Hi Robert, You are not the first to get stalled at this point. I've been there and been unable to get past using the conventional math outlined by Waugh. The mathematical treatment by Rohr (pg 76-81) shows the step change in complexity necessary to solve for reclining declining dials.
To help understand why this is so, have a look at the armillary sphere on Luke Coletti's site: http://www.gcstudio.com/ Go to the bigger picture (gcdial_b.jpg) by right clicking on the image. The time lines are the brass meridian circles. Your position is the center of the universe. Your zenith is straight up from the center. The style is the axis of rotation through the center at the poles. Your co-latitude is the angle from the pole to the zenith. The blue ring is the horizontal plane. You can see the hour angles of a horizontal sundial as lines from the center to the intersection of the ring (plane) with the meridian time lines. I consider this to be an excellent visualization of how to design a sundial. Thanks Luke for producing and posting the drawing. In your mind, rotate the blue ring. Change the inclination of the dial from horizontal to vertical. If you rotate it on the east west axis, maintaining the declination south, you will see that the inclination correction is a simple adjustment in the design latitude. When the plane is equal to the latitude, you have an equatorial design. When the ring is vertical, the design is as expected with co-latitude substituted for latitude. Rotating the vertical ring shows the effect of declination changes. The math is still simple. So far the geometry still involves right angle triangles. Now rotate the ring to a plane that both inclines and declines. The intersection of the blue ring with the time lines still gives the appropriate hour angles. It solution seems simple enough but it is not. The solution now involves the solution of spherical triangles that do not contain a right angle. The usual solution of such triangles involves the cosine rule and this brings in all those complicated looking trig functions. This is why the usual treatments describing sundial design do not used the mathematical approach for reclining declining dials. The equations you are seeking do exist but are too complex to reproduce in this e-mail format. Have a look at the equation section of the BSS Glossary at http://www.sundialsoc.org.uk/glossary/index.html . They show the equations for Substyle Height, Substyle Distance and Hour Angles for the common but limited case of a dial roughly south facing reclining slightly from the vertical. This Glossary is also in the NASS Repository CD. When I got stalled at this point, I jumped past the problem by abandoning my own spreadsheet based programs and starting to use the excellent programs written by Francois Blateyron and Fer de Vries. They are very effective in producing all sorts of sundial designs. Fer has posted an outline of his mathematical techniques using stepwise transformations at http://home.iae.nl/users/ferdv/compute.htm . This outline is very useful in understanding his techniques and providing the mathematical logic for others to use in their own programs. Your simple question does not have a simple answer. I hope this helps. Roger Bailey Walking Shadow Designs N 51 W 115 -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Behalf Of Shadow Maker Sent: March 23, 2001 10:59 AM To: sundial@rrz.uni-koeln.de Subject: Reclining/Declining I have built some horizonal, vertical and vertical/declining dials and I am now ready to build a reclining/declining dial, that will compensate for standard time (that part is easy). I have read what Mayall/Mayall, Waugh and Rohr have written about the layout process. I would prefer to use the trig formulas for SD, SH and HourAngles because I can use them in a PostScript file that will render the dial face and print out the values of all of the angles. Rohr does show the formulas for SD and SH but, like the others, he recommends a graphic layout method instead of a computational method for the hour line angles. I was hoping that all of the trig formulas would be available, but can't find them. Waugh says that older editions of the Encyclopedia Britannica had all the trig formulas but dropped them a hundred years ago. Can anyone help? Robert Hough [EMAIL PROTECTED] _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com
