Steve—
…
I know you said you wanted a link, not instructions, but people have been
suggesting how to achieve dial-autocorrection to Local True Solar Time
(LTST) at the standard-meridian, instead of one’s own meridian. So I felt
that it would be justified to comment about it.
…
…even though that autocorrection wouldn’t bring any convenience for the
user, who’d still need a correction (for Eqt), & even though it would
create an inconvenience for anyone who wanted genuinely local LTST, because
they’d have to uncorrect the longitude-correction.
…
First I suggested solution of 3 simultaneous nonlinear equations, written
via coordinate-transformation formulas, with three unknown variables:
Initial horizontal dial-plate rotation about the vertical axis through its
center, the place on the dial-plate circumference for placing the wedge, &
the amount to tip the plate with that wedge.
…
…in order to get the style pointed at the celestial-pole, with the
noon-line in the meridianal-plane. 3 equations in 3 unknowns.
…
Undeniably that would solve the problem, but 3 nonlinear equations would be
a bit of work. That work is unnecessary, because it can be solved
analytically.
…
I described how one could find how to tip the dial-plate:
…
1) The point at the top of the sphere having a great-circle that coincides
with the circumference of the dial-plate, has, in the
equatorial-coordinate-system, a declination equal to the latitude of the
dial. It has an hour-angle (equatorial longitude) of zero.
…
2) Say you’re 7 degrees east of your standard meridian. Rotating the dial 7
degrees westward about the polar axis, the axis of the equatorial-system,
changes the top-point’s equatorial coordinates to (Lat, 7).
…
3) Transform those new equatorial coordinates to the horizontal coordinate
system, to get the altitude & azimuth of the top-point.
…
4) Place the wedge at the edge of the circular dial-plate 180 degrees from
the calculated azimuth of the top point. Tip the dial-plate up, there, by
an angle equal to the complement of the calculated altitude of the
top-point.
…
Now the dial-plate is tipped as it would be if the dial had been rotated 7
degrees westward in equatorial-longitude, hour-angle, about the polar-axis.
But the dial’s noon-line might not be in the meridianal-plane.
…
One way to fix that:
…
5) Rotate the dial-plate in the plane of the dial-face, until the dial
reads the correct LTST at the standard-meridian.
…
That would require carefully marking where the edge of the dial place is,
at several circumference-positions on the table-surfa ce, marking where the
wedge with respect to dial & table-surface, & marking where the dial-plate
touches the wedge.
…
Then lift the dial-plate a bit off the wedge & rotate the dial-plate in the
plane of its dial-surface, & set it back down, making sure that the
dial-plate & wedge are at their original marks.
…
Do that till the dial reads the LTST at the standard-meridian.
…
That dial-rotation sounds laborious & awkward, doing it after the tipping,
with all the position marking & keeping. …especially with the wedge under
the dial-plate.
…
Another way:
…
6) Before the tipping, the style is pointing at the celestial-pole.
Transform that position to the get the pole’s pre-tip coordinates in the
coordinate system whose axis is a horizontal line perpendicular to the
direction in which the direction in which the dial is going to be tipped. Now
add the complement of the calculated top-point-altitude to the longitude in
that system with the horizontal axis. That gives the style’s
pointing-direction’s new longitude in the system with the horizontal tip-axis.
So now you have both of its new coordinates in that system.
…
7) Transform that position to either the horizontal (altazimuth)
coordinate-stem, to get the altitude & azimuth of the style’s new
pointing-direction…or instead to the equatorial-system to get the style’s
pointing-direction, as declination & hour-angle in the equatorial-system.
…
8) That tells you how much the style’s pointing-direction is off, in terms
of its altitude, or its azimuth, or its declination. …whichever of those
you want to use. Its azimuth should of course be zero. Its altitude should
of course equal your latitude, & its declination of course should be 90
degrees. The altitude is probably not a good choice to use, because it
changes more slowly with change in the dial plate rotation. I’d probably
use the declination, because its formula is simpler than that of the
azimuth.
…
9) So, find out how much thequantity for the style’s pointing-direction
that you’re using, say the declination, needs to change, to put it where
it should be. Another coordinate-transformation will tell you how much the
dial-plate would have to rotate in the plane of the dial-face, to achieve
that. That’s the desired dial-plate-rotation.
…
10) So, before tipping, you rotate the dial-plate in its own plane, by
that amount, before you
