Quick Conversion on GIS Metrics

Posted by Bruce Liu on November 19, 2019 , updated on May 12, 2022

🏷️ Tags: GIS , metrics conversion

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.

Arc Length Calculator

Reference:


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