MAP PROJECTIONS In the construction of maps, one has to consider how a portion of spherical See also:

• SURFACE

Perhaps the next idea which would occur would be to derive the meridians and parallels in some other See also:

• SIMPLE

Projections having this property are said to be equal-area projections or See also:

• EQUIVALENT

Especially has harm been caused by this idea when dealing with the See also:

• GROUP

It is clear that, when the points of See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

Take two points, as d and g, which are 90° apart, and let ik be their projections on the equator; then i is the pole of the meridian which passes through k. This meridian is of course an ellipse, and is described with reference to i exactly as the equator was described with reference to P. Produce io to 1, and make lo equal to See also:

• HALF

Let kplV (fig. 7) be a great circle through the point of vision, and ors the trace of the plane of projection. Let c be the centre of a small circle whose radius is cp=cl; the straight line pl represents this small circle in v orthographic projection. FIG. 7. We have first to show that the stereographic projection of the small circle p1 is itself a circle; that is to say, a straight line through V, moving along the circumference of pl, traces a circle on the plane of projection ors. This line generates an oblique cone See also:

• STANDING

A tangent plane to the surface at t cuts the plane of See also:

• PRO

To construct a stereographic projection of the sphere on the horizon of a given place. Draw the circle alit?. (fig. 9) with the diameters ' A. Germain, Traite See also:

• DES

op=taniu; op'=cotlu; om=cotu; mn=cosec u cot co. Again, for the parallels, take Pb = Pc equal to the co-latitude, say c, of the parallel to be projected; join vb, vc cutting lr in e, d. Then ed is the diameter of the circle which is the required projection; its centre is of course the middle point of ed, and the lengths of the lines arc od=tan2(u-c); oe=tan%(u+c). The line sn itself is the projection of a parallel, namely, that of which the co-latitude c =180° —u, a parallel which passes through the point of vision. Notwithstanding the facility of construction, the stereo-graphic projection is not much used in map-making. It is sometimes used for maps of the hemi- See also:

• SPHERES, MUSIC OF THE

The product act' gives the exaggeration of areas. With respect to the alteration of angles we have = (h+ cos u)/ (1+k cos u), and the greatest alteration of angle is = sin \h-+-Itan22/ This vanishes when h =1, that is if the projection be stereographic; or for u=o, that is at the centre of the map. At a distance of 90° from the centre, the greatest alteration is 90°—2 cot—' d h. (See Phil. Meg. 1862.) Clarke's Projection.—The constants h and k can be determined, so that the total misrepresentation, viz.: M=f Pt(a—1)2+(a'—1)2} sin udu, shall be a minimum, R being the greatest value of is, or the spherical radius of the map. On substituting the expressions; for a and a' the integration is effected without difficulty. Put , a=(1—cos fl)/(h+ cos /3) ; v=(h—I)X, H=v—(h+1) See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

Thus we find that if it be required to make the best possible perspective representation of a hemisphere, the values of h and k are h=1.47 and k =2.034; so that in this case 2.034 sin u p 1.625 + cos u For See also:

• ASIA

The following table contains the computed co-ordinates for a map of See also:

• AFRICA
• AFRICA, ROMAN

The central projection is useful for the study of direct routes by See also:

• SEA (in O. Eng. sae, a common Teutonic word; cf. Ger. See, Dutch Zee, &c.; the ultimate source is uncertain)
• SEA, COMMAND OF THE

The use of this term in the past has p P caused much confusion. Two selected parallels are represented by concentric cir- cular ~jCO t°r r arcs of their true lengths; the meridians are their radii. The degrees along the meridians are represented by n'co.,at.r' their true lengths; and the other parallels are circular arcs through points so deter-See also:

• MINED

In fig. 15 let Cm and C'm' represent the extreme parallels of the map, and let the co-latitudes of these parallels be c and c', then any one of the following conditions may be fulfilled: (a) The errors of scale of the extreme parallels may be made equal and may be equated to the See also:

• ERROR (Lat. error, from errare, to wander, to err)

If this is applied to the case of a map of South Africa between the limits 15° S. and 35° S. (see fig. 16) it will be found that the parallel of maximum error is 25° 20'; the errorless parallels, to the nearest degree, are those of 18° and 32°. The greatest scale error in this case is about 0.7 %. In the above account the earth has been treated as a sphere. Of course its real shape is approximately a spheroid of revolution, and the values of the axes most commonly employed are those of Clarke or of See also:

• BESSEL, FRIEDRICH WILHELM (1784-1846)

Where no great refinement is required it will be sufficient to take the errorless parallels as those distant from the extreme parallels about one-See also:

• SIXTH

It is clear that this class of projection is accurate, simple and useful. vmritgr,.1D d 60 lUIi 7 I , 31 LAN 18E 70 70 JO 36E 8 (From See also:

• TEXT (Lat. textum, woven fabric, from texere, to weave)

The modification (d) of this projection was selected for the 1:1,000,000 map of India and Adjacent Countries under publication by the Survey of India. An account of this is given in a pamphlet produced by that department in 1903. The limiting° parallels are 8° and 4o° N., and the parallel of greatest error is 23' 40' 51" The errors of scale are I.8, 2.3, and 1.9 %. It is not as a rule desirable to select this form of the projection. If the surface of the map is everywhere equally valuable it is clear that an arrangement by which errors of scale are larger towards the pole than towards the equator is unsound, and it is to be noted that in the case quoted the great bulk of the land is in the north of the map. Projection (a) would for the same region have three equal maximum scale errors of 2 %. It may be admitted that the practical difference between the two forms is in this case insignificant, but linear scale errors should be reduced as much as possible in maps intended for general use, f. In the fifth form of the projection, the total area of the projection between the extreme parallels and any two meridians is equated to the area of the portion of the sphere which it represents, and the errors of scale of the extreme parallels are equated. Then it is easy to show that _ (c' sin c—c sin c')((sin c'—sin c); h= (cos c—cos c')/(c'—c)ltz-1-; (c+c')}. It can also be shown that any other zone of the same range in latitude will have the same scale errors along its limiting parallels. For instance, a See also:

• SERIES (a Latin word from serere, to join)

But the curvatures of these parallels will be different, and two adjacent zones will not See also:

• FIT

This projection has no merits as compared FIG. 17. with the group just described. The errors of scale along the parallels increase rapidly as the selected parallel is departed from, the parallels on paper being always too large. As an example we may take the case of a map of South Africa of the same range as that of the example given in (a) above, viz. from 15° S. to 35° S. Let the selected parallel be 25° S.; the radius of this parallel on paper (taking the radius of the sphere as unity) is cot 25°; the radius of parallel°35° S.=radius of 25° —meridian distance between 25 and 3$ =cot 250—104180=1.970. Also h=sin of selected latitude=sin 25°, and length on paper along parallel 35° of co°=whX1.97o=wX1.970Xsin 25°, but length on sphere of w=w cos 35°, hence scale error = 19 cos 350 5° — = i •6 %, an error which is more than twice as great as that obtained by method (a). Bonne's Projection.—This projection, which is also called the " modified conical projection," is derived from the simple conical, just described, in the following way: a central meridian is chosen and drawn as a straight line; degrees of latitude spaced at the true rectified distances are marked along this line; the parallels are concentric circular arcs drawn through the proper points on the central meridian, the centre of the arcs being fixed by describing one chosen parallel with a radius of v cot as before; the meridians on each side of the central meridian are drawn as follows: along each parallel distances are marked equal to the true lengths along the parallels on sphere or spheroid, and the See also:

• CURVE (Lat. curvus, bent)

The projection has the property of equal areas, since each small See also:

• ELEMENT (Lat. elementum)

The parallels on paper then become incomplete circular arcs of which the pole is the centre. The central meridian, is still a straight line which is cut by the parallels at true distances. The projection (after Johann Werner, 1468-1528), though interesting, is practically useless. Polyconic Projections. These pseudo-conical projections are valuable not so much for their See also:

• INTRINSIC (through Fr. intrinsique, from Lat. intrinsecus, inwardly; inter, within, secus, following, from root of sequi, to follow)

The real merit of the projection is that each particular parallel has for every map the same absolute radius, and it is thus easy to construct tables which shall be of universal use. This is especially valuable for the projection of single sheets on comparatively large scales. A See also:

• SHEET

20. CC' = C'U' — CU — UU' = tan (z+dz) —tan z—dz=tan 2zdz. The tangents CQ, C'Q' will intersect at q, and in the triangle CC'q the perpendicular from C on C'q is (omitting small quantities of the second See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

For if we suppose 2w to be the longitude of the required point Q, US is by construction=w sin z, and the angle subtended by SU at C is tan -1 j sin zl =tan -1 (w cos z) =a, 1\ tan z// and therefore UCQ=2a as it should be. The advantages of this method are that with a remarkably simple and convenient mode of construction we have a map in which the parallels and meridians intersect at right angles. Fig. 22 is a representation of this system of the continents of See also:

• EUROPE

Both are sensibly perfect representations. The rectangular polyconic is occasionally used by the topographical section of the general staff. Zenithal Projections. Some point on the earth is selected as the central point of the map; great circles radiating from this point are represented by straight lines which are inclined at their true angles at the point of intersection. Distances along the radiating lines vary according to any law outwards from the centre. It follows (on the spherical See also:

• ASSUMPTION, FEAST OF

The errors are smaller than might be supposed. There are no scale errors along the meridians, and along the parallels the scale error is (z/ sin x)—I, where z is the co-latitude of the parallel. On a parallel ro° distant from the pole the error of scale is only o.5 General Theory of Zenithal Projections.—For the See also:

• SAKE

A small square situated in co-latitude z, having one side in the direction of the meridian—the length of its side being i—is represented by a rectangle whose sides are is and ia'; its area consequently is i2aa'. If it were possible to make a perfect representation, then we should have a = r, a' = r throughout. This, however, is impossible. We may make a- = r throughout by taking p = z. This is the Equidistant Projection just described, a very simple and effective method of representation. Or we may make a'= I throughout. This gives p = sin z, a perspective projection, namely, the Orthographic. Or we may require that areas be strictly represented in the development. This will be effected by making aa'= 1, or pdp=sin zdz, the integral of which is p = 2 sin2z, which is the Zenithal Equal-area Projection of See also:

• LAMBERT [alias NICHOLSON], JOHN (d. 1538)
• LAMBERT, DANIEL (1770-1809)
• LAMBERT, FRANCIS (c 1486-153o)
• LAMBERT, JOHANN HEINRICH (1728–1777)
• LAMBERT, JOHN (1619-1694)

For instance, a small square on the spherical surface is to be represented as a small square in the development. This condition will be attained by making a = a', or dp/p = dz/sin z, the integral of which is, c being an arbitrary constant, p=c tan 2z. This, again, is a perspective projection, namely, the Stereographic. In this, though all small parts of the surface are represented in their correct shapes, yet, the scale varying from one part of the map to another, the whole is not a similar representation of the original. The scale, a= 2csec2z, at any point, applies to all directions round that point. These two last projections are, as it were, at the extremes of the scale; each, perfect in its own way, is in other respects objectionable. We may avoid both extremes by the following considerations. Although we cannot make a = i and a'= 1, so as to have a perfect picture of the spherical surface, yet considering a — i and a'— i as the local errors of the representation, we may make (a — 1)2+ (a'—I)2 a minimum over the whole surface to be represented. To effect this we must multiply this expression by the element of surface to which it applies, viz. sin zdzdp, and then integrate from the centre to the (circular) limits of the map. Let 13 be the spherical radius of the segment to be represented, then the total misrepresentation is to be taken as l o a - (dz— I) 2+ (sin z - - 2 sin zdz, which is to be made a minimum. Putting p = z+y, and giving to y only a variation subject to the condition by =o when z=o, the equations of solution—using the ordinary notation of the calculus of See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

This method of development is due to 'See also:

• SIR

Then the azimuth is represented unaltered, and any spherical distance z is represented by p. Thus we get all the points of intersection transferred to the representation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation. Thus treating the earth as a sphere and applying the Zenithal Equal-area Projection to the case of Africa, the central point selected being on the equator, we have, if 0 be the spherical distance of any point from the centre, cis, a the latitude and longitude (with reference to the centre), of this point, cos 0= cos 4 cos a. If A is the azimuth of this point at the centre, tan A = sin a cot O. On paper a line from the centre is drawn at an azimuth A, and the distance 0 is represented by 2 sin 10. This makes a very good projection for a single-sheet equal-area map of Africa. The exaggeration in such systems, it is important to remember, whether of linear scale, area, or angle, is the same for a given distance from the centre, whatever be the azimuth; that is, the exaggeration is a function of the distance from the centre only. General Theory of Conical Projections. Meridians are represented by straight lines drawn through a point, and a difference of longitudew is represented by an angle hw. The parallels of latitude are circular arcs, all having rig as centre the point of divergence of the meridian lines. It is clear that perspective and zenithal projections are particular See also:

• GROUPS
• GROUPS, THEORY

Consider the in- finitely small space on on the sphere contained r' by two consecutive meridians, the difference of of whose longitude is dµ, and two See also:

• CON

The representation is See also:

• FAN (Lat. vannus; Fr. eventail)

This distance of the parallels may be expressed in the form r=a (sin ¢+; sin 3sb+1 sin 54-+ ...), showing that near the equator r is nearly proportional to the latitude. As a consequence of the similar representation of small parts, a curve drawn on the sphere cutting all meridians at the same angle—the loxodromic curve—is projected into a straight line, and it is this property which renders Mercator's chart so valuable to See also:

• SEAMEN, LAWS RELATING TO

(i.) W, Mercator's projection, although indispensable at sea, is of little value for land maps. For topographical sheets it is obviously unsuitable; and in cases in which it is required to show large areas on small scales on an orthomorphic projection, that form should be chosen which gives two standard parallels (Lambert's conical orthomorphic). Mercator's projection is often used in atlases for maps of the world. It is not a good projection to select for this purpose on account of the great exaggeration of scale near the poles. The misconceptions arising from this exaggeration of scale may, however, be corrected by the juxtaposition of a map of the world on an equal-area projection. It is now necessary to revert to the general consideration of conical projections. It has been shown that the scales of the projection (fig. 23) as compared with the sphere are p'q'/pq =dp/dz = a- along a meridian, and p'r'/pr'=ph/sin z=o' at right angles to a meridian. Now if vo' =1 the areas are correctly represented, then hpdp=sin zdz, and integrating lhp'=C—cos z; (i.) this gives the whole group of equal-area conical projections. As a special case let the pole be the centre of the projected parallels, then when z=o, p=o, and const=l, we have p=2 sin Zz/Sh (ii.) Let zi be the co-latitude of some parallel which is to be correctly represented, then 2h sin Zz,/Ih=sin z,, and h=cos' 1z,; putting this value of h in equation (ii.) the radius of any parallel =p=2 sin lz sec Iz, (iii.) This is Lambert's conical equal-area projection with one standard parallel, the pole being the centre of the parallels. If we put z, =8, then h= i, and the meridians are inclined at their true angles, also the scale at the pole becomes correct, and equation (iii.) becomes p=2 sin iz; this is the zenithal equal-area projection. Reverting to the general expression for equal-area conical projections p=al [2(C—cos z)/h} (i.) we can dispose of C and h so that any two selected parallels shall be their true lengths; let their co-latitudes be z, and z2, then 2h (C —cos z,) = sin' z, (v.) 2h (C —cos z2) = sin' z2 (vi.) from which C and h are easily found, and the radii are obtained from (i.) above.

This is H. C. Albers' conical equal-area projection with two standard parallels. The pole is not the centre of the parallels. Projection by Rectangular Spheroidal Co-ordinates. If in the simple conical projection the selected parallel is the equator, this and the other parallels become parallel straight lines and the meridians are straight lines spaced at equatorial distances, cutting the parallels at right angles; the parallels are their true distances apart. This projection is the simple cylin- drical. If now we imagine the touching cylinder turned through a right-angle in such a way as to touch the sphere along any meridian, a projection is obtained exactly similar to the last, except that in this case we represent, not parallels and meridians, but small circles parallel to the given meridian and great circles at right angles to it. It is clear that the projection is a special case of conical projection. The position of any point on the earth's surface is thus referred, on this projection, tq a selected meridian as one axis, and any great circle at right angles to it as the other. Or, in other words, any point is fixed by the length of the perpendicular from it on to the fixed meridian and the distance of the See also:

• FOOT

Such a great circle clearly diverges from the parallel; the exact difference in latitude and longitude between the point and the foot of the perpendicular can be at once obtained by ordinary geodetic formulae, putting the azimuth=9o°. Approximately the difference of latitude in seconds is x' tan . cosec I'/2pv where x is the length of the perpendicular, p that of the radius of curvature to the meridian, v that of the normal terminated by the minor axis, . the latitude of the foot of the perpendicular. The difference of longitude in seconds is approximately x sec p cosec I'/v. The resulting error consists principally of an exaggeration of scale north and south and is approximately equal to sec x (expressing x in arc) ; it is practically See also:

• INDEPENDENT

If the whole world is represented on the spherical assumption, the equator is twice the length of the central meridian. Each elliptical meridian has for one axis the central meridian, and for the other the intercepted portion of the equally divided equator. It follows that the meridians go° east and west of the central meridian form a circle. It is easy to show that to preserve the property of equal areas the distance of any parallel from the equator must be 112 sin 6 where 7r sin =2b+sin 26, (b being the latitude of the parallel. The length of the central meridian from pole to pole=2 d2, where the radius of the sphere is unity. The length of the equator =4 112. The following equal-area projections may be used to exhibit the entire surface of the globe: Cylindrical equal area, Sinusoidal equal area and Elliptical equal area. Conventional or Arbitrary Projections. These projections are devised for simplicity of drawing and not for any special properties. The most useful projection of this class is the globular projection. This is a conventional N S representation of a hemisphere in which the equator and central meridian are two equal straight lines at right angles, their inter-section being the centre of the circular boundary. The meridians divide the equator into equal parts and are arcs of circles passing through points so determined and the poles.

The parallels are arcs of circles which divide the central and extreme meridians into equal parts. Thus in fig. 26 NS = EW and each is divided into equal parts (in this case each division is ro°); the circumference NESW is also divided into ro° spaces and circular arcs are drawn through the corresponding points. This is a simple and effective projection and one well suited for conveying ideas of the (iv.) general shape and position of the See also:

• CHIEF (from Fr. chef, head, Lat. caput)

27 shows the method of plotting the projection for a field sheet. Such a projection is usually called a graticule. In this case ABC is the central meridian; the true meridian lengths of 30' spaces are marked on this meridian, and to each of these, such as AB, the figure (in this case representing a square half degree), such as ABED, is applied. Thus the point D is the intersection of a circle of radius AD with a circle of radius BD, these lengths being taken from geodetic tables. The method has no merit except that of convenience. See also:

• SUMMARY

4. General external perspective. 5. Minimum error „ (Clarke's). 6. Central. 7. Conical, with rectified meridians and two standard parallels (5 forms). 8. Simple conical. 9. Simple cylindrical (a special case of 8).

10. Modified conical equal-area (Bonne's). II. Sinusoidal (Sanson's). 12. Werner's conical Conical 13. Simple polyconic. 14. Rectangular polyconic. 15. Conical orthomorphic with 2 standard parallels (Lambert's, commonly called Gauss's). 16.

Cylindrical orthomorphic (Mercator's). 17. Conical equal-area with one standard parallel. 18. „ two parallels. 19. Projection by rectangular spheroidal co-ordinates. 20. Equidistant zenithal. 21. Zenithal equal-area. J.22.

Zenithal projection by balance of errors (Airy's). 1 23. Elliptical equal-area (Mollweide's). 124. Globular (conventional). 25. Field sheet graticule. Of the above 25 projections, 23 are conical or quasi-conical, if zenithal and perspective projections be included.. The projections may, if it is preferred, be grouped according to their properties. Thus in the above See also:

• LIST (O.E. lisle, a Teutonic word, cf. Dut. lust, Ger. Leiste, adapted in Ital. lista and Fr. lisle)
• LIST, FRIEDRICH (1789-1846)

If the map is intended for statistical purposes to show areas, See also:

• DENSITY (Lat. densus, thick)

Simple Polyconic and Rectangular Polyconic maps on scales of I : 1,000,000, I : 500,000, I : 250,000 and I : 125,000 of the topographical section of the General Staff, including all maps on these scales of British Africa. A rectilinear approximation to the simple polyconic is also used for the topographical sheets of the Survey of India. The simple polyconic is used for the I in. maps of the See also:

• MILITIA (Fr. milice, Ger. Miliz, from Lat. miles, soldier, militia, military service)

See also:

• BERLIN
• BERLIN, CONGRESS AND TREATY OF
• BERLIN, ISAIAH (1725–1799)