RE: About the circumpolar parabola

[Warren Thom]

Dear Anselmo,   You have caught my attention big time.  (Forgive the pun.)   I am aware of the “circumpolar circle’ – because I did a construction suggested by Fer de Vries of a Hemispherium.  I urge the construction for better understanding.  I used a 4” plastic spherical Christmas ornament and a compass.  See  “http://home.iae.nl/users/ferdv/hemisph.htm”   I have a few questions though about the parabola.   How do you draw iti and bab lines on the parabola that are tangents, how are they located, and labeled? Why are 16 hour iti and bab the same line? All parabolas are similar shape – so how can a parabola be used for all the different latitudes?   Thank you for your posting – I find it very interesting,   Warren Thom  (42N  88W )   -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Anselmo Pérez Serrada Sent: Saturday, March 16, 2002 4:54 PM To: Sundial, Mailinglist 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