h(z + b) h(z + c)
1 - -------- = -------- - 1 whence h is found. (iii.)
sin b sin c
For the errorless parallels of co-latitudes [gamma] and [gamma]´ we
have
h = (z + [gamma])/sin [gamma] = (z + [gamma]´)/sin [gamma]´.
If this is applied to the case of a map of South Africa between the
limits 15° S. and 35° S. (see fig. 16) it will be found that the
parallel of maximum error is 25° 20´; the errorless parallels, to the
nearest degree, are those of 18° and 32°. The greatest scale error in
this case is about 0.7%.
In the above account the earth has been treated as a sphere. Of course
its real shape is approximately a spheroid of revolution, and the
values of the axes most commonly employed are those of Clarke or of
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 a1, a2, 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 = {(a2 - a1)/d}180/[pi],
and the radius on paper of parallel, a1 is a1d/(a2 - a1), and the
radius of any other parallel = radius of a1 ± 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 = L²/50,000, 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 0.1%;[29]
there is no error along any meridian. It is immaterial with this
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.
[Illustration: (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 1 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 + ½c + ½c´) - sin ½(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
1:1,000,000 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 + ½(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 quasi-geometrical way of regarding conical projections
is legitimate in this instance.
[Illustration: FIG. 17.]
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 [phi],
where r is the radius of the sphere and [phi] is the latitude; or if
the spheroidal shape is taken into account, the radius of the parallel
on paper will be [nu] cot [phi] where [nu] is the normal terminated by
the minor axis (the value [nu] 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° - 10[pi]/180 = 1·970.
Also h = sin of selected latitude = sin 25°, and length on paper along
parallel 35° of [omega]° = [omega]h × 1.970 = [omega] × 1.970 × sin
25°,
but length on sphere of [omega] = [omega] cos 35°,
1.970 sin 25°
hence scale error = ------------- - 1 = 1.6%,
cos 35°
an error which is more than twice as great as that obtained by method
(a).
_Bonne's Projection._--This projection, which is also called the
"modified conical projection," is derived from the simple conical, just
described, in the following way: a central meridian is chosen and drawn
as a straight line; degrees of latitude spaced at the true rectified
distances are marked along this line; the parallels are concentric
circular arcs drawn through the proper points on the central meridian,
the centre of the arcs being fixed by describing one chosen parallel
with a radius of [nu] cot [phi] 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).
[Illustration: FIG. 18.]
This system is that which was adopted in 1803 by the "Dépôt 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 [phi]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 [rho] be the radius of the parallel of
latitude [phi], we have [rho] = cot [phi]0 + [phi]0 - [phi]. Also, if
S be a point on this parallel whose co-ordinates are x, y, so that VS
= [rho], and [theta] be the angle VS makes with the central meridian,
then [rho][theta] = [omega] cos [phi]; and x = [rho] sin [theta], y =
cot [phi]0 - [rho] cos [theta].
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.
[Illustration: FIG. 19.--Sinusoidal Equal-area Projection.]
_Sinusoidal Equal-area Projection._--This projection, which is
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.
_Werner's Projection._--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 [phi] and is then unrolled, the parallel will on
paper have a radius of [nu] cot [phi], where [nu] 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 [nu] cot [phi].
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 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.
_Rectangular Polyconic._--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 right angles.
[Illustration: FIG. 20.]
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 + dz);
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[alpha], UC´Q´ = 2([alpha] + d[alpha]), then the
small angle CQC´, or the angle between the tangents at QQ´, will =
2d[alpha]. Now
CC´ = C´U´- CU - UU´ = tan (z + dz) - tan z - dz = tan² z dz.
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
tan² z dz sin 2[alpha] = -2 tan zd [alpha].
-tan z dz = 2d[alpha] / sin 2[alpha],
which is the differential equation of the meridian: the integral is
tan [alpha] = [omega] cos z, where [omega], a constant, determines a
particular meridian curve. The distance of Q from the central
meridian, tan z sin 2[alpha], is equal to
2 tan z tan [alpha] 2[omega] sin z
------------------- = -------------------------
1 + tan² [alpha] 1 + [omega]² cos² [alpha]
[Illustration: FIG. 21.]
At the equator this becomes simply 2[omega]. Let any equatorial point
whose actual longitude is 2[omega] be represented by a point on the
developed equator at the distance 2[omega] 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. 1°
30´ to radius sin z, and with the centre S and radius SU describe a
circular arc, which will intersect the parallel in the required point
Q. For if we suppose 2[omega] to be the longitude of the required
point Q, US is by construction = [omega] sin z, and the angle
subtended by SU at C is
/[omega] sin z\
tan^(-1) ( ------------- ) = tan^(-1) ([omega] cos z) = [alpha],
\ tan z /
and therefore UCQ = 2[alpha] 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.
[Illustration: FIG. 22.]
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 0.95
and 1.13, and the area 1.08, the diagonals intersecting at 90° ± 9° 56´.
In Clarke's perspective projection a square of unit side occupying the
same position, when transformed to a rectangle, has its sides 1.02 and
1.15, its area 1.17, and its diagonals intersect at 90° ± 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.
_Equidistant Zenithal Projection._--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.