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Bible Encyclopedias
Map Projections
1911 Encyclopedia Britannica
or 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.
Cylindrical Equal Area Projection
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.
Orthographic Projection
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.
Stereographic 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 Vr
Vp, being =Vo sec kVp
Vk cos kVp= Vo
Vk, is equal to Vs
Vl; 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.
External Perspective Projection
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=1 47 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.
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