[rho] = [root]{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 z1 and z2, then
2h(C - cos z1) = sin² z1 (v.)
2h(C - cos z2) = sin² z2 (vi.)
from which C and h are easily found, and the radii are obtained from
(i.) above. This is H. C. Albers' _conical equal-area projection with
two standard parallels_. The pole is not the centre of the parallels.
_Projection by Rectangular Spheroidal Co-ordinates._
If in the simple conical projection the selected parallel is the
equator, this and the other parallels become parallel straight lines and
the meridians are straight lines spaced at equatorial distances, cutting
the parallels at right angles; the parallels are their true distances
apart. This projection is the _simple 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 x² tan [phi] cosec 1´´ /
2[rho][nu] where x is the length of the perpendicular, [rho] that of
the radius of curvature to the meridian, [nu] that of the normal
terminated by the minor axis, [phi] the latitude of the foot of the
perpendicular. The difference of longitude in seconds is approximately
x sec [rho] cosec 1´´ / [nu]. 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 1/2,500 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 1-in., ½ in. and ¼ 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 0.1%.
_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 [root]2 sin [delta] where [pi] sin [phi] = 2[delta] + sin
2[delta], [phi] being the latitude of the parallel. The length of the
central meridian from pole to pole = 2 [root]2, where the radius of the
sphere is unity. The length of the equator = 4 [root]2.
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.
[Illustration: FIG. 26.--Globular Projection.]
_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 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 10°); the
circumference NESW is also divided into 10° 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 general shape
and position of the chief land masses; it is better for this purpose
than the stereographic, which is commonly employed in atlases.
[Illustration: (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 1 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.
_Summary._