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Subject: Re: horizon visibility puzzle
From: Paul Hirose (paulhirose@XXX.XXX)
Date: Thu Jun 21 2001 - 18:58:04 EDT
I got the same result as Hal and Aubrey. The low hill blocks the view
of the horizon.
The formula I used was distance to the horizon, from Bowditch but
converted to metric: d = 3.92 * sqrt(h), where d is km and h is
meters.
Here is how I reasoned out the problem. From the 777 m hill the
horizon is 109 km away. From the 401 m hill it's 79 km away, or
(adding the 35 km between hills) 114 km from the rear hill. This is
further than you can see from the rear hill, so the forward one is
blocking your line of sight to the horizon.
I was delighted at the two responses I got. Exactly what I was hoping
for. All three of us used the same tool: distance to the horizon. We
all thought in terms of the horizon distance from each hill. However,
each person utilized different imagery.
Hal pegged a point on the horizon from the high hill, then went out
there and looked back at the low hill. Aubrey started with the hills
beside each other, then separated them until the horizons coincided. I
kept the hills 35 km apart, but adjusted the horizon distances to a
common frame of reference. It's fascinating and comforting that
individuality emerges even when solving a cut and dried mathematical
problem with the same formula.
On the sci.engr.surveying newsgroup, where I first saw this, the one
worked solution I saw was in terms of depression angles of the lines
of sight to the horizon and the forward hill. That's a surveyor's
mind-set at work.
Speaking of surveying, one of my textbooks gives a formula which
closely approximates the amount a horizontal line of sight rises above
a level surface. In "universal" form (same units for height and
distance), it's
h = 6.75e-8 * d * d
I.e., in 10 km your line of sight rises 6.75 meters. This takes the
curvature of Earth and refraction into account.
You can use this to solve the "two hills problem". Imagine yourself on
the ocean, in line with the two hills. With your eye at water level
you can barely see the top of the 777 m hill. (Ignore the other one
for the moment.) Your distance must be such that h = 777. Solving the
formula, d = 107 km. (Just 2% different from Bowditch formula.)
The 401 m hill is 35 km closer, so its distance is 72 km. Your
horizontal line of sight rises only 353 m at this distance, so it hits
the hillside.
Intuition tells me this formula is closely related to the ones for dip
and dip short. I've fiddled around trying to rearrange it to look like
the dip formula, but so far have not been able to get the plug into
the socket.
--paulhirose@XXX.XXX (Paul Hirose)
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