Hello Frank and all, You can rest assured: there are easy solutions using plane geometry to solve vertical declining sundials. You just have to sort out which plane the angles are measured on. The attached slide shows the vertical declining tetrahedron made up of four planes. Knowing the style is parallel to the earth's axis and two of the angles, usually latitude and wall declination, you can solve all the angles in the triangles forming the VD tetrahedron. Substyle height and distance are required to build the dial and place the style at the proper angles.
The solution for these angles using plane trigonometry and a scientific calculator are the following; Substyle Height: Sin SH = Cos Dec x Cos Lat Substyle Distance: Tan SD = Sin Dec / Tan Lat Time and longitude are uniform and symmetrical around the polar style. They are best described on the equatorial disc, a plane perpendicular to the polar style. Solve for the Difference in Longitude as Tan DL = Tan Dec / Sin Lat. This is the angle in time or longitude from the noon meridian to the from the substyle line on the wall. The hour lines are calculated using the Polar Time Angle P where P = DL + t. that equatorial plane intersects the vertical declining or wall plane. Solve for the equivalent dial hour lines as Tan HA = Tan P x Sin SH. Going back to your example, a dial at your latitude 55 and wall declination 30 has a SH = 28.73, SD = 21.04 and DL = 35.18. Moving to 73 latitude for the same wall declination the angles are SH = 14.18, SD = 18.24 and DL = 31.12. Note the difference in the differences in latitude for your location and farther north is 35.18 - 31.12 or 4.06 degrees. There are an infinite number of intermediate points as you move your wall north. All the parameters change: SH, SD and DL. Note for every latitude or declination there is a unique difference in Longitude as Tan DL = Tan Dec / Sin Lat. The line defined by these latitude and longitude points is a smooth curve. I think it is a great circle as you suggest but cannot prove this using plane geometry. The attached sketch is from a presentation "Designing a Sundial from Scratch". I have posted the rest of the slides but not the pictures on the following website: http://groups.msn.com/WalkingShadow I invite you and others to click on this link, join the group and download the presentation from the Walking Shadow documents folder. I hope this is an easy way to make available such information. Also included in the folder is instruction manual, "Sundial Design with a Programmable Scientific Calculator". This manual was written for a workshop at the NASS Chicago conference and was included as a digital bonus in the December NASS Compendium. Roger Bailey Walking Shadow Designs N 48.6 W 123.4 -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Behalf Of Frank Evans Sent: November 25, 2005 3:00 AM To: Sundial Subject: declining dials Greetings, fellow dialists, Following Tony Moss's query about dials on declining walls I am tormented with some fairly pointless thoughts and a question on vertical sundials. A south facing vertical sundial is fairly easily comprehended with its symmetrical hour lines showing the time from six to six and its vertical gnomon inclined to the wall at an angle of ninety degrees minus the latitude so that a dial at the equator would have its gnomon horizontal and one at the pole vertical. But there is a little subtlety in a south facing dial and even more in a declining one. First, the six o'clock hour lines show sunrise and sunset only at the equinoxes, when the sun rises and sets east-west. In the summer the rising and setting points are behind the wall since the sun rises and sets north of east-west and in the winter the sun rises later and sets earlier than six. So in the winter, if those hour lines were not there we would hardly miss them except for interpolation with the hours of 7 am and 5 pm. With the south facing dial the time lines are, as noted, symmetrically grouped around twelve noon. But more fundamentally, they are also symmetrically grouped around the gnomon. Fundamentally, because time lines (not necessarily hour lines) are always grouped symmetrically around the gnomon, whether the dial faces south or not. In a dial which does not face south the style or shadow edge of the gnomon points (almost) to the pole star. The angle between the style and the ground, if the style was long enough to reach the ground, is the latitude. This angle is converted, or resolved, by dialists into two angles known in the hermetic jargon of the fraternity as style height and substyle distance. Presumably these terms descend to us from the days when dialling brethren worked in chords, or lines drawn across a circle, instead of using protractors and pocket calculators or computers as we do today. The substyle distance is the angular deflection of the gnomon from the vertical as would be seen in a full-face photograph. Around it the time lines cluster symmetrically. But what of the style height? It is an angle sticking out of the wall at right angles to the plane of the dial plate but inclined away from the dial plate by the amount of the substyle distance. The combination of style height and substyle distance defines the style and reflects the direction of the earth's axis. Now it has been said that a dial may tell the correct time anywhere in the world where the sun shines so long as the dial plate and gnomon are in the same relation as in the place for which they were made. Perhaps you remember the photograph that Mike Cowham once displayed of the English dial on the wall of the parish church in Stellenbosch in South Africa. People laughed at the ignorance of the vicar who was given the dial from his parish church on the English south coast and set it up there. Of course, the sun goes round anticlockwise in the southern hemisphere so the hour lines read absurdly. But if the dial were to be turned approximately horizontally to be in the plane of southern England it would tell the time well enough. What then of the style height. We have said that the gnomon of a south facing dial sticks out of the wall at an angle of ninety degrees minus the latitude. With a non-south facing dial it is at different angle. For instance, in my latitude, 55 deg. north the style height is 35 deg with a south wall but with a wall declining 30 deg from south it is about 17 deg. Note that the longitude where I live is close to 0 deg. Let us now convey our 30 deg declining dial to latitude 73 deg north and make the gnomon vertical (substyle distance = 0deg) Now the time lines are clustered symmetrically around the gnomon as in a south-facing dial but they are telling the wrong time. The error is the substyle distance, 31 deg, in hours and minutes. Can we correct this? Simple. Just move the dial to 31 deg west and it’s telling the right solar time, corrected for longitude. My question, Chairman, is, are there intermediate points between 55 deg north 0 deg and 73 deg north, 31 deg west where the dial can also be made to tell the right time, and do these points lie on a great circle. Please hasten with your replies as I am not sleeping at all well because of it all. Frank 55N 1W -- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.1.362 / Virus Database: 267.13.7/182 - Release Date: 24/11/2005 -
VD Tetra.pdf
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