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

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