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Dip formula derivation
From: Paul Hirose
Date: 2001 Jun 22, 2:46 PM

Yesterday I said the following formula, adapted from one I found in a
surveying textbook, gives the rise of a horizontal line of sight above
a level surface over a known distance:

rise = 6.75e-8 * d * d

I said it would work for any units, i.e., for distance in feet, rise
is in feet. That was wrong. The formula is correct for meters only.

A little thought experiment will prove that formula can't possibly
work for all units. Suppose it's correct for yards. We compute rise
over a certain distance. Now do it again, same distance but expressed
in feet. The answer should be 3x bigger, but since distance is squared
on the right side, the number you actually get is 9x bigger.

Once I noticed that blunder, it was straightforward to derive a dip
formula. There are two key ideas: 1) distance to the sea horizon is
the distance at which "rise" equals height of eye, which implies 2)
the horizontal line of sight of an observer h meters high is 2 * h
meters high when it crosses the horizon.

If the horizon is d meters away, the dip (in radians) to practical
accuracy is (2*h)/d. We know h (height of eye) and can compute d from
the surveyor's formula. After cleaning up, the final formula is: dip =
.02977 * sqrt(h), where dip is in degrees and h is meters.
The Nautical Almanac formula is .0293 * sqrt(h), about 1.6% different.

Dip short is a little easier. For practical purposes, dip short (in
radians) equals (rise + h) / d. We already know h and d. Rise is easy
to calculate. In units of arc minutes, feet and nautical miles:

dip short = .4298 * d + .5658 * h / d

Compared to the table in Bowditch, this agrees within 0.2'. I am now
happy. Distance to the horizon, geographic range, dip, and dip short
all stem from one equation.

--

paulhirose---.net (Paul Hirose) Browse Files

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