Next message: Frank Reed: "Re: dip, dip short, distance off with buildings, etc."
I wrote:
> With a sextant we
> are measuring the altitude of point B from point A so that means that we
> know the angle in the triangle at point A, let's call that gamma.
Bill, you asked:
"Which triangle? The small oblique or large oblique?"
The only triangle I defined <g>. You have point C at the center of the
Earth, point A at height h above the Earth's surface, and point B some distance
away at height H above the Earth's surface.Those three points make a big
triangle ABC. Our GOAL is to find the angle in the triangle at point C (since, if
we multiply that by the Earth's radius, we get the distance). Now what do we
measure with our sextant or theodolite? Well, we're at point A, and we measure
altitudes above the horizontal which is, by definition, the plane
perpendicular to side AC of the triangle. Now how is that related to the angle at point
A in the big triangle (the angle "CAB")? Clearly that's just 90 degrees plus
the measured altitude. So we KNOW the angle CAB. This is equivalent to the
measured parameter (or at minimum, it's related to the measured parameter by a
simple relationship). So we know the corner angle at point A (the observer),
which I named "gamma", and we are seeking the corner angle at point C (the
center of the Earth), which I named "phi". But, uh-oh, we're stuck with another
angle --the one at point B. Wait... no we're not. Since it's a simple plane
triangle, all three angles must add up to 180 degrees. That means that the
angle in the big triangle at point B is NECESSARILY equal to 180-(gamma+phi).
Ok so far?? If you haven't drawn a picture of this yet, you can't possibly be
ok here <g> so please make sure you've got a picture of this. And if you
don't want to draw your own picture, see image 488 in the archive (see below).
Now we are in a position where we can apply the law of sines (the ordinary
plane trig law of sines...) to the big triangle ABC. Set it up as
sin("angle at A")/(R+H)=sin("angle at B")/(R+h).
And work from there. You'll need to remember that
sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b).
That's enough triangles for now, I think!
I don't know if anyone will get a kick out of it or not, but if you like
this kind of math/physics, I've photographed some of my notes. This is RAW
material. You can download it at www.HistoricalAtlas.com/lunars/refnotes.zip. It's
about 5.7 megabytes. By the way, try to bookmark or record these addresses
with the "HistoricalAtlas" part intact. Yes, it currently points to one of my
other addresses "clockwk.com" but that's changing soon.
-FER
42.0N 87.7W, or 41.4N 72.1W.
www.HistoricalAtlas.com/lunars
