Copyright 2005, Bent Dalager. The following table shows the distance from an observer to the horizon, assuming no intervening obstacles, on Kulthea. |

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.

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).

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 ...

Eye height | Distance to horizon | ||
---|---|---|---|

0.0 | m | 0.0 | m |

1.0 | cm | 370 | m |

10 | cm | 1.2 | km |

20 | cm | 1.7 | km |

30 | cm | 2.0 | km |

40 | cm | 2.3 | km |

50 | cm | 2.6 | km |

60 | cm | 2.9 | km |

70 | cm | 3.1 | km |

80 | cm | 3.3 | km |

90 | cm | 3.5 | km |

1.0 | m | 3.7 | km |

1.1 | m | 3.9 | km |

1.2 | m | 4.1 | km |

1.3 | m | 4.2 | km |

1.4 | m | 4.4 | km |

1.5 | m | 4.5 | km |

1.6 | m | 4.7 | km |

1.7 | m | 4.8 | km |

1.8 | m | 5.0 | km |

1.9 | m | 5.1 | km |

2.0 | m | 5.3 | km |

2.1 | m | 5.4 | km |

2.2 | m | 5.5 | km |

2.3 | m | 5.6 | km |

2.4 | m | 5.8 | km |

2.5 | m | 5.9 | km |

3.0 | m | 6.4 | km |

3.5 | m | 6.9 | km |

4.0 | m | 7.4 | km |

4.5 | m | 7.9 | km |

5.0 | m | 8.3 | km |

6.0 | m | 9.1 | km |

7.0 | m | 9.8 | km |

8.0 | m | 11 | km |

9.0 | m | 11 | km |

10 | m | 12 | km |

11 | m | 12 | km |

12 | m | 13 | km |

13 | m | 13 | km |

14 | m | 14 | km |

15 | m | 14 | km |

20 | m | 17 | km |

25 | m | 19 | km |

30 | m | 20 | km |

35 | m | 22 | km |

40 | m | 23 | km |

45 | m | 25 | km |

50 | m | 26 | km |

60 | m | 29 | km |

70 | m | 31 | km |

80 | m | 33 | km |

90 | m | 35 | km |

100 | m | 37 | km |

110 | m | 39 | km |

120 | m | 41 | km |

130 | m | 42 | km |

140 | m | 44 | km |

150 | m | 45 | km |

200 | m | 53 | km |

250 | m | 59 | km |

300 | m | 64 | km |

350 | m | 69 | km |

400 | m | 74 | km |

500 | m | 83 | km |

600 | m | 91 | km |

700 | m | 98 | km |

800 | m | 110 | km |

900 | m | 110 | km |

1.0 | km | 120 | km |

1.5 | km | 140 | km |

2.0 | km | 170 | km |

2.5 | km | 190 | km |

3.0 | km | 200 | km |

3.5 | km | 220 | km |

4.0 | km | 230 | km |

4.5 | km | 250 | km |

5.0 | km | 260 | km |

6.0 | km | 290 | km |

7.0 | km | 310 | km |

8.0 | km | 330 | km |

9.0 | km | 350 | km |

10 | km | 370 | km |

11 | km | 390 | km |

12 | km | 410 | km |

13 | km | 420 | km |

14 | km | 440 | km |

15 | km | 460 | km |

20 | km | 530 | km |

25 | km | 590 | km |

30 | km | 640 | km |

35 | km | 700 | km |

40 | km | 740 | km |

45 | km | 790 | km |

50 | km | 830 | km |

60 | km | 910 | km |

70 | km | 990 | km |

80 | km | 1100 | km |

90 | km | 1100 | km |

100 | km | 1200 | km |

1 m is 100 cm 1 km is 1000 m |

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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