34 PUBLICATIONS of the CELESTIAL CARTOGRAPHY By

34 PUBLICATIONS OF THE CELESTIAL CARTOGRAPHY By Roy S. Farmer Abstract.—The purpose of this discussion of celestial cartography is to increase the appreciation of the proper application of maps to the demonstration or interpretation of problems relating to the distribution of celestial objects. The divisions of celestial mapping considered in the paragraphs that follow may be classified into five groups: (1) The principles of the projection of maps on to the developable surfaces of planes, cylinders, cones, and mathematical figures ; (2) the relevant features of selected map projections ; (3) the uses of celestial maps ; (4) obtaining the desired co-ordinate system of maps of the celestial sphere ; (5) a few suggestions concerning the construction of maps; (6) bibliography. PRINCIPLES OF MAP PROJECTIONS Since the continuously curving surface of the imaginary celestial sphere cannot be developed graphically or mathemati- cally into a continuous plane surface (map) without distortion of one or more of several involved irreconcilables, it is apparent that any sphere-to-plane projection must be a compromise between the elements of the projection. Certain types of maps may show correctly the scale or area or azimuth of plotted objects. But a single map cannot present, in a wholly undistorted form, all three elements of the ideal map. For this reason, the choice of a projection to be used for any one map is arbitrary and may be determined only when a desirable balance is found between the type of data to be shown, the area to be covered in one map, and the most relevant feature (scale, area, or azimuth) to be demonstrated. The most simple types of projection, which are adapted only to the presentation of small angular inclusions of a few degrees in declination and right ascension, are those of a plane tangent to, and receiving the projection from, the celestial sphere. Fig- ure 1 represents four projections fulfilling such a set of conditions. In each of the examples shown, the plane tangent to the sphere receives the projection in a manner that is determined by the location of the point of origin of the projection. The Gnomic projection, which has its point of origin at the center of the sphere, projects the true distances from the sphere to © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System ASTRONOMICAL SOCIETY OF THE PACIFIC 35 the plane in a distorted manner so they become too large when measured on the plane, as will be seen by the difference in length between the tangent solid line (representing a tangent plane) METHODS OF PROJECTION \SX/ /a/ // \ ' u 1 \ > ' 1 V/ / GNOMIC STEREOGRAPHIC \ V// ι ! Κ \\ i / U \ i / / / \ L' CLARKE'S ORTHOGRAPHIC Fig. I.—For simplifying the conditions under which tangent plane projections may be demonstrated, the tangent solid lines in this figure represent two-dimensional planes in contact with a pole of the spheres. In practice, the sphere is imaginary and the pole of projection (point of tangency) may be placed at any point on the imaginary sphere. The out- lined bar above each method of projection indicates the true length of the projected arc when measured on the surface of the sphere. The Gnomic projection distorts co-ordinates most, while Clarke's projection is one of minimum error. © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System 36 PUBLICATIONS OF THE and the hollow line just above it which represents the true linear arc distance when measured on the surface of the sphere of the projection. The Stereographic projection places the point of origin at the sphere's surface opposite to the pole of projection between sphere and plane. Distances projected by the Stereographic method are still too great, but the error of dis- crepancy is decreasing, and in Clarke's projection is found the minimum distortion of scale. The fourth type of projection shown in Figure 1 is the Orthographic, which is based on the use of parallel lines to carry the projection from sphere to plane. In this method of construction the projected distances are smaller than their true lengths on the sphere. From a consideration of the trigonometric functions of a plane tangent to a sphere, it is obvious that the errors of scale in such a projection relationship are dependent upon the distance of the boundary of the projection from the point of tangency (pole of projection). It is for this reason that the use of a plane tangent to a sphere is limited to the mapping of small areas of say fifty square degrees placed concentrically about the pole of projection. A simple expedient to increase the area of a map that may be included within the limits of an allowable distortion boundary is the forming of a tangent plane into a cylinder rolled about the sphere, usually with its line of tangency congruent with the equator of the sphere, as shown in the left half of Figure 2. The use of the tangent cylinder extends the length of practicable projection through 360° of right ascension, although it does not increase the range of allowable error of projection in declination. A cylindrical projection will permit an area of 7° in declination by 360° in right ascension to be mapped with a scale error in declination of only one-half of one per cent. With a more generous error allowance, an area of 3600 square degrees (10° X 360°) to 18,000 square degrees (50° X 360°) may be shown on a single map. Should the scale errors of a tangent cylinder projection exceed two or three per cent, the cylinder may advantageously be placed secant to the sphere as .is shown in the right half of Figure 2. This method of placing the cylinder creates two © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System ASTRONOMICAL SOCIETY OF THE PACIFIC 37 standard parallels along which projected scale errors are zero and between which the scale is too small and outside of which the scale is too large. When a cylindrical projection is carried into high latitudes the projection becomes so distorted that, to restore conformality, an adaption of the cylindrical surface becomes desirable and the cylinder is replaced by a cone or by several cones with radii that are successively shorter as higher and higher latitudes are mapped. In the left half of Figure 3 is shown a single cone tangent to the sphere. In the right half of the same figure the cone is placed secant to the sphere. The use of several adjoining cones instead of the single one shown in the figure permits the surfaces which receive the projection to conform more closely CYLINDRICAL PROJECTION CONICAL PROJECTION Fig. 2 Fig. 3 Fig. 2.—The cylinder which is to receive a projection from a sphere may be placed tangent (left) or secant (right) to the imaginary celestial sphere. The secant projection creates two standard parallels along which scale errors in right ascension are zero. Mercator's projection is based on the use of a tangent cylinder and the Gnomic method (Fig. 1) of carry- ing co-ordinates from the undevelopable surface of the sphere to the developable surface of the cylinder. Fig. 3.—The use of a cone instead of a cylinder for projections of areas including high latitudes is a particular purpose adaption of the cylinder and may be considered as such regarding the placing of the cone, or cones, in relation to the imaginary sphere which they will enclose. © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System 38 PUBLICATIONS OF THE MOLLWEIDE . SINUSOIDAL Fig. 4.—The Mollweide and Sinusoidal (sometimes called Sanson- Flamsteed) projections of the entire sphere. The parallels are straight, equidistant lines and the meridians are located by dividing each parallel into the same number of parts equally spaced between , the V axis of the projection and the bounding ellipse. The Sinusoidal projection has straight, equidistant parallels crossed by sine curves of differing ampli- tudes which represent the meridians of right ascension. © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Dâta System ASTRONOMICAL SOCIETY OF THE PACIFIC 39 to the surface of the sphere. This arrangement produces some of the most accurate projections possible between an undevelopable surface (sphere) and a developable one (cone). Since two adjoining cones of differing radii will be in complete edge contact only when they are formed as cones, it is necessary, upon their being flattened into planes (maps), to adjust the scale of the projection on each cone until the adjacent edges of succes- sive flattened cones are in contact throughout their length. AITOFF EQUAL-AREA Fig. 5.—AitoiFs Equal-Area projection preserves the equality of area between the spherical co-ordinate quadrangles and the quadrangles of the projected representation of the sphere. The co-ordinate network of this projection is constructed from formulae which are solved and assembled in usable form in Special Publication 68 of the Coast and Geodetic Sur- vey. The network in printed form may also be purchased from the Survey. To the astronomer interested in stellar distribution in areas of limited size, not greater than a hemisphere, the methods of the plane, cylinder, cone, or polyconical map construction will furnish co-ordinate networks fulfilling his requirements. Very often, however, an analysis of distribution demands one continuous unserrated representation of the entire celestial sphere ex- tending 180° in declination and 360° in right ascension, a total of approximately 41,250 square degrees. To construct a map that © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System 40 PUBLICATIONS OF THE COMPARISON OF SELECTED QUADRANGLES NORMAL AITOFF EQUAL-AREA SINUSOIDAL Fig.

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