Start with a familiar sphere: the Earth (assume for the moment that it is spherical), spinning around an axis.
The North & South Poles are where this axis meets the Earth's surface. The equator lies midway between them.
The equator is an example of a great circle:
one whose plane passes through the centre of the sphere.
Every
great circle has two poles. We can define these:
(a)
as the points which are 90° away from the circle, on the surface
of the sphere.
(b) as the points
where the perpendicular to the plane of the great circle cuts the
surface of the sphere.
These two definitions are equivalent.
The length of a great-circle arc on the
surface of a sphere
is the angle between its end-points, as seen
at the centre of the sphere,
and is expressed in degrees (not
miles, kilometres etc.).
A great circle is a geodesic (the shortest
distance between two points) on the surface of a sphere,
analogous
to a straight line on a plane surface.
To describe a location X on the surface of the Earth,
we use latitude and longitude
(two coordinates,
because the surface is two-dimensional).
Draw a great circle from pole to pole, passing through location X: this is a meridian of longitude.
The latitude of X is the angular distance
along this meridian from the equator to X,
measured from -90°
at the South Pole to +90° at the North Pole.
The co-latitude of X is the angular distance from the North Pole to X = 90° - latitude.
There is no obvious point of origin for measuring
longitude;
for historical reasons, the zero-point is the meridian
which passes through Greenwich (also called the Prime Meridian).
The longitude of X is the angular distance along the
equator from the Prime Meridian to the meridian through X.
It may
be measured east or west 0° to 360°, or both ways 0° to
180°.
Small circles parallel to equator are parallels of
latitude.
The circumference of a small circle at any given
latitude is 360 x cos(latitude)
degrees.
The length of arc of a small circle between
two meridians of longitude is
(difference
in longitude) x cos(latitude).
The
great-circle distance is always less than this, as we shall see in
the next section.
Note that a position on the surface of the Earth is
fixed using
one fundamental circle (the equator)
and
one fixed point on it (the intersection with Greenwich
Meridian).
Celestial navigation used at sea (and in the air)
involves spherical trigonometry,
so the results are in angular
measure (degrees).
These must be converted to linear measure for
practical use.
We define the nautical mile as 1 arc-minute
along a great circle on Earth's surface.
This comes out about 15%
greater than the normal "statute" mile (6080 feet instead
of 5280 feet).
Exercise:
Alderney,
in the Channel Islands, has longitude 2°W, latitude 50°N.
Winnipeg, in Canada, has longitude 97°W, latitude 50°N.
How far apart are they, in nautical miles, along a parallel of
latitude?
Click here for the answer.
Note: terrestrial coordinates are actually more
complicated than this, because the Earth is not really a sphere.
One source where you
can find out more about this is the Ordnance Survey's "Guide
to coordinate systems in Great Britain".
Previous section: Introduction
Next section: Spherical
Trigonometry
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