Distance to The Horizon

Copyright 2005, Bent Dalager.
Creative Commons License
This work is licensed under a Creative Commons License.

The following table shows the distance from an observer to the horizon, assuming no intervening obstacles, on Kulthea.

How to use the table

There are basically two ways to use the table: one is the obvious use - finding out how far away the horizon is. This is a simple table lookup. The other is a to find out if an observer can spot some tall object across the horizon. This entails two table lookups and some comparation of numbers.

Distance to horizon

In order to determine how far away the horizon is for any given observer, first determine how far above sea level the observer's eyes are. If the observer is standing on the surface of the ocean, his eyes might typically be somewhere between 140 and 200 centimeters above sea level. Look this number up in the first column in the below table. The second column then tells you how far away the horizon is.

Note that obstructions or poor visibility might further restrict how far an observer can see. The distance to the horizon is simply an upper limit to how far away it is physically possible to see - other circumstances may very well restrict this much further.

Example Ut'kikk has been placed in the tops of a sailing ship to keep a look out. The skies are clear and visibility is excellent. Ut'kikk is currently located up in the main mast, 12 metres above sea level. He can see as far away as 13 kilometers (approximately).

Determine visibility of some structure

In order to determine if an observer can (theoretically) spot some tall structure, the following procedure can be used:

First, determine the actual distance to the target structure (we'll refer to this as "the target distance" in the following). Then determine the distance to the horizon from the observer's position. If the distance to the horizon is larger than the target distance, then the target might be seen by the observer.

If the target distance is beyond the horizon, the target might still be visible. If it is tall enough, parts of it may be "sticking up" over the horizon. Find out in the following manner: determine the height of the highest part of the target in meters above sea level. Using this as the height, look up the target's horizon distance in the normal manner (see above). If the target's horizon distance plus the observer's horizon distance is more than the target distance, then the target might be spotted by the observer.

Example Ut'kikk is placed 12 meters above sea level in the main mast of a sailing ship. There is another ship on the ocean. It is 43 kilometers distant from Ut'kikk's ship. Its highest point is some sails that are 16 meters above the surface of the sea. Ut'kikk's horizon distance (based on a 12 meters height) is 13 km. The target's horizon distance (based on a 16 meters height) is 14.5km (extrapolating a bit since the table doesn't actually list "16m"). 13 km plus 14.5 km is 27.5 km. Since this is much less than the distance between the two ships (43 km), Ut'kikk cannot see the other ship yet. Some time later, when the distance between the two ships has closed to 25 km, however, the ship is within viewing distance. At this time, the GM will start rolling Observation maneuvers to see if Ut'kikk spots it.

Note that while Ut'kikk can now spot the other vessel in the above example, it is not given that the other vessel can spot Ut'kikk's ship. If their lookout has been placed at a low position, he will have a far shorter distance to the horizon than what Ut'kikk has. You really need your lookout to be as high up in the tops as possible if you want to get the drop on the enemy. Obviously, what you really want is aerial reconnaisance ...

Distance to horizon table (metric)

Eye height Distance to horizon
0.0m 0.0m
1.0cm 370m
10cm 1.2km
20cm 1.7km
30cm 2.0km
40cm 2.3km
50cm 2.6km
60cm 2.9km
70cm 3.1km
80cm 3.3km
90cm 3.5km
1.0m 3.7km
1.1m 3.9km
1.2m 4.1km
1.3m 4.2km
1.4m 4.4km
1.5m 4.5km
1.6m 4.7km
1.7m 4.8km
1.8m 5.0km
1.9m 5.1km
2.0m 5.3km
2.1m 5.4km
2.2m 5.5km
2.3m 5.6km
2.4m 5.8km
2.5m 5.9km
3.0m 6.4km
3.5m 6.9km
4.0m 7.4km
4.5m 7.9km
5.0m 8.3km
6.0m 9.1km
7.0m 9.8km
8.0m 11km
9.0m 11km
10m 12km
11m 12km
12m 13km
13m 13km
14m 14km
15m 14km
20m 17km
25m 19km
30m 20km
35m 22km
40m 23km
45m 25km
50m 26km
60m 29km
70m 31km
80m 33km
90m 35km
100m 37km
110m 39km
120m 41km
130m 42km
140m 44km
150m 45km
200m 53km
250m 59km
300m 64km
350m 69km
400m 74km
500m 83km
600m 91km
700m 98km
800m 110km
900m 110km
1.0km 120km
1.5km 140km
2.0km 170km
2.5km 190km
3.0km 200km
3.5km 220km
4.0km 230km
4.5km 250km
5.0km 260km
6.0km 290km
7.0km 310km
8.0km 330km
9.0km 350km
10km 370km
11km 390km
12km 410km
13km 420km
14km 440km
15km 460km
20km 530km
25km 590km
30km 640km
35km 700km
40km 740km
45km 790km
50km 830km
60km 910km
70km 990km
80km 1100km
90km 1100km
100km 1200km
1 m is 100 cm
1 km is 1000 m

Technical details

The distance to the horizon can be calculated, in metric units, using the formula d = sqrt(h(2r+h)), where sqrt is the square root, h is the height of the observer (or rather his eyes) above sea level and r is the radius of the planet. All units are in meters. The distance (d) you get out of this formula is the distance in direct line from the observer's eyes to the horizon. It is not the distance along the curvature of the planet. The difference between these two figures is, however, very small for moderate values of h (i.e., values below 100 kilometers). If you really, absolutely, positively must know the distance along the curvature of the planet, the formula to use is d = r * arccos(r/(r+h)) where arccos is the inverse cosine, the slash (/) is division, the star (*) is multiplication and the other symbols are as above.

On Kulthea, r = 4,300 miles = 6,900 km (these figures are only accurate to two digits - the above calculations reflect this).

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