Hello John Davis and sundial list members, I think I should give a short comment since I worked on this problem (or at least on a closely related one) years back. I quote from the mail of John Davis.
>The solution, I think is the suggestion of Fer de Vries to use the polar dial >designed originaly by his namesake. This has a polar dialplate and a >gnomon that is >shaped on its UNDERSIDE, I think (from memory) that the shape required is >a cycloid. >This was shown in the NASS Compendium a few years back. The gnomon only >touches the >dialplate on the split noon lines with a gap increasing under it. Thus >the distance >of the part of the gnomon that casts the indicating shadow moves further >from the >dialplate, and in the E'W directions, for increasing times from noon. The >hourlines >are straight and equi-spaced so the whole dial can be rotated to account >for EoT, BST >and longitude. A polar sundial (without EOT correction) is mentioned at the end of Section 3 of my old report http://www.math.tut.fi/~ruohonen/SundialRep.pdf (It can also be found in the NASS Repository CD (March 2001).) The gnomon consists of four identical parts shaped as astroidal cylinders (not cycloidal). And see also my article "Sundials and Mathematical Surfaces" in the Compendium (Vol. 8, March 2001). Using a shifted time (which can then be corrected for by rotating the dial face by a corresponding angle) it should be possible to get a someways simpler gnomon surface (see Section 6 of the report). >I will be interested to see if anyone can come up with a more conventional >"hourglass" gnomon solution - I believe it is impossible (there's a >challenge!). Yes, this is a challenge! The construct I came up with has a gnomon surface divided into eight parts to achieve an EOT correction. These eight surfaces are quite complicated and certainly not of the "hourglass shape" any more, see Figure 46 in Section 7 of the report. Using a shifted time (countermanded by rotation of the dial face) one needs only two gnomon surfaces but they are truly complicated, and even difficult to plot, see Figure 47. One difficulty is that basically exact EOT correction, using "hourglass- shaped" gnomons and straight equispaced hour lines, is impossible already for equitorial sundials, as pointed out in Section 5 of the report, and only approximate solutions can be found. The problem is then much more difficult for polar and other "tilted" sundials. Regards, Keijo Ruohonen -
