On cone gnomons,ita-bab lines, sidereal time and umbrella dials

[Anselmo P�rez Serrada]

Hi diallists, hi Chris,   Thanks a lot for your explanation. I was writing something like that for the list, but you've done it much better than I would.   Let me add a small thing: Where does all this come from?   As I told before, in an equatorial plane the circumpolar 'parabola'  is now a circle and the ita-bab lines are tangent to it and perpendicular to the corresponding true solar time lines (I to I, II to II, etc). Why? Because if we stand in this equatorial plane, the Sun spins around us and the circumpolar circle is just above us, with the ita-bab circles tangent to it. Then if you make a gnomonical projection you get an arrangement of circles and straight lines. Now we can make the conical gnomon being its basis an equatorial circle tangent to the ground. It is very easy to see that the ita-bab lines intersect the ground at a distance Radius*tan(HourAngle/2) from the contact point, and so we can determine the ita-bab hours from the shadow of the cone over an horizontal sundial, as you (Chris) said.   Here in Spain we call this elegant dial Bores' Dial, because it was first reported by J. Moreno Bores, but I don't know if there are precedents elsewhere like, for instance, the 'umbrella dial' described by Chris. By the way, with this umbrella you can also make a nice bifilar sundial.   A similar idea can be used to determine the (straight) lines of sidereal time. Now the circles of equal sidereal time are tangent to the tropical minor circles (Cancer and Capricornius) and so their projections are evolutes of the solsticial hyperbolae. Once again, have a look at them in ZW2000.   And finally, we could have deduced many of these results in a very crisp and elegant way using the theory of central projection sundials as shown by Bruno Ernst in a delicious article you can find in (guess it!) De Zonnewijzerkring. It's one of the best articles in gnomonics I have EVER seen.   Greetings to all,   Anselmo Perez Serrada ----- Original Message ----- From: Chris Lusby.Taylor To: sundial@rrz.uni-koeln.de Sent: Monday, March 18, 2002 1:10 PM Subject: Re: About the circumpolar parabola The sundial park at Genk in Belgium has a dial where the gnomon is this very cone. See dial number nine at <http://www.biol.rug.nl/maes/genk/welcome-e.htm>. Its edges cast shadows on the horizontal dial that it rests on. They directly indicate Italian and Babylonian hours. Interestingly, these hour lines are identical to the half hour lines of a horizontal dial. So, if part of the cone were replaced by a polar gnomon the same lines could be used to show Italian (sunset-related), Babylonian (sunrise-related) and common (midnight/midday-related) local apparent time. They would merely need different numbering for each time system. Indeed, you don't need the entire conical surface - any complete section would do. So, the cone could be reduced to an equatorial circle or, in latitudes <45 degrees, a vertical ellipse. Thus, adding an equatorial circle to any horizontal dial allows it to indicate Babylonian and Italian hours, with no extra lines, but noting that every conventional hour line represents a two hour period - so 10am should be labelled as 20:00 Italian, 11am as 22:00 Italian, 1pm as 02:00 Babylonian, 2pm as 04:00 Babylonian... Chris Lusby Taylor 51.4N 1.3W