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relatively small area, since you are
trying to show a 3 dimensional
object on a 2 dimensional surface.
The further from the centre of projection
you go, the more the distortion is
but, to all intents and purposes, it
can mostly be ignored. You can see
the problem if you flatten a globe:
There are many ways of adjusting for
this, and each suits a different
purpose, so lines drawn on maps
based on different projections will
not cross through the same places.
The quality of orthomorphism, that all
charts should strive for, means the
scale must be correct on all
directions within a very small area.
In addition, parallels must always
cross meridians at right angles.
Otherwise, no chart is perfect, as
you will find when you fold them:
Lambert's Conformal
Imagine the Earth with a light
shining at the centre, then place a
cone on the top. Where the cone
meets the earth, the shadows of the
land formations will be accurate, but
will be out of shape the further
North and South you go. This is the
conic projection, the basis of the
316 Canadian Professional Pilot Studies
Lambert Conformal, and what most of
today's charts are based on, as the
meridians will be straight, even if
they converge towards the North:
Great circles are assumed to be
straight lines (actually they are very
shallow curves), and rhumb lines will
be curves concave to the nearer pole.
Johannes Lambert overcame the
problem of scale expansion in the
18th century by pushing the
imaginary cone further into the
Earth's surface, to cut in two places:
This gives it two Standard Parallels, or
points where scale is correctly
shown. To be sure, there is a slight
contraction between them, but this is
considered insignificant (1% or less)
if two-thirds of the chart are
between the Parallels.
Mercator
The Mercator projection does things
differently. Instead of a cone, the
Earth is surrounded with a vertical
cylinder, touching at the Equator.
Meridians now do not converge, so
rhumb lines will be accurate, but
distance between latitude lines
increases away from the centre (not
significant below about 300 nm, but
always use the scale near the distance
to be measured).
Again, shapes will be accurate where
the cylinder touches the surface, but
distortion will be much greater the
further away (as a point of interest,
the Mercator was the first chart to be
used for maritime navigation in the
16th century). Since rhumb lines on
this projection are straight lines, it
follows that great circles must be
curved, in this case, concave to the
Equator, that is, the rhumb line is
always nearer the Equator.
The rhumb line looks shorter than
the great circle because of scale
expansion (it actually expands as the
secant of the latitude, or the
reciprocal of the cosine). The
relevance of this lies with plotting
radio bearings, because radio waves
take the shortest way (e.g. great
Navigation 317
circles), so long distances need the
conversion angle to be applied
(luckily, not in the exam) to plot
them as straight rhumb lines – in
fact, an ABAC scale on the chart will
do this for you. Complications also
arise from whether the plot is done
at the aircraft (ADF) or the station
(VOR/VDF), but we won't go into
that here. Radio bearings on Mercator
charts must be converted to rhumb lines.
The Mercator projection is the one
mostly used for plotting charts, as
constant headings are easier to use.
Transverse Mercator
This is a horizontal cylinder
projection, and a straight line still
represents a great circle. The Central
Meridian (CM), where the cylinder
touches the sphere, coincides with
the relevant latitude, so True North
and Grid North are the same along
it. However, because rectangular grid
lines are drawn based on the CM,
moving East or West means
applying some sort of grivation (see
below). A scale factor also has to be
applied as you move around the map
to convert ground distances to
measured distances. To reduce this,
the projection uses two North-South
lines with a scale factor of 1, so in
the centre the correction is less
than 1 (0.9996 for the UTM), while
the outer parts have it greater than 1.
The WAC at 1:1,000,000 and the
VNC at 1:500,000 are based on a
two-parallel Lambert Conformal
Conic, whereas the larger scale VTA
at 1:250,000 uses Transverse
　
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