----- Original Message -----
Sent: Saturday, March 16, 2002 11:53
PM
Subject: About the circumpolar
parabola
Hi dialists,
This is something I learnt from Prof.
Fernando M. Box and hadn't seen as this in any
of my books on gnomonics.
Most of you know about the 'circumpolar circle',
which is, for a given latitude, the
celestial circle for which the stars 'above' it
are seen all over the year (in that latitude).
This circle is parallel to the celestial equator
and its declination equals the colatitude.
From the observer's place it touches the horizon
circle just in the northern tip of the
meridian line (so as to say).
Now it is easy to see that the gnomonical
projection of this circle is a parabola with
the following parameters:
-- Axis: The meridian line. The
curve opens towards the South, of course.
-- Vertex: At H*cotg(2*Lat) from the
foot of the (vertical) gnomon, where H is its
height over the ground.
-- Focal distance: equals
(H/2)*tan(Lat)
[HINT: Remember that an sphere always casts a
conical-curve shadow whose
focus lies just in the point of contact with the
ground]
The most interesting property of this
parabola is that the straight lines tangent to it coincide
with these of
italic and babilonical hours. Why? Well, because
by the definition of these hours, its corresponding
(great) circles are necessarily tangent to the circumpolar circle.
This gives an easy way to draw these two
groups of lines (remember that the line for 16-ita is the
same for 16-bab and that the cross points define
the equinoctial straight line).
We could as well make a stereographic projection
instead of a gnomonical one to get all the lines
as circular arcs, or define the circumpolar cone,
which has also quite interesting properties,
but I'll leave these things for later
on.
Isn't it beautiful?
Greetings,
Anselmo Perez Serrada
