Map PROJECTIONS In the construction of maps, one has to consider how a portion of spherical surface, or a configuration traced on a sphere, can be represented on a plane. If the area to be represented bear a very small ratio to the whole surface of the sphere, the matter is easy: thus, for instance, there is no difficulty in making a map of a parish, for in such cases the curvature of the surface does not make itself evident. If the district is larger and reaches the size of a county, as Yorkshire for instance, then the curvature begins to be sensible, and one requires to consider how it is to be dealt with. The sphere cannot be opened out into a plane like the cone or cylinder; consequently in a plane representation of configurations on a sphere it is impossible to retain the desired proportions of lines or areas or equality of angles. But though one cannot fulfil all the requirements of the case, we may fulfil some by sacrificing others; we may, for instance, have in the representation exact similarity to all very small portions of the original, but at the expense of the areas, which will be quite misrepresented. Or we may retain equality of areas if we give up the idea of similarity. It is therefore usual, excepting in special cases, to steer a middle course, and, by making compromises, endeavour to obtain a representation which shall not involve large errors of scale.
A globe gives a perfect representation of the surface of the earth; but, practically, the necessary limits to its size make it impossible to represent in this manner the details of countries. A globe of the ordinary dimensions serves scarcely any other purpose than to convey a clear conception of the earth's surface as a whole, exhibiting the figure, extent, position and general features of the continents and islands, with the intervening oceans and seas; and for this purpose it is indeed absolutely essential and cannot be replaced by any kind of map.
The construction of a map virtually resolves itself into the drawing of two sets of lines, one set to represent meridians, the other to represent parallels. These being drawn, the filling in of the outlines of countries presents no difficulty. The first and most natural idea that occurs to one as to the manner of drawing the circles of latitude and longitude is to draw them according to the laws of perspective. Perhaps the next idea which would occur would be to derive the meridians and parallels in some other simple geometrical way.
Let us suppose a model of the earth to be enveloped by a cylinder in such a way that the cylinder touches the equator, and let the plane of each parallel such as PR be prolonged to intersect the N cylinder in the circle pr. Now unroll the cylinder and the projection will appear as in fig. 2. The whole world is now represented as a rectangle, each E Q parallel is a straight line, and its total length is the same as that of the equator, the distance of each parallel from the equator is sin 1 (where l is the latitude and the radius of the model earth is taken as unity). The meridians are parallel straight lines spaced at equal distances.
FIG. I.
6 5 4_ [MAP Projections This projection possesses an important property. From the elementary geometry of sphere and cylinder it is clear that each N, N N r' Q strip of the projection is equal in area to the zone on the model which it represents, and that each portion of a strip is equal in area to the corresponding portion of a zone. Thus, each small four-sided figure (on the model) bounded by meridians and parallels is represented on the projection by a rectangle which is of exactly the same area, and this applies to any such figure however small. It therefore follows that any figure, of any shape on the model, is correctly represented as regards area by its corresponding figure on the projection. Projections having this property are said to be equal-area projections or equivalent projections; the name of the projection just described is " the cylindrical equal-area projection." This projection will serve to exemplify the remark made in the first paragraph that it is possible to select certain qualities of the model which shall be represented truthfully, but only at the expense of other qualities. For instance, it is clear that in this case all meridian lengths are too small and all lengths along the parallels, except the equator, are too large. Thus although the areas are preserved the shapes are, especially away from the equator, much distorted.
The property of preserving areas is, however, a valuable one when the purpose of the map is to exhibit areas. If, for example, it is desired to give an idea of the area and distribution of the various states comprising the British Empire, this is a fairly good projection. Mercator's, which is commonly used in atlases, preserves local shape at the expense of area, and is valueless for the purpose of showing areas.
Many other projections can be and have been devised, which depend for their construction on a purely geometrical relationship between the imaginary model and the plane. Thus projections may be drawn which are derived from cones which touch or cut the sphere, the parallels being formed by the intersection with the cones of planes parallel to the equator, or by lines drawn radially from the centre. It is convenient to describe all projections which are derived from the model by a simple and direct geometrical construction as " geometrical projections." All other projections may be known as " non-geometrical projections." Geometrical projections, which include perspective projections, are generally speaking of small practical value. They have loomed much more largely on the map-maker's horizon than their importance warrants. It is not going too far to say that the expression " map projection " conveys to most well-informed persons the notion of a geometrical projection; and yet by far the greater number of useful projections are nongeometrical. The notion referred to is no doubt due to the very term " projection," which unfortunately appears to indicate an arrangement of the terrestrial parallels and meridians which can be arrived at by direct geometrical construction. Especially has harm been caused by this idea when dealing with the group of conical projections. The most useful conical projections have nothing to do with the secant cones, but are simply projections in which the meridians are straight lines which converge to a point which is the centre of the circular parallels. The number of really useful geometrical projections may be said to be four: the equal-area cylindrical just described, and the following perspective projections - the central, the stereographic and Clarke's external. Perspective Projections. In perspective drawings of the sphere, the plane on which the representation is actually made may generally be any plane perpendicular to the line joining the centre of the sphere and the point of vision. If V be the point of vision, P any point on the spherical surface, then p, the point in which the straight line VP intersects the plane of the representation, is the projection of P.
In this projection the point of vision is at an infinite distance and the rays consequently parallel; in this case the plane of the drawing may be supposed to pass through the centre of the sphere. Let the circle (fig. 3) represent the plane of the equator on which we propose to make an orthographic representation of meridians and parallels. The centre of this circle is clearly the projection of the pole, and the parallels are projected into circles having the pole for a common centre. The diameters aa', bb being at right angles, let the semicircle bab be divided into the required number of equal parts; the diameters drawn through these points are the projections of meridians. The distances of c, of d and of e from the diameter aa' are the radii of the successive circles representing the parallels. It is clear that, when the points of division are very close, the parallels will be very much crowded towards the outside of the map; so much so, that this projection is not much used.
For an orthographic projection of the globe on a meridian plane let qnrs (fig. 4) be the meridian, ns the axis of rotation, then qr is the projection of the equator. The parallels will be represented by straight lines passing through the points of equal division; these lines are, like the equator, perpendicular to ns. The meridians will in this case be ellipses described on ns as a common major axis, the distances of c, of d and of e from ns being the minor seir,iaxes.
FIG. 4. FIG. 5.
Let us next construct an orthographic projection of the sphere on the horizon of any place.
Set off the angle aop (fig. 5) from the radius oa, equal to the latitude. Drop the perpendicular pP on oa, then P is the projection of the pole. On ao produced take ob =pP, then ob is the minor semiaxis of the ellipse representing the equator, its major axis being qr at right angles to ao. The points in which the meridians meet this elliptic equator are determined by lines drawn parallel to aob through the points of equal subdivision cdefgh. 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 half the shortest chord that can be drawn through i; then lo is the semi-axis of the elliptic meridian, and the major axis is the diameter perpendicular to iol. For the parallels: let it be required to describe the parallel whose colatitude is u; take pm = pi/ =u, and let m'n be the projections of m and n on oPa; then m'n' is the minor axis of the ellipse representing the parallel. Its centre is of course midway between in and n', and the greater axis is equal to mn. Thus the construction is obvious. When pm is less than pa the whole of r Q S g FIG. 2.
E FIG. 3.
FIG. 6. - Orthographic Projection.
the ellipse is to be drawn. When pm is greater than pa the ellipse touches the circle in two points; these points divide the ellipse into two parts, one of which, being on the other side of the meridian plane aqr, is invisible. Fig. 6 shows the complete orthographic projection.
In this case the point of vision is k.?° on the surface, and the projection is made on the plane of the great circle whose pole is V. 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 orthographic projection. FIG. 7.
We have first to show that the stereographic projection of the small circle pl 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 standing on a circular base, its axis being cV (since the angle pVc =angle cVl); this cone is divided symmetrically by the plane of the great circle kpl, and also by the plane which passes through the axis Vc, perpendicular to the plane kpl. Now VrVp, being =Vo sec kVpVk cos kVp= VoVk, is equal to VsVl; therefore the triangles Vrs, Vlp are similar, and it follows that the section of the cone by the plane rs is similar to the section by the plane pl. But the latter is a circle, hence also the projection is a circle; and since the representation of every infinitely small circle on the surface is itself a circle, it follows that in this projection the representation of small parts is strictly similar. Another inference is that the angle in which two lines on the sphere intersect is represented by the same angle in the projection. This may otherwise be proved by means of fig. 8, where Vok is the diameter of the sphere passing through the point of vision, fgh the plane of projection, kt a great circle, passing of course through V, and ouv the line of intersection of these two planes. A tangent plane to the surface at t cuts the plane of projection in the line rvs perpendicular to ov; tv is a tangent to the circle kt at t, tr and is are any two tangents to the surface at t. Now the angle vtu (u being the projection of t) is 90° - otV = 90° - oVt= ouV=tuv, FIG.8. therefore tv is equal to uv; and since tvs and uvs are right angles, it follows that the angles vts and vus are equal. Hence the angle its also is equal to its projection rus; that is, any angle formed by two intersecting lines on the surface is truly represented in the stereographic projection.
In this projection, therefore, angles are correctly represented and every small triangle is represented by a similar triangle. Projections having this property of similar representation of small parts are called orthomorphic, conform or conformable. The word orthomorphic, which was introduced by Germain' and adopted by Craig, 2 is perhaps the best to use.
Since in orthomorphic projections very small figures are correctly represented, it follows that the scale is the same in all directions round a point in its immediate neighbourhood, and orthomorphic projections may be defined as possessing this property. There are many other orthomorphic projections, of which the best known is Mercator's. These are described below.
We have seen that the stereographic projection of any circle of the sphere is itself a circle. But in the case in which the circle to be projected passes through V, the projection becomes, for a great circle, a line through the centre of the sphere; otherwise, a line anywhere. It follows that meridians and parallels are represented in a projection on the horizon of any place by two systems of orthogonally cutting circles, one system passing through two fixed points, namely, the poles; and the projected meridians as they pass through the poles show the proper differences of longitude.
To construct a stereographic projection of the sphere on the horizon of a given place. Draw the circle vlkr (fig. 9) with the diameters 1 A. Germain, Traite des Projections (Paris, 1865).
2 T. Craig, A Treatise on Projections (U.S. Coast and Geodetic Survey, Washington, 1882).
kv, lr at right angles; the latter is to meridian. Take koP equal to the co-latitude of the given place, say u; draw the diameter Por n, and vP, vP' cutting lr in pp' : these are the projections of the poles, through which all the circles representing meridians have to pass. All their centres then will be in a line smn which crosses pp' at right angles through its middle point m. Now to describe the meridian whose west longitude is w, draw pn making the angle opn =90° - w, then n is the centre of the required circle, whose direction as it passes through p will make an angle opg=w with pp'. The lengths of the several lines are op= tan2u; op'=cot2u; 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 are od=tan2(u - c); oe=tan2(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 stereographic projection is not much used in map-making. It is sometimes used for maps of the hemispheres in atlases, and for star charts.
We now come to the general case in which the point of vision has any position outside the sphere. Let abcd (fig. Do) be the great circle section of the sphere by a plane passing through c, the central point of the portion of surface to be represented, and V the point of vision. Let pj perpendicular to Vc be the plane of representation, join FIG. 10.
mV cutting pj in f, then f is the projection of any point m in the circle abc, and ef is the representation of cm. Let the angle com = u, Ve = k, Vo = h, of =p; then, since ef: eV = mg: gV, we have p = k sin u/(h+cos u), which gives the law connecting a spherical distance u with its rectilinear representation p. The relative scale at any point in this system of projection is given by a =dp/du, a' =p/sin u, = k(I +h cos u)/(h+cos u) 2; v' = k/(h+cos u), the former applying to measurements made in a direction which passes through the centre of the map, the latter to the transverse direction. The product aa gives the exaggeration of areas. With respect to"the alteration of angles we have E = (h+ cos u)/(1 +k cos u), and the greatest alteration of angle is tan2. = sin -'(h+I 2 / \II 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-' i I h. (See Phil. Mag. 1862.) Clarke's Projection. - The constants h and k can be determined, so that the total misrepresentation, viz.: M=. s{ (v-1)2+(v'- 1)2) sin udu, shall be a minimum, 1 3 being the greatest value of u, or the spherical radius of the map. On substituting the expressions, for v and a' the integration is effected without difficulty. Put X = (I - cos f3)/(h+ cos a); v = (h - 1)X, H= v - (h +I) log e (X+I), H' =X(2 - v+30)/(h +I). Then the value of M is M =4 sin e 213+2kH+k2H'.
When this is a minimum, dM/dh=o; dM/dk=o .'. kH'+H =o; 2dH/dh+kdhH'/dh =o. Therefore M =4 sin e z s - H'/H', and h must be determined so as to make H 2: H' a maximum. In any particular case this maximum can only be ascertained by trial, that is to say, log H 2 - log H' must be calculated for certain equidistant values of h, and then the represent the central 1 L 'p' ' FIG. 9.
9
c particular value of h which corresponds to the required maximum can be obtained by interpolation. 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=147 and k =2.034; so that in this case 2.034 sin U P- 1.47 + cos u' For a map of Africa or South America, the limiting radius R we may take as 40 0; then in this case 2.543 sin u P 1.625 + cos u' For Asia, /3=54, 54, and the distance h of the point of sight in this case is 1.61. Fig. II is a map of Asia having the meridians and parallels laid down on this system.
FIG. I I.
Fig. 12 is a perspective representation of more than a hemisphere, the radius /3 being 108°, and the distance h of the point of vision, 1.40.
The co-ordinates xy of any point in this perspective may be expressed in terms of latitude and longitude of the corresponding FIG. 12.-Twilight Projection. Clarke's Perspective Projection for a Spherical Radius of 108°.
point on the sphere in the following manner. The co-ordinates originating at the centre take the central meridian for the axis of y and a line perpendicular to it for the axis of x. Let the latitude of the point G, which is to occupy the centre of the map, be y; if 0, w be the latitude and longitude of any point P (the longitude being reckoned from the meridian of G), u the distance PG, and i the azimuth of P at G, then the spherical triangle whose sides are 90° - y, 90°- 4), and u gives these relations sin u sin i =cos 0 sin w, sin u cos µ = cos -y sin 41-sin y cos 4) cos co, cos u =sin y sin 4+cos y cos 4) cos w.
Now x= p sin µ, y = p cos µ, that is, x_ cos 0 sin w k h, + sin y sin 4) + cos y cos 4) cos w' y_ cos -y sin 0sin y cos 4) cos w k h + sin y sin 4) + cos -y cos 0 cos w' by which x and y can be computed for any point of the sphere. If from these equations we eliminate w, we get the equation to the parallel whose latitude is 0; it is an ellipse whose centre is in the central meridian, and its greater axis perpendicular to the same. The radius of curvature of this ellipse at its intersection with the centre meridian is k cos 0/(h sin y+sin 0).
The elimination of 0 between x and y gives the equation of the meridian whose longitude is w, which also is an ellipse whose centre and axes may be determined.
The following table contains the computed co-ordinates for a map of Africa, which is included between latitudes 40° north and 40 south and 40° of longitude east and west of a central meridian.
0
Values of x and y.
w=0°
w =IO°
w = 20°
w =30°
w =40°
o°
=
9'69
1 9.43
2 9.2 5
39.17
y= 0.00
000
000
000
000
Io °
x= 000
y = 9' 6 9
9.60
9.75
1924
9.92
2895
10.21
38'76
10.63
20 °
x = 0.00
y=19.43
9.32
19'54
18.67
19.87
28.07
20.43
37.53
21.25
30°
x = 000
y=2 9' 2 5
8.84
29.40
17.70
2 9.8 7
26.5 6
3 0 ' 6 7
35'44
31'83
40°
x= 000
8.15
16.28
24.39
32.44
y=39.17
39'36
39'94
4 0 '93
42'34
Central or Gnomonic (Perspective) Projection.-In this projection the eye is imagined to be at the centre of the sphere. It is evident that, since the planes of all great circles of the sphere pass through the centre, the representations of all great circles on this projection will be straight lines, and this is the special property of the central projection, that any great circle (i.e. shortest line on the spherical surface) is represented by a straight line. The plane of projection may be either parallel to the plane of the equator, in which case the parallels are 'represented by concentric circles and the meri dians by straight lines radiating from the common centre; or the plane of projection may be parallel to the plane of some meridian, in which case the meridians are parallel straight lines and the parallels are hyperbolas; or the plane of projection may be inclined to the axis of the sphere at any angle X.
In - the latter case, which is the most general, if 0 is the angle any meridian makes (on paper) with the central meridian, a the longitude of any point P with reference to the central meridian, 1 the latitude of P, then it is clear that the central meridian is a straight line at right angles to the equator, which is also a straight line, also tan 0 =sin A tan a, and the distance of p, the projection of P, from the equator along its meridian is (on paper) m sec a sin l / sin (l+x), where tan x =cot X cos a, and m is a constant which defines the scale.
The three varieties of the central projection are, as is the case with other perspective projections, known as polar, meridian or horizontal, according to the inclination of the plane of projection.
MAP PROJECTIONS]
Fig. 14 is an example of a meridian central projection of part of the Atlantic Ocean. The term " gnomonic " was applied Conical Projections. Conical projections are those in which the parallels are represented by concentric circles and the meridians by equally spaced radii. There is no necessary connexion between a conical projection and any touching or secant cone. Projections for instance which are derived by geometrical construction from secant cones are very poor projections, exhibiting large errors, and they will not be discussed. The name conical is given to the group embraced by the above definition, because, as is obvious, a can be bent round to form a cone. The same time, one of the most useful forms of the following: with Rectified Meridians and Two Standard books this has been, most unfortunately, conical," on account of the fact that there are two parallels of the correct length. The use of this term in the past has caused much confusion. Two selected parallels are represented by concentric circular arcs of their true lengths; the meridians are their radii. The degrees along the meridians are represented by their true lengths; and the other parallels are circular arcs through points so determined and are concentric with the chosen FIG. 15. parallels.
Thus in fig. 15 two parallels Gn and G'n' are represented by their true lengths on the sphere; all the distances along the meridian PGG', pnn' are the true spherical lengths rectified.
Let -y be the co-latitude of Gn; -y' that of Gn'; w be the true difference of longitude of PGG' and pnn'; ho) be the angle at 0; and OP =z, where Pp is the representation of the pole. Then the true length of parallel Gn on the sphere is w sin y, and this is equal to the length on the projection, i.e. w sin y = hw(z+y); similarly w sin -y' =hw(z+y'). The radius of the sphere is assumed to be unity, and z and y are expressed in circular measure. Hence h = sin y/(z+y) = sin y'(z+y'); from this h and z are easily found.
In the above description it has been assumed that the two errorless parallels have been selected. But it is usually desirable to impose some condition which itself will fix the errorless parallels. There are many conditions, any one of which may be imposed. 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 error of scale of the parallel of maximum error (which is near the mean parallel).
(b) Or the errors of scale of the extreme parallels may be equated to that of the mean parallel. This is not so good a projection as (a).
(c) Or the absolute errors of the extreme and mean parallels may be equated.
(d) Or in the last the parallel of maximum error may be considered instead of the mean parallel.
(e) Or the mean length of all the parallels may be made correct. This is equivalent to making the total area between the extreme parallels correct, and must be combined with another condition, for example, that the errors of scale on the extreme parallels shall be equal.
We will now discuss (a) above, viz. a conical projection with rectified meridians and two standard parallels, the scale errors of the extreme parallels and parallel of maximum error being equated. Since the scale errors of the extreme parallels are to be equal, h(z+c) h(z +c') c' sin c - c sin c' 1 = 1, whence z= (i.) sin c sin c sin c - sin c The error of scale along any parallel (near the centre), of which the co-latitude is b is 1 - {h(z+b)/sin b}. (ii.) This is a maximum when tan b - b=z, whence b is found.
h(z+c) Also I - sin b sin c 1, whence h is found. (iii.) For the errorless parallels of co-latitudes y and y we have h= (z-{-y)/sin y = (z+y')/sin y'. If this is applied to the case of a map of South Africa between the limits 15° S. and 35 0 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 Bessel. For the spheroid, formulae arrived at by the same principles but more cumbrous in shape must be used. But it will usually be sufficient for the selection of the errorless parallels to use the simple spherical formulae given above; then, having made the selection of these parallels, the true spheroidal lengths along the meridians between them can be taken out of the ordinary tables (such as those published by the Ordnance Survey or by the U.S. Coast and Geodetic Survey). Thus, if a l, a 2 , are the lengths of 1 ° of the errorless parallels (taken from the tables), d the true rectified length of the meridian arc between them (taken from the tables), h= { (as - a l)/d} 180r, and the radius on paper of parallel, a l is a i d/(a 2 - a l), and the radius of any other parallel = radius of a l t the true meridian distance between the parallels.
This class of projection was used for the 1 /1,000,000 Ordnance map of the British Isles. The three maximum scale errors in this case work out to 0.23%, the range of the projection being from 50° N. to 61° N., and the errorless parallels are 59°31' and 51°44'.
Where no great refinement is required it will be sufficient to take the errorless parallels as those distant from the extreme parallels about one-sixth of the total range in latitude. Thus suppose it is required to plot a projection for India between latitudes 8° and 40° N. By this rough rule the errorless parallels should be distant from the extreme parallels about 32°/6, i.e. 5° 20'; they should therefore, to the nearest degree, be 13° and 35° N. The maximum scale errors will be about 2%.
The scale errors vary approximately as the square of the range of latitude; a rough rule is, largest scale error =L 2 /50,000, where L is the range in the latitude in degrees. Thus a country with a range of 7° in latitude (nearly 500 m.) can be plotted on this projection with a maximum linear scale error (along a parallel) of about o1%;1 there is no error along any meridian. It is immaterial with this 1 This error is much less than that which may be expected from contraction and expansion of the paper upon which the projection is drawn or printed.
to this projection because similar problem to that of the projection of the meridians is a the graduation of a sun-dial. It is, ° E however, better to use the - 3u term " central, which explains itself. The cen- ° tral projection is useful for the study of direct routes by sea and land. The United States Hydrographic Department has published some charts on this projection. False notions of the direction of shortest lines, which are engendered by a study of maps on Mercator's projection, may be corrected by an inspection of maps drawn on the central projection.
There is no projection which accurately possesses the property of showing shortest paths by straight lines when applied to the spheroid; one which very results from the intersection of CapeTevr,, (From Text Book of Topographical Surveying, by permission of the Controller of H. M. Stationery Office.) FIG. 14. - Part of the Atlanticl Ocean on a Meridian Central Pro-j jection. The shortest path between.; any two points is shown on this; projection by a straight_line.
nearly does so is that which terrestrial normals with a plane.
We have briefly reviewed the most important projections which are derived from the sphere by direct geometrical construction, and we pass to that more important branch of the subject which deals with projections which are not subject to this limitation.
projection so drawn simplest and, at the conical projection is Conical Projection Parallels. - In some termed the " secant 0 projection (or with any conical projection) what the extent in longitude is. It is clear that this class of projection is accurate, simple and useful.
40 352 s (From Text Book of Topographical Surveying, by permission of the Controller of H. M. Stationery Office.) FIG. 16. - South Africa on a conical projection with rectified meridians and two standard parallels. Scale 800 m. to I in.
In the projections designated by (c) and (d) above, absolute errors of length are considered in the place of errors of scale, i.e. between any two meridians (c) the absolute errors of length of the extreme parallels are equated to the absolute error of length of the middle parallel. Using the same notation h (z+c) - sin :c =h (z+c') - sin c' = - h (z-}-2c-+2c') - sin z (c+-c').
L. Euler, in the Acta Acad. Imp. Petrop. (1778), first discussed this projection.
If a map of Asia between parallels 10° N. and 70° N. is constructed on this system, we have c =20°, c' =80°, whence from the above equations z =66.7° and h=6138. The absolute errors of length along parallels 10°, 40° and 70° between any two meridians are equal but the scale errors are respectively 5, 6.7, and 15%.
The modification (d) of this projection was selected for the :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 40° N., and the parallel of greatest error is 23° 40' 51". The errors of scale are 1.8, 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 z= (c' sin c - c sin c')/(sin c' - sin c); h= (cos c - cos c')/(c' - c){z--+z (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 series of projections may be constructed for zones, each having a range of 10° of latitude, from the equator to the pole. Treating the earth as a sphere and using the above formulae, the series will possess the following properties: the meridians will all be true to scale, the area of each zone will be correct, the scale errors of the limiting parallels will all be the same, so that the length of the upper parallel of any zone will be equal to that of the lower parallel of the zone above it. But the curvatures of these parallels will be different, and two adjacent zones will not fit but will be capable of exact rolling contact. Thus a very instructive flat model of the globe may be constructed which will show by suitably arranging the points of contact of the zones the paths of great circles on the sphere. The flat model was devised by Professor J. D. Everett, F.R.S., who also pointed out that the projection had the property of the equality of scale errors of the limiting parallels for zones of the same width. The projection may be termed Everett's Projection. Simple Conical Projection. - If in the last group of projections the two selected parallels which are to be errorless approach each other indefinitely closely, we get a projection in which all the meridians are, as before, of the true rectified lengths, in which one parallel is errorless, the curvature of that parallel being clearly that which would result from the unrolling of a cone touching the sphere along the parallel represented. And it was in fact originally by a consideration of the tangent cone that the whole group of conical projections came into being. The quasigeometrical way of regarding conical projections is legitimate in this instance.
The simple conical projection is therefore arrived at in this way: imagine a cone to touch the sphere along any selected parallel, the radius of this parallel on paper (Pp, fig. 17) will be r cot 0., where r is the radius of the sphere and 49 is the latitude; or if the spheroidal shape is taken into account, the radius of the parallel on paper will be v cot 49 where v is the normal terminated by the minor axis (the value v can be found from ordinary geodetic tables). The meridians are generators of the cone and every parallel such as HH' is a circle, concentric with the selected parallel Pp and distant from it the true rectified length of the meridian arc between them.
This projection has no merits as compared 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 35°=cot 25° - IOir/180=1970. Also h = sin of selected latitude = sin 25°, and length on paper along parallel 35° of co° = wh X1.970 = co X 1.970 X sin 25°, but length on sphere of co=w cos 35°, hence scale error = 1.970 s 35° o° 1 =1.6%, COS an error which is more than twice as great as that obtained by method (a).
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 4 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 curve through corresponding points so fixed are the meridians (fig. 18).
This system is that which was adopted in 1803 by the " Depot de la Guerre " for the map of France, and is there known by the title of Projection de Bonne. It is that on which the ordnance survey map of Scotland on the scale of 1 in. to a mile is constructed, and it is frequently met with in ordinary atlases. It is ill-adapted for countries having great extent in longitude, as the intersections of the meridians and parallels become very oblique - as will be seen on examining the map of Asia in most atlases.
If 0 0 be taken as the latitude of the centre parallel, and co-ordinates be measured from the intersection of this parallel with the central meridian, then, if p be the radius of the parallel of latitude cp, we have p = cot c° +0 0 - 0. Also, if S be a point on this parallel whose co-ordinates are x, y, so that VS = p, and 0 be the angle VS makes with the central meridian, then p6 = w cos 4); and x = p sin 0, y= cot 4 ° - p cos 0.
The projection has the property of equal areas, since each small element bounded by two infinitely close parallels is equal in length and width to the corresponding element on the sphere or spheroid. Also all the meridians cross the chosen parallel (but no other) at right angles, since in the immediate neighbourhood of that parallel the projection is identical with the simple conical projection. Where an equal-area projection is required for a country having no great extent in longitude, such as France, Scotland or Madagascar, this projection is a good one to select.
sometimes known as Sanson's, and is also sometimes incorrectly called Flamsteed's, is a particular case of Bonne's in which the selected parallel is the equator. The equator is a straight line at right angles to the central meridian which is also a straight line. Along the central meridian the latitudes are marked off at the true rectified distances, and from points so found the parallels are drawn as straight lines parallel to the equator, and therefore at right angles to the central meridian. True rectified lengths are marked along the parallels and through corresponding points the meridians are drawn. If the earth is treated as a sphere the meridians are clearly sine curves, and for this reason d'Avezac has given the projection the name sinusoidal. But it is equally easy to plot the spheroidal lengths. It is a very suitable projection for an equal-area map of Africa.
This is another limiting case of Bonne's equal-area projection in which the selected parallel is the pole. 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 intrinsic merits as for the fact that they lend themselves to tabulation. There are two forms, the simple or equidistant polyconic, and the rectangular polyconic. The Simple Polyconic. - If a cone touches the sphere or spheroid along a parallel of latitude 4) and is then unrolled, the parallel will on paper have a radius of v cot 4), where v is the normal terminated by the minor axis. If we imagine a series of cones, each of which touches one of a selected series of parallels, the apex of each cone will lie on the prolonged axis of the spheroid; the generators of each cone lie in meridian planes, and if each cone is unrolled and the generators in any one plane are superposed to form a straight central meridian, we obtain a projection in which the central meridian is a straight line and the parallels are circular arcs each of which has a different centre which lies on the prolongation of the central meridian, the radius of any parallel being v cot 4). So far the construction is the same for both forms of polyconic. In the simple polyconic the meridians are obtained by measuring outwards from the central meridian along each parallel the true lengths of the degrees of longitude. Through corresponding points so found the meridian curves are drawn. The resulting projection is accurate near the central meridian, but as this is departed from the parallels increasingly separate from each other, and the parallels and meridians (except along the equator) intersect at angles which increasingly differ from a right angle. 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 sheet of a degree square on a scale of 1: 250,000 projected in this manner differs inappreciably from the same sheet projected on a better system, e.g. an orthomorphic conical projection or the conical with rectified meridians and two standard parallels; there is thus the advantage that the simple polyconic when used for single sheets and large scales is a sufficiently close approximation to the better forms of conical 2 tan z tan a _ 2w sin z I + tan 2 a I+ w 2 cos 2a At the equator this becomes simply 2w. Let any equatorial point whose actual longitude is 2w be represented by a point on the developed equator at the distance 2w from the central meridian, then we have the following very simple construction (due to O'Farrell of the ordnance survey). Let P (fig. 21) be the pole, U any point in the central meridian, QUQ' the represented parallel whose radius CU =tan z. Draw SUS' perpendicular to the meridian through U; then to determine the point Q, whose longitude is, say, 3°, lay off US equal to half the true length of the arc of parallel on the sphere, i.e. I° 30' to S,. u radius sin z, and with the centre S and FIG. 21.
radius SU describe a circular arc, which will intersect the parallel in the required point Q. 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 w sin z - tan 1 (w cos z) = a, (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 Europe and Africa, for which it is well suited. For Asia this system would not do, as in the northern latitudes, say along the parallel of 70°, the representation is much cramped.
With regard to the distortion in the map of Africa as thus constructed, consider a small square in latitude 40° and in 40° longitude east or west of the central meridian, the square being so placed as to be transformed into a rectangle. The sides, originally unity, became o95 and I13, and the area 1.08, the diagonals inter secting at 9c pc ' 9° 56'. In Clarke's perspective FIG. 19. - Sinusoidal Equal-area Projection.
projection. The simple polyconic is used by the topographical section of the general staff, by the United States coast and geodetic survey and by the topographical division of the U. S. geological survey. Useful tables, based on Clarke's spheroid of 1866, have been published by the war office and by the U.S. coast and geodetic survey.
In this the central meridian and the parallels are drawn as in the simple polyconic, but the meridians are curves which cut the parallels at righ t angles.
In this case, let P (fig. 20) be the north pole, CPU the central meridian, U, U' points in that meridian whose co-latitudes are z and z+dz, so that UU'=dz. Make PU=z, UC =tan z, U'C' =tan (z-Fdz); and with CC' as centres describe the arcs UQ, U'Q', which represent the parallels of co-latitude z and z+dz. Let PQQ' be part of a meridian curve cutting the parallels at right angles. join CQ, C'Q'; these being perpendicular to the circles will be tangents to the curve. Let UCQ = 2 a, UC'Q' = 2 (a -l-da), then the small angle CQC', or the angle between the tangents at QQ', will = 2da. Now 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 order) equal to either side of the equation tan 2 zdz sin 2a = - 2 tan zda.
- tan zdz=2da/sin 2a, which is the differential equation of the meridian: the integral is tan a = w cos z, where co, a constant, determines a particular meridian curve. The distance of Q from the central meridian, tan z sin 2a, is equal to FIG. 20.
C P FIG. 22.
a projection square of unit side occupying the same position, when trans formed to a rectangle, has its sides I 02 and 115, its area 117, and its diagonals intersect at 9Q0 = 7° 6'. The latter projection is therefore the best in point of " similarity," but the former represents areas best. This applies, however, only to a particular part of the map; along the equator towards 30° or 40 longitude, the polyconic is certainly inferior, while along the meridian it is better than the perspective - except, of course, near the centre. Upon the whole the more even distribution of distortion gives the advantage to the perspective system. For single sheets on large scales there is nothing to choose between this projection and the simple polyconic. 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 assumption), that circles of which the selected point is the centre are also circles on the projection. It is obvious that all perspective projections are zenithal.
In this projection, which is commonly called the " equidistant projection," any point on the sphere being taken as the centre of the map, great circles through this point are represented by straight lines of the true rectified lengths, and intersect each other at the true angles.
In the general case if z l is the co-latitude of the centre of the map, z the co-latitude of any other point, a the difference of longitude of the two points, A the azimuth of the line joining them, and c the spherical length of the line joining them, then the position of the intersection of any meridian with any parallel is given (on the spherical assumption) by the solution of a simple spherical triangle.
Thus let tan 0 = tan z cos a, then cos c = cos z sec 0 cos (z - 0), and sin A = sin z sin a cosec c. The most useful case is that in which the central point is the pole; the meridians are straight lines inclined to each other at the true angular differences of longitude, and the parallels are equidistant circles with the pole as centre. This is the best projection to use for maps exhibiting the progress of polar discovery, and is called the polar equidistant projection. 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 distant from the pole the error of scale is only 0.5%.
For the sake of simplicity it will be at first assumed that the pole is the centre of the map, and that the earth is a sphere. According to what has been said above, the meridians are now straight lines diverging from the pole, dividing the 360° into equal angles; and the parallels are represented by circles having the pole as centre, the radius of the parallel whose co-latitude is z being p, a certain function of z. The particular function selected determines the nature of the projection.
Let Ppq, Prs (fig. 23) be two contiguous meridians crossed by parallels rp, sq, and Op'q', Or's the straight lines representing these meridians. If the angle at P is d i e, this also is the value of the angle at O. Let the co-latitude Pp= z, Pq=z-1-dz; op' =, Oq'=p+dp, the circular arcs p'r', q's representing the parallels pr, qs. If the radius of the sphere be unity, p'q' = dp; p 'r' = pdp, pq =dz; pr = sin zdjs. Put a=dp/dz; a'=p/sin z, then p'q' = apq and p'r' = a'pr. That is to say, a, a' may be regarded as the relative scales, at co-latitude z, of the representation, a applying to meridional measurements, a' to measurements perpendicular to the meridian. 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 12aa'. If it were possible to make a perfect representation, then we should have a = 1, a' = I throughout. This, however, is impossible. We may make a = i 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' = i, or pdp = sin zdz, the integral of which is 2 sin2z, which is the Zenithal Equal-area Projection of Lambert, sometimes, though wrongly referred to as Lorgna's Projection after Antonio Lorgna (b. 1736). In this system there is misrepresentation of form, but no misrepresentation of areas.
Or we may require a projection in which all smallparts are to be represented in their true forms i.e. an orthomorphic projection. 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= Z csec zz, 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' = i, 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' - 0 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 be the spherical radius of the segment to be represented, then the total misrepresentation is to be taken as ß + dz I) 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 Sy =o when z=o, the equations of solution - using the ordinary notation of the calculus of variations - are N - ddP) = o; P,l3 =o, Pa being the value of 2p sin z when z=0. This gives 2 d (d) sin, 2 z d d z +sin z cos z d z - y =z - sin z a = o. This method of development is due to Sir George Airy, whose original paper - the investigation is different in form from the above, which is due to Colonel Clarke - will be found in the Philosophical Magazine for 1861. The solution of the differential equation leads to this result p=2 cot Zz log e sec Zz + C tan 2z, C =2 cot e 2N log e sec g.
The limiting radius of the map is R =2C tan 11. In this system, called by Sir George Airy Projection by balance of errors, the total misrepresentation is an absolute minimum. For short it may be called Airy's Projection. Returning to the general case where p is any function of z, let us consider the local misrepresentation of direction. Take any indefinitely small line, length =i, making an angle a with the meridian in co-latitude z. Its projections on a meridian and parallel are i cos a, i sin a, which in the map are represented by is cos a, icr' sin a. If then a' be the angle in the map corresponding to a, tan a' _ (o f /o-) tan a.
a'/a = pdz/sin zd p = E, and the error a' - a of representation = e, then (s - r) tan a tan e = I +/ tan' Put cot 2 i', then e is a maximum when a=i', and the corresponding value of e is Put e= 27r -2?.
For simplicity of explanation we have supposed this method of development so applied as to have the pole in the centre. There is, however, no necessity for this, and any point on the 1tcl " FIG. 23.
surface of the sphere may be taken as the centre. All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meridians and parallels within the space which is to be represented in a plane. 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, 45, 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 q5. On paper a line from the centre is drawn at an azimuth A, and the distance 0 is represented by 2 sin 1-0. 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 longitude w is represented by an angle ho.). The parallels of latitude are circular arcs, all having dw as centre the point of divergence of the meridian lines. It is clear that perspective and zenithal projections are particular groups of conical projections.
Let z be the co-latitude of a parallel, and help. p, a function of z, the radius of the circle O' q representing this parallel. Consider the infinitely small space on the sphere contained by two consecutive meridians, the difference s of whose longitude is dµ, and two con FIG. 24. secutive parallels whose co-latitudes are z and z+dz. The sides of this rectangle are pq = dz, pr =sin zdp; in the projection p'q'r's these become p'q'=dp, and p'r' = phdp,.
The scales of the projection as compared with the sphere are p'q'/pq = dp/dz = the scale of meridian measurements =u,say, and p'r'/pr = phd a/sin zd 1 s = ph/sin z = scale of measurements perpendicular to the meridian =a', say.
Now we may make a = I throughout, then p = z+const. This gives either the group of conical projections with rectified meridians, or as a particular case the equidistant zenithal. We may make a = a' throughout, which is the same as requiring that at any point the scale shall be the same iq all directions. This gives a group of orthomorphic projections. In this case dp/dz = ph/sin z, or dp/p = hdz/sin z.
Integrating, p= k (tan -1z), (i.) where k is a constant.
Now h is at our disposal and we may give it such a value that two selected parallels are of the correct lengths. Let z,, z 2 be the co-latitudes of these parallels, then it is easy to show that h _ log sin z i - log sin z2 (ii.) log tan 2 z l - log tan J2--z2 This projection, given by equations (i.) and (ii.), is Lambert's orthomorphic projection - commonly called Gauss's projection; its descriptive name is the orthomorphic conical projection with two standard parallels. The constant k in (i.) defines the scale and may be used to render the scale errors along the selected parallels not nil but the same; and some other parallel, e.g. the central parallel may then be made errorless.
The value h=i, as suggested by Sir John Herschel, is admirably suited for a map of the world. The representation is fan-shaped, with remarkably little distortion (fig. 24).
If any parallel of co-latitude z is true to scale hk(tan 2zi) h =sin z, if this parallel is the equator, so that z i =90°, kh =1, then equation (i.) becomes p= (tan 2 z) h /h, and the radius of the equator = i/h. The distance r of any parallel from the equator is i /h - (tan zz) h /h = (i/h){i - (tan 2z)5}.
If, instead of taking the radius of the earth as unity we call it a, r = (a/h){-1 - (tan zz) h }. When h is very small, the angles between the meridian lines in the representation are very small; and proceeding to the limit, when h is zero the meridians are parallel - that is, the vertex of the cone has removed to infinity. And at the limit when h is zero we have r=a log e cot 2z, which is the characteristic equation of Mercator's projection.
N S FIG. 25. - Elliptical equal-area Projection, showing the whole surface of the globe.
From the manner in which we have arrived at this projection it is clear that it retains the characteristic property of orthomorphic projections - namely, similarity of representation of small parts of the surface. In 1Vlercator's chart the equator is represented by a straight line, which is crossed at right angles by a system of parallel and equidistant straight lines representing the meridians. The parallels are straight lines parallel to the equator, and the distance of the parallel of latitude 4 from the equator is, as we have seen above, r =a log, tan (45°+-1 4)). In the vicinity of the equator, or indeed within 30° of latitude of the equator, the representation is very accurate, but as we proceed northwards or southwards the exaggeration of area becomes larger, and eventually excessive - the poles being at infinity. This distance of the parallels may be expressed in the form r=a (sin 4)41 sin 3 0+k sin 5 4)-F ...), 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 seamen. For instance: join by a straight line on the chart Land's End and Bermuda, and measure the angle of intersection of this line with the meridian. We get thus the bearing which a ship has to retain during its course between these ports. This is not great-circle sailing, and the ship so navigated does not take the shortest path. The projection of a great circle (being neither a meridian nor the equator) is a curve which cannot be represented by a simple algebraic equation.
If the true spheroidal shape of the earth is considered, the semiaxes being a and b, putting e= (a 2 - b 2) /a, and using common logarithms, the distance of any parallel from the equator can be shown to be (a/M){log tan (45°+10) - e 2 sin 4)-10 sin 30...
where M, the modulus of common logarithms, =0.434294. Of course Mercator's projection was not originally arrived at in the manner above described; the description has been given to show that Mercator's projection is a particular case of the conical orthomorphic group. The introduction of the projection is due to the fact that for navigation it is very desirable to possess charts which shall give correct local outlines (i.e. in modern phraseology shall be orthomorphic) and shall at the same time show as a straight line any line which cuts the meridians at a constant angle. The latter condition clearly necessitates parallel meridians, and the former a continuous increase of scale as the equator is departed from, i.e. the scale at any point must be equal to the scale at the equator X sec. latitude. In early days the calculations were made by assuming that for a small increase of latitude, say the scale was constant, then summing up the small lengths so obtained. Nowadays (for simplicity the earth will be taken as a sphere) we. should say that a small length of meridian ado is represented in this projection by a sec 4dc/), and the length of the meridian in the projection between the equator and latitude q, ,Vo a sec odq =a log, tan (45°+10), which is the direct way of arriving at the law of the construction of this very important projection.
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 = Q along a meridian, and p'r'/pr' = ph/sin z =a' at right angles to a meridian.
Now if vo-' = i the areas are correctly represented, then hpdp= sin zdz, and integrating Zhp 2 =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 =0, p = o, and const =I, we have p= 2 sin 2z/3h (ii.) Let z 1 be the co-latitude of some parallel which is to be correctly represented, then 2h sin 2z1/S11---sin z 1, and h=cos t 2z 1; putting this value of h in equation (ii.) the radius of any parallel =p= 2 sin Zz sec zzl (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 1 =B, 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 2z; (iv.) this is the zenithal equal-area projection. Reverting to the general expression for equal-area conical projections p=-41{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 1 and z 2, then 2h (C - cos z 1) =sin 2 z 1 (v.) 2h(C - cos z 2) = sin' z 2 (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 cylindrical. 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, to 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 foot of the perpendicular from some fixed point on the meridian, these spherical or spheroidal co-ordinates being plotted as plane rectangular co-ordinates.
The perpendicular is really a plane section of the surface through the given point at right angles to the chosen meridian, and may be briefly called a great circle. 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 =90°. Approximately the difference of latitude in seconds is x 2 tan cp 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, 4 the latitude of the foot of the perpendicular. The difference of longitude in seconds is approximately x sec p cosec 0/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 independent of the extent in latitude.
It is on this projection that the I /2,500 Ordnance maps and the 6-in. Ordnance maps of the United Kingdom are plotted, a meridian being chosen for a group of counties. It is also used for the I-in., a in. and 4 in. Ordnance maps of England, the central meridian chosen being that which passes through a point in Delamere Forest in Cheshire. This projection should not as a rule be used for topographical maps, but is suitable for cadastral plans on account of the convenience of plotting the rectangular co-ordinates of the very numerous trigonometrical or traverse points required in the construction of such plans. As regards the errors involved, a range of about 150 miles each side of the central meridian will give a maximum error in scale in a north and south direction of about oi %.
Elliptical Equal - area Projection. In this projection, which is also called Mollweide's projection the parallels are parallel straight lines and the meridians are ellipses, the central meridian being a straight line at right angles to the equator, which is equally divided. 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 90° 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 ,J 2 sin b where 71sin 2S-I-sin 25, being the latitude of the parallel. The length of the central meridian from pole to pole = 2 A/2, where the radius of the sphere is unity. The length of the equator =4 j2.
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 FIG. 26. - Globular Projection.
representation of a hemisphere in which the equator and central meridian are two equal straight lines at right angles, their intersection 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 Io°); the circumference Nesw is also divided into io° 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 Wo general shape and position of the chief land masses; it is better for this purpose than the stereographic, which is commonly employed in atlases.
(From Text Book of Topographical Surveying, by permission of the Controller of H.M. Stationery Office.) FIG. 27.-Plane Table Graticule, dimensions in inches, for a scale of 4 in. to I_m.
Projections for Field Sheets. Field sheets for topographical surveys should be on conical projections with rectified meridians; these projections for small areas and ordinary topographical scales-not less than 1/500,000 -are sensibly errorless. But to save labour it is customary to employ for this purpose either form of polyconic projection, in which the errors for such scales are also negligible. In some surveys, to avoid the difficulty of plotting the flat arcs required for the parallels, the arcs are replaced by polygons, each side being the length of the portion of the arc it replaces. This method is especially suitable for scales of 1:125,000 and larger, but it is also sometimes used for smaller scales.
Fig. 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.
The following projections have been briefly described: I. Cylindrical equal-area.
2. Orthographic.
3. Stereographic (which is orthomorphic).
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).
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.
22. Zenithal projection by balance of errors (Airy's). 23.Elliptical equal-area (Mollweide's).
L 24. 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 list 8 are equal-area, 3 are orthomorphic, i balances errors, I represents all great circles by straight lines, and in 5 one system of great circles is represented correctly.
Among projections which have not been described may be mentioned the circular orthomorphic (Lagrange's) and the rectilinear equal-area (Collignon's) andja considerable number of conventional projections, which latter are for the most part of little value.
The choice of a projection depends on the function which the map is intended to fulfil. If the map is intended for statistical purposes to show areas, density of population, incidence of rainfall, of disease, distribution of wealth, &c., an equal-area projection should be chosen. In such a case an area scale should be given. At sea, Mercator's is practically the only projection used except when it is desired to determine graphically great circle courses in great oceans, when the central projection must be employed. For conveying good general ideas of the shape and distribution of the surface features of continents or of a hemisphere Clarke's perspective projection is the best. For exhibiting the progress of polar exploration the polar equidistant projection should be selected. For special maps for general use on scales of I/I,000,000 and smaller, and for a series of which the sheets are to fit together, the conical, with rectified meridians and two standard parallels, is a good projection. For topographical maps, in which each sheet is plotted independently and the scale is not smaller than 1/500,000, either form of polyconic is very convenient.
The following are the projections adopted for some of the principal official maps of the British Empire Conical, with Rectified Meridians and Two Standard Parallels.-The 1,000,000 Ordnance map of the United Kingdom, special maps of the topographical section, General Staff, e.g. the 64-mile map of Afghanistan and Persia. The i: 1,000,000 Survey of India series of India and adjacent countries.
Modified Conical, Equal-area (Bonne's).-The i in., 2 in., 4 in. and in. Ordnance maps of Scotland and Ireland. The I: 800,000 map of the Cape Colony, published by the Surveyor-General. 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 Militia Department of Canada.
Zenithal Projection by Balance of Errors (Airy's).-The io-mile to 1 in. Ordnance map of England.
Projection by Rectangular Spheroidal Co-ordinates.-The i : 2500 and the 6 in. Ordnance sheets of the United Kingdom, and the i in., 2 in. and 4 in. Ordnance maps of England. The cadastral plans of the Survey of India, and cadastral plans throughout the empire./n==Authorities==-See Traite des projections des cartes geographiques, by A. Germain (Paris, 1865) and A Treatise on Projections, by T. Craig, United States Coast and Geodetic Survey (Washington, 1882). Both Germain and Craig (following Germain) make use of the term projections by development, a term which is apt to convey the impression that the spherical surface is developable. As this is not the case, and since such projections are conical, it is best to avoid the use of the term. For the history of the subject see d'Avezac, " Coup d'oeil historique sur la projection des cartes geographiques," Societe de geographie de Paris (1863).
J. H. Lambert (Beitreige zum Gebrauch der Mathematik, u.s.w. Berlin, 1772) devised the following projections of the above list: 1, 15, 17, and 21; his transverse cylindrical orthomorphic and the transverse cylindrical equal-area have not been described, as they are seldom used. Among other contributors we mention Mercator, Euler, Gauss, C. B. Mollweide (1774-1825), Lagrange, Cassini, R. Bonne (1727-1795), Airy and Colonel A. R. Clarke. (C. F. CL.; A. R. C.)
From Britannica 11th Edition (1911) 
ZWI signed:
KSF