The terrestrial sphere

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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