Hello Helmut, Good question, how to proof this. There are a number of things I know and I have read about this but when and where? I'l try to find it back.
I started in a special brochure with all kind of stuff about analemmatic sundials, written in 1993 by Marinus J. Hagen, fouder of our sundial society, in Dutch. A paragraph is written about these arcs, also called "Lambert's Circles". Noted is that: Details may be found in an article by René R.J. Rohr in our bulletin nr. 2 of 1989. This article is in German. The same is written in the BSS bulletin july 1989 in English. Lambert ( 1728 - 1777 ) has written about these circles in: Beyträge zum Gebrauch der Mathematik und deren Anwendungen, Berlin 1770, 2. Teil, S 314. This is what Hagen wrote. In Rohr's articles we may read the proof of these circles and as I see I noted that Rohr's article is also published in Schriften der Freunde alte Uhren Band XXVIII, 1989. I guess you know the SFAU. And indeed, there also is Rohr's article with the title Der Lambertsche Kreis, page 129-137. As I noted in that article there is a typing error on page 133. A formula should read R cos phi, not R / cos phi. Hope this information is good enough to find you asked for. Best wishes, Fer. Fer J. de Vries [EMAIL PROTECTED] http://www.iae.nl/users/ferdv/ Eindhoven, Netherlands lat. 51:30 N long. 5:30 E ----- Original Message ----- From: "Sonderegger Helmut" <[EMAIL PROTECTED]> To: <sundial@rrz.uni-koeln.de> Sent: Saturday, January 12, 2002 1:59 PM Subject: Re: Seasonal Sunrise Marker > Hello Fer, hello Roger, > your ideas on seasonal sunrise markers are very interesting. After having > compared your constructions I have 2 questions: > 1. Fer, where can I find the proof, that your construction ist exact? > 2. Roger, I think the maximal error in your "linear approximation" would be > a bit smaller, if you don't use the summer solstice for fixing the > intersection point (you called this point "Sunrise" in your attached > pdf-file) on the horizontal axis, but a day with smaller declination, may be > half a month later or so. Is that correct? It looks so, when the points of > the dates in Fer's construction are conected with the corelating points of > sunset. Does anybody know which day would be the best? I suppose it could > depend on the latitude. > Thanks > Helmut > > Helmut Sonderegger, A-6800 Feldkirch > Email: [EMAIL PROTECTED] > URL: http://webland.lion.cc/vorarlberg/280000/sonne.htm > ----- Original Message ----- > From: "fer j. de vries" <[EMAIL PROTECTED]> > To: "Roger Bailey" <[EMAIL PROTECTED]>; "Sundial Mail List" > <sundial@rrz.uni-koeln.de>; "Steve Lelievre" <[EMAIL PROTECTED]> > Cc: "Mike Deamicis-Roberts" <[EMAIL PROTECTED]> > Sent: Friday, January 11, 2002 10:04 PM > Subject: Re: Seasonal Sunrise Marker > > > Hello Roger, > > To have an accurate reading for the times of sunset and sunrise on an > analemmatic sundial draw arcs through the focus points of the ellipse and a > date point. > You may see this in the attached picture. > > Best wishes, Fer. > > Fer J. de Vries > [EMAIL PROTECTED] > http://www.iae.nl/users/ferdv/ > Eindhoven, Netherlands > lat. 51:30 N long. 5:30 E > > ----- Original Message ----- > From: "Roger Bailey" <[EMAIL PROTECTED]> > To: "Sundial Mail List" <sundial@rrz.uni-koeln.de>; "Steve Lelievre" > <[EMAIL PROTECTED]> > Cc: "Mike Deamicis-Roberts" <[EMAIL PROTECTED]> > Sent: Friday, January 11, 2002 5:35 PM > Subject: Seasonal Sunrise Marker > > > > The original copy of this note did not get to/from the sundial mailing > list. > > I have changed the address on this one and hope for better results. I > > apologize if you happen to receive multiple copies. If you do, keep one > and > > pass the other on to your grandchildren. > > > > The original note follows. Roger Bailey > > > > Hi Steve et al, > > > > You guessed correctly on the sunrise marker. Have a look at the little pdf > > file attached showing the seasonal sunrise marker on Mike > Deamicis-Roberts' > > analemmatic dial. > > > > The seasonal sunrise marker is a point on the east west axis of the > > analemmatic dial that is used in combination with the date line of the > > Zodiac table to show when and where the sun rises throughout the year. > Stand > > on the sunrise marker point and view across date marks on the zodiac to > see > > where the sun will rise on that date. Or stand on the date mark on the > > Zodiac and view past the sunrise marker to see the time of sunrise on that > > date. Use a string from the date through the marker to the hour ellipse to > > convert the dial into a sunrise calculator. What could be easier? > > > > The red line on the sketch shows the summer solstice sunrise at 4:49 AM at > N > > 60.2 East for Mike's latitude of N 36.8. In your mind, rotate the red line > > around the marker point to determine the time and direction of other > sunrise > > dates. For sunsets, use the marker on the east side of the dial. Things > are > > symmetrical. These two marker points provide an excellent new feature for > > analemmatic sundials, the ability to show where and when the sun rises and > > sets and how this changes throughout the year. > > > > This idea started with Mike's "Seasonal Sundial" posting to the sundial > > mailing list last fall. He was looking for a sundial design that would > show > > the cycle of the seasons through the year. I proposed an analemmatic dial > > with distant solstice sunrise markers like a "medicine wheel". This did > not > > suit Mike's topography. He proposed a marker point within the sundial that > > could be used with the date table to show sunrise phenomenon. I had not > > heard of such a point but did the math to reduce the idea to practice. As > > you can see, Mike's brilliant idea works! > > > > Here are the steps to calculate where to put the seasonal markers on any > > analemmatic dial. All you have to do is determine where the red line > crosses > > the axis. This calculation could be done for any date but the error is > least > > if you use the solstice, either the summer or winter (they are > symmetrical). > > > > 1. Calculate the azimuth of the solstice sunrise for your latitude. When > the > > altitude is zero (sunrise), the azimuth (Az) given by Cos (Az) = Sin (Dec) > / > > Cos (Lat). In Mike's case Cos (Az) = Sin 23.44º /Cos 36.8º = .497 so the > > sunrise azimuth, east of north is Az = 60.2º. > > > > 2. Solve the right angle triangle between the two axes and the red line to > > find the marker point on the E/W axis. Start with the zodiac distance on > the > > N/S axis which is size (or the semimajor axis) x Cos Lat x Tan Dec. In > > Mike's case of a 9 meter dial, the semimajor axis is 4.5, so the solstice > > zodiac distance is 4.5 x Cos 36.8º x Tan 23.44º = 1.562 meters. From the > > triangle geometry, the distance to the marker on the E/W axis is 1.562 x > Tan > > (Az) or 2.727 meters. > > > > How accurate is it? The mathematics are not exact as the trig > relationships > > are not the same, but they are pretty close. Both the zodiac distance are > > functions of latitude and declination but not the same functions. The > > declination distance relationship of the zodiac is slightly different for > > the azimuth derivation. From the layout method, the error is zero at the > > solstices. It is also zero at the equinoxes when the sun rises due east. > > There is a sinusoidal periodic error for dates in between. This error > > increases with latitude. In Mike's case the maximum error is only +/-2.3%. > > At my latitude, N51, the maximum error increases to +/-6.5% so the > > relationship is only approximately correct. With sundials we are used to > > this level of accuracy. Corrections for the equation of time and leap > years > > are of similar magnitude. > > > > All these calculations are based on the theoretical sunrise when the > > calculated altitude of the sun is zero. Refraction and semidiameter affect > > the real view. If you have a perfect horizon (ocean view), allow the sun > to > > rise one full diameter, from the horizon to the lower limb to correct for > > semidiameter and average refraction. For other locations, you will have to > > correct for the horizon pollution. In Mike's case there is a devilish > range > > of mountains affecting his horizon by up about 5 degrees. We are working > at > > corrections for this. > > > > My conclusion is that the simple addition of these markers to the design > of > > analemmatic dials adds a lot to their function of demonstrating the cycles > > of the sun with the seasons. > > > > Roger Bailey > > Walking Shadow Designs > > N 51 W 115 > > > > -----Original Message----- > > From: Steve Lelievre [mailto:[EMAIL PROTECTED] > > Sent: January 8, 2002 7:03 PM > > To: Roger Bailey > > Subject: Re: Garden/Human Sundial > > > > > > Roger, > > > > What precisely is the "the sunrise seasonal marker proposed by Mike > > Deamicis-Roberts"? I'm guessing it's some sort of mark or curve on the > dial, > > which gives a line from today's place on the date scale to the a place on > > the ellipse showing the corresponding sunrise time, but I've not heard of > it > > before. > > > > Thanks, Steve > > > > > > > > >
