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# FM Latitude

## 2.8 Shortest Distance between Points on a Parallel

Note: if you cannot remember what parallels of longitude are, revise notes for 2.6 Modelling the World.

Note: if you cannot remember how to find the shortest distance between points on great and small circles, revise notes for 2.5 Distance Between Points on a Sphere.

### Shortest Distance between Points on the Equator

• Locations of equal latitude exist on the same parallel of longitude. By taking the shorter section of that parallel with the locations as the endpoints, we can extract an arc representing the shortest path between the two locations. The length of that path can be found using the longitudinal coordinates of the locations.
• The plane of the equator intersects the centre of the world and so it is a great circle.
• We can modify the equation for arc length (from notes 2.1 Circles and Arcs) for this situation. There are 3 situations with different formulas:
• If both locations are given in the same units (oE or oW) or one or both lie on the Greenwich meridian, the formula is:
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## 2.7 Shortest Distance to the Poles or Equator

Note: if you cannot remember what a meridian is, revise notes for 2.6 Modelling the World.

### Finding the Shortest Distance using the Meridian

• The shortest path between any point on the surface of the earth to the north or south pole, or the equator is along the meridian intersecting that point. By taking the section of the meridian between the relevant pole or equator and the point being solved for, we find an arc whose length is the shortest distance between the pole or equator and the position. The angle of that arc can be found using the latitude of that location.
• We can modify our formula for arc length (from 2.1 Circles and Arcs) to be specific for this situation:

s_{\text {north pole }}=\frac{\pi 6400\left(90-l_{N}\right)}{180}=\frac{\pi 640\left(90-l_{N}\right)}{18}

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## 2.6 Modelling the World

### The World as a Sphere

• The world can be approximated to be a sphere of radius 6400km.

### Meridians

• Meridians of longitude are circles drawn around the world with the same radius as the world and intercepting the north and south poles.

Example

In the above spherical representation of the world, a meridian of longitude is shown in red.

### Parallels

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