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 the quantity 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 do the tipping, when the dial-plate is sitting horizontal on the table. … Knowing the diameter & circumference of the dial-plate, you could just directly do that rotation on the table before the tipping. That would, again, require pre-marking the position of the dial-plate-edge at several circumferential-positions on the table, but it would be easier than rotating the dial while tipped. Then, when you tip the dial, it should be entirely oriented right for LTST at the standard-meridian. … 11) But maybe you don’t want to have to do that position-marking & maintaining. Then, by another coordinate transformation, determine how much the dial’s time reading should change, from its correct current LTST reading, when you rotate the still-horizontal dial-plate on the table by the desired amount. … Then rotate the dial until the time changes by that calculated amount. Because horizontal-dials aren’t marked linearly, preventing good linear time-reading interpolation, it would be necessary to use the dial’s hour-line construction-formula to calculate exactly where the style-shadow should be when the desired horizontal rotation has been done. … Anyway, when you’ve done that horizontal rotation, the tip the dial as described. … So, each refinement, to make the dial-rotation easier, avoiding some marking & position-maintaining work, adds a few more coordinate-transformations to do. For instance, if one doesn’t mind the more awkward position-marking & keeping, one could just do operations 1 thru 5. On Sun, Mar 26, 2023 at 5:30 PM Steve Lelievre < steve.lelievre.can...@gmail.com> wrote: > Hi, > > Can anyone point me to an existing online calculator for making a wedge > to adjust a horizontal dial to a new latitude and longitude? > > I am not asking for an explanation of how to do the calculation; I just > want to be able to point people to a calculator that has already been > proved on the internet. It should use the original location (latitude > and longitude) and the new location to calculate the angle of slope of > the wedge and the required rotation from the meridian. > > Many thanks, > > Steve > > > --------------------------------------------------- > https://lists.uni-koeln.de/mailman/listinfo/sundial > >
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