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.
[Illustration: FIG. 1.]
_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 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 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 l (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.
[Illustration: FIG. 2.]
This projection possesses an important property. From the elementary
geometry of sphere and cylinder it is clear that each 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 [symbol] is represented on
the projection by a rectangle [symbol] 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 non-geometrical.
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.
[Illustration: FIG. 3.]
_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 semiaxes.
[Illustration: FIG. 4.]
[Illustration: 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 l, and make lo equal to half the shortest chord that can be drawn
through i; then lo is the semiaxis of the elliptic meridian, and the
major axis is the diameter perpendicular to iol.
[Illustration: FIG. 6.--Orthographic Projection.]
For the parallels: let it be required to describe the parallel whose
co-latitude is u; take pm = pn = 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 m´
and n´, and the greater axis is equal to mn. Thus the construction is
obvious. When pm is less than pa the whole of 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.
[Illustration: FIG. 7.]
_Stereographic Projection._--In this case the point of vision is 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.
[Illustration: FIG. 8.]
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 ts
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, 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 rts 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[27] and adopted by Craig,[28] 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.
[Illustration: FIG. 9.]
To construct a stereographic projection of the sphere on the horizon
of a given place. Draw the circle vlkr (fig. 9) with the diameters
kv, lr at right angles; the latter is to represent the central
meridian. Take koP equal to the co-latitude of the given place, say u;
draw the diameter PoP¹, 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 [omega], draw pn making
the angle opn = 90° - [omega], then n is the centre of the required
circle, whose direction as it passes through p will make an angle opg
= [omega] with pp´. The lengths of the several lines are
op = tan ½u; op´ = cot ½u; om = cot u; mn = cosec u cot [omega].
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 = tan ½(u - c); oe = tan ½(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.
[Illustration: FIG. 10.]
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. 10) 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 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, ef = [rho]; then, since ef: eV
= mg:gV, we have [rho] = k sin u/(h + cos u), which gives the law
connecting a spherical distance u with its rectilinear representation
[rho]. The relative scale at any point in this system of projection is
given by
[sigma] = d[rho]/du, [sigma]´ = [rho]/sin u,
[sigma] = k(1 + h cos u)/(h + cos u)²; [sigma]´ = 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 [sigma][sigma]´ gives the exaggeration of areas. With
respect to the alteration of angles we have [Sigma] = (h + cos u)/(l +
kcos u), and the greatest alteration of angle is
/h - 1 u \
= sin^(-1) ( ----- tan² --- ).
\h + 1 2 /
This vanishes when h = 1, that is if the projection be stereographic;
or for u = 0, that is at the centre of the map. At a distance of 90°
from the centre, the greatest alteration is 90° - 2 cot^(-1) [root]h.
(See _Phil. Mag._ 1862.)
_Clarke's Projection._--The constants h and k can be determined, so
that the total misrepresentation, viz.:
_
/ [beta]
M = | {([sigma] - 1)² + ([sigma]´ - 1)²} sin u du,
_/ 0
shall be a minimum, [beta] being the greatest value of u, or the
spherical radius of the map. On substituting the expressions for
[sigma] and [sigma]´ the integration is effected without difficulty.
Put
[lambda] = (1 - cos [beta])/(h + cos [beta]); [nu] = (h - 1)[lambda],
H = [nu] - (h + 1) log_e ([lambda] + 1), H´ = [lambda](2 - [nu] + (1/3)[nu]²)/(h + 1).