps, upon the projected shdpes and positions of meridians and paral?cls. A yreater effect is the translat'on of latitude and longitude for all point,s 011 a liap, due to a change in datiii:i, that chan!,zs the position of +.he ellipsoid re1,qtive to the Larth. For this reason, the natation in tbe corsrier o+' !IS@, quadrangles stating "North nrerican Pat.uin 1927'' or "1933" is as important. ds the i.iardiiltltCrs of the vidp projection in defirii ng the basis of these naps. PR! NC IPAL PROJCCTIt!NS Polyconic Project icn About 18?0, Hdssler begdn to proi:iot.e the edsi ly const.ructed Polyconic ORIGINAL PAGE IS OF POOR QuAm c 0 .c Y 227 Gktiii4AL PAGE 13 OF POOR QUALITY projection as the basis of large-scale mapping. The USGS used this projec- tion for the earliest quadrangles, only changing in the 1959’s to other projections, although relabeling the map legend lagged considerably behind the change. The Polyconic is neither equal-area nor conformal. For 7-1/2- and 15-min. quadranyles, however, the distortion is negligible. Along the central meridian, it is free of distortion. Each parallel is also true to scale, but the other meridians are too long, and constantly change scale. The projection is not recoirunended for maps of considerable east-west extent cod, in fact, should not be seriously used for any new maps in view of other projections available. The parallels of latitude are circular arcs spaced at their true distances along the central meridian, but with radii equal to the length of the element of a cone tangent at the particular parallel.