Quick Conversion on GIS Metrics
The formula for arc length of a circle:
$$ arc length=2\pi r\left ( \frac{\theta }{360} \right )=p\left ( \frac{\theta }{360} \right ) $$
If the equatorial plane is looked as a circle, then the arc length at the equator is calculated by:
$$ arc length=P\left ( \frac{longitude}{360} \right )=40075* \left ( \frac{longitude}{360} \right )=111*longitude $$
Note taht the radius (R) and perimeter (P) of the earth at the equator is 6378 km, 40075km separately.
And the arc length on a specific parallel/latitude is calculated by:
$$ \begin{eqnarray}arc length & = & 2\pi R\cos(latitude)\left ( \frac{longitude}{360} \right ) \\ & = & P\left ( \frac{longitude}{360} \right )\cos(latitude) \\ & = & 40075* \left ( \frac{longitude}{360} \right )\cos(latitude) \\ & = & 111*longitude \cos(latitude) \end{eqnarray} $$
| longitude
|
quick check of arc length at (latitude) | |||
|---|---|---|---|---|
| equator | 25° | 36° | 44° | |
| 1 arc degree | 111.32 km | 100 km | 90 km | 80 km |
| 0.5 arc degree (30 arc minutes) | 55.66 km | 50 km | 45 km | 40 km |
| 0.25 arc degree (15 arc minutes) | 27.83 km | 25 km | 23 km | 20 km |
| 0.1 arc degree (6 arc minutes) | 11.13 km | 10 km | 9 km | 8 km |
| \(0.08\dot{3}\) arc degree (5 arc minutes) | 9.28 km | 8.5 km | 7.5 km | 6.7 km |
| 0.05 arc degree (3 arc minutes) | 5.566 km | 5 km | 4.5 km | 4 km |
| \(0.01\dot{6}\) arc degree (1 arc minutes) | 1.855 km | 1.7 km | 1.5 km | 1.3 km |
| 0.01 arc degree (36 arc seconds) | 1.113 km | 1 km | 900 m | 800 m |
| \(0.008\dot{3}\) arc degree (30 arc seconds) | 928 m | 850 m | 750 m | 670 m |
| \(0.0041\dot{6}\) arc degree (15 arc seconds) | 464 m | 425 m | 375 m | 335 m |
| \(0.001\dot{6}\) arc degree (6 arc seconds) | 186 m | 170 m | 150 m | 130 m |
| 0.001 arc degree (3.6 arc seconds) | 111 m | 100 m | 90 m | 80 m |
| \(0.0002\dot{7}\) arc degree (1 arc seconds) | 31 m | 30 km | 25 km | 22 km |
坐地日行八万里。By毛泽东
The arc length i.e. perimeter of the Earth is about 40,000 km, which equals 80,000 Li (Chinese mile). Our planet rotates per day. According to elative motion, a people stays at the surface will have a 360 arc degrees motion. In some way, the people moves 40,000 km per day. Roughly, 1 arc degree equals 100 km at the surface of earth.
Reference: