When this is a minimum,
dM/dh = 0; dM/dk = 0
:. kH´ + H = 0; 2 dH/dh + k dh H´/dh = 0.
Therefore M = 4 sin² ½[beta] - H²/H^1, and h must be determined so as
to make H²:H´ a maximum. In any particular case this maximum can only
be ascertained by trial, that is to say, log H² - log H´ must be
calculated for certain equidistant values of h, and then the
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
[rho] = ------------.
1.47 + cos u
For a map of Africa or South America, the limiting radius [beta] we
may take as 40°; then in this case
2.543 sin u
[rho] = -------------.
1.625 + cos u
For Asia, [beta] = 54, and the distance h of the point of sight in
this case is 1.61. Fig. 11 is a map of Asia having the meridians and
parallels laid down on this system.
[Illustration: FIG. 11.]
Fig. 12 is a perspective representation of more than a hemisphere, the
radius [beta] being 108°, and the distance h of the point of vision,
1.40.
[Illustration: FIG. 12.--Twilight Projection. Clarke's Perspective
Projection for a Spherical Radius of 108°.]
The co-ordinates xy of any point in this perspective may be expressed
in terms of latitude and longitude of the corresponding 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 [gamma]; if [phi],
[omega] be the latitude and longitude of any point P (the longitude
being reckoned from the meridian of G), u the distance PG, and [mu]
the azimuth of P at G, then the spherical triangle whose sides are 90°
- [gamma], 90° - [phi], and u gives these relations--
sin u sin [mu] = cos [phi] sin [omega],
sin u cos [mu] = cos [gamma] sin [phi] - sin [gamma] cos [phi] cos [omega],
cos u = sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega].
Now x = [rho] sin [mu], y = [rho] cos [mu], that is,
x cos [phi] sin [omega]
--- = -------------------------------------------------------------,
k h + sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega]
y cos [gamma] sin [phi] - sin [gamma] cos [phi] cos [omega]
--- = -------------------------------------------------------------,
k h + sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega]
by which x and y can be computed for any point of the sphere. If from
these equations we eliminate [omega], we get the equation to the
parallel whose latitude is [phi]; 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 [phi] / (h sin [gamma] + sin [phi]).
The elimination of [phi] between x and y gives the equation of the
meridian whose longitude is [omega], 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.
+-----+------------------------------------------------------------------------------+
| | Values of x and y. |
|[phi]+--------------+---------------+---------------+---------------+---------------+
| | [omega] = 0° | [omega] = 10° | [omega] = 20° | [omega] = 30° | [omega] = 40° |
+-----+--------------+---------------+---------------+---------------+---------------+
| 0° | x = 0.00 | 9.69 | 19.43 | 29.25 | 39.17 |
| | y = 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| | | | | | |
| 10° | x = 0.00 | 9.60 | 19.24 | 28.95 | 38.76 |
| | y = 9.69 | 9.75 | 9.92 | 10.21 | 10.63 |
| | | | | | |
| 20° | x = 0.00 | 9.32 | 18.67 | 28.07 | 37.53 |
| | y = 19.43 | 19.54 | 19.87 | 20.43 | 21.25 |
| | | | | | |
| 30° | x = 0.00 | 8.84 | 17.70 | 26.56 | 35.44 |
| | y = 29.25 | 29.40 | 29.87 | 30.67 | 31.83 |
| | | | | | |
| 40° | x = 0.00 | 8.15 | 16.28 | 24.39 | 32.44 |
| | y = 39.17 | 39.36 | 39.94 | 40.93 | 42.34 |
+-----+--------------+---------------+---------------+---------------+---------------+
[Illustration: FIG. 13.]
_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 meridians 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 [lambda].
In the latter case, which is the most general, if [theta] is the angle
any meridian makes (on paper) with the central meridian, [alpha] the
longitude of any point P with reference to the central meridian, l 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 [theta] = sin [lambda] tan [alpha], and the distance of
p, the projection of P, from the equator along its meridian is (on
paper) m sec [alpha] sin l / sin (l + x), where tan x = cot [lambda]
cos [alpha], 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.
[Illustration: (From _Text Book of Topographical Surveying_, by
permission of the Controller of H. M. Stationery Office.)
FIG. 14.--Part of the Atlantic Ocean on a Meridian Central Projection.
The shortest path between any two points is shown on this projection by
a straight line.]
Fig. 14 is an example of a _meridian central projection_ of part of the
Atlantic Ocean. The term "gnomonic" was applied to this projection
because the projection of the meridians is a similar problem to that of
the graduation of a sun-dial. It is, however, better to use the term
"central," which explains itself. The central 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 nearly does so is that which results from the
intersection of 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.
_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
projection so drawn can be bent round to form a cone. The simplest and,
at the same time, one of the most useful forms of conical projection is
the following:
[Illustration: FIG. 15.]
_Conical Projection with Rectified Meridians and Two Standard
Parallels._--In some books this has been, most unfortunately, termed the
"secant 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 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 [gamma] be the co-latitude of Gn; [gamma]´ that of Gn´; [omega] be
the true difference of longitude of PGG´ and pnn´; h[omega] be the
angle at O; and OP = z, where Pp is the representation of the pole.
Then the true length of parallel Gn on the sphere is [omega] sin
[gamma], and this is equal to the length on the projection, i.e.
[omega] sin [gamma] = h[omega](z + [gamma]); similarly [omega] sin
[gamma]´= h[omega](z + [gamma]´).
The radius of the sphere is assumed to be unity, and z and [gamma] are
expressed in circular measure. Hence h = sin [gamma]/(z + [gamma]) =
sin [gamma]´(z + [gamma]´); 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