Dear David, Well, here is a bit more on Cycloid Dials and in particular the book form I mentioned. I thank Fred Sawyer and Thys de Vries for their information on the basic polar cycloid dial.
The actual curve is not quite a Cycloid, nor an Astroid, but actually an Evolute of an Ellipse, since even the rotation and orbit of the earth is very slightly non-circular. A Cycloid, being easier to construct, is extremely close in shape and works quite well as the gnomon for a sundial. It's main property for our current purpose is that when placed in the equatorial plane in contact with a tangent placed at an equal distance from it's cusps and the pair of lines rotated within that plane the shadow cast by the convex portion of the cycloid on the tangent line marks out evenly spaced points for evenly spaced angles of rotation. ( Fred Sawyer says this much better!) When placed with the cycloid in the equatorial plane and the tangent perpendicular to a line that is parallel to the polar axis, the apparent motion of the sun at 15 degrees per hour will make a demarcation between the light and shadow that is constantly spaced. If the cycloid and tangent are drawn up along the axis, making a flat sheet from the tangent and a curved sheet from the cycloid, and the then hour lines marked on the flat sheet we have a sundial with evenly spaced parallel hour lines. I first saw this many years ago in an article by Thys de Vries of the Netherlands. It is brought into current light by an article by Fred Sawyer III in the NASS Compendium of December 1998 called "Cycloid Polar Sundial". The Parametric equation for a cycloid is x=r(t-sin(t)) and y=r(1-cos(t)) or y=(a^2/3-x^2/3)^3/2 and is among other things the locus of a point on a circle which rotates without slippage along a straight line. It has other properties and can be described by other sets of loci. The parametric equation for an evolute of an ellipse is x=((a^2- b^2)*cos(t)^3)/a and y=((a^2-b^2)*sin(t)^3)/b with a and b the axes of the ellipse. The hours of one half of the sundial above start at 6 AM with no shadow going to all shadow at 12 noon. ( assuming Local Apparent Time) The other half of the sundial starts at 12 noon with all shadow and goes to 6 PM with no shadow. Each of these halves can be separated, rotated 90 degrees and used as east and west facing vertical dials as mentioned in Fred Sawyer's article. In which case they start with all shadow at the 6AM or 6PM points and each have no shadow at noon. If we take these two dials and place them with the two flat sheets in the same place, The hour numbers would be the same on both sides of the sheet. My idea was to continue the curves on around to become pages in an open book where no continued portion of them struck a shadow on the marked hour lines. This would already be great looking book dial with one page sticking out from the center of the book with the hours marked on it. To go further I would mark on the curves themselves where the shadows of the other curves would fall by extending a tangent from one curve through the flat sheet and hour line to the other curve and so mark it. The center sheet could then be removed leaving an open book whose pages would make shadows on each other telling the time. If the book were then rotatable around the polar axis, correction for longitude variation, the equation of time and daylight savings time could be set. It is too bad we can't send drawings through the list, which would make the idea clearer. I believe there are other book sundials that work in a similar fashion, but using the cycloid curve it becomes easier to design them, or so it seems to me. Enjoy the Light! Edley McKnight [43.126N 123.357W] > Dear Edley, > > I am very interested in the cycloid dial. Could you send me a > mathematical description of this type of dial? > > Thanks > > David Pratten > www.sunlitdesign.com > > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] On Behalf Of Edley > Sent: Saturday, 15 December 2001 8:22 AM > To: [email protected] > Subject: Polar Cycloid Book Dial > > > Dear Membership, > > Polar Cycloid Book Dial. > > In the NASS Compendium of December 1998 Page 24, which is part of > Fred Sawyer's article on the Cycloid Polar Dial, it mentions making a > vertical dial facing east or west with the noon line now occurring when > the surface is fully exposed to the sun. The idea of both an east and > west cycloid polar dial mounted together with the flat surface mounted > between them occurred to me. The resulting shapes reminded me of a number > of dials shaped like books that I'd seen when a child. If the curves were > then taken back from the (now) 6 O'Clock edges to form the two leaves of a > book, how neat it would be. I then thought how easy it would be to let > each cycloid's shadow fall on the other, and how easy it would be to > project the new hour lines on to each surface, ie. removing the flat > sheet. (The hour lines are no longer equal spaced and parallel of course). > An easy to design polar cycloid book dial emerges! This may well not be > a first thought of such a design, but it seems quite a neat idea for > places of learning. > Tilted at the proper polar angle it would keep quite good Local > Solar Time. If it were arranged to rotate around the polar axis it > could be set to show standard mean time if set whenever the equation of > time, longitude correction and daylight savings time required it. The two > pages themselves could include whatever text would be appropriate. > > What do you think? > > Edley McKnight > >
