I've been quietly gnawing on two sundial puzzles for a while now. And
rather than suffer silently, a prisoner of my own inadequacies, I
decided to 'throw myself on the mercy of the court' so to speak.
Puzzle the first. (I'll leave the second puzzle for another day when I
have the time to set it out with a bit of clarity...)
I've been wondering, idly, for about a year, after having made a mock-up
bifilar dial, what the locus of the intersection of the N-S and E-W
threads is, during the day, and over the course of the year.
More recently, partly inspired by the fantastic graphics on Fabio
Savian's webpage (www.nonvedolora.it/bifilare.htm) (I finally figured
out what a wonderful pun "non vedo lora" is; or I _think_ I have). So.
I got out my trigonometry books and tried to figure out what the
equations for x and y would be. I couldn't get Fabio's equations 2 and
3 for x and y to work for me.
So I went back to the diagram in Fred Sawyer's article "Bifilar
Gnomonics" _Journal of the British Astronomical Association_, Jun 1978,
88(4):334-351. and _Bulletin of the British Sundial Society_, Feb 1993,
93(1):36-44, also Feb 1995, 95(1):18-27. (Thanks, once again, Fred for
sending me a copy). After some stumbling around I derived equations for
x and y. And was both pleased, and mortified, to discover that Fred had
presented the same equations later on in his article:
x = g1 sin t/(sin theta tan delta + cos theta cos t)
(10)
y = g2(sin theta cos t - cos theta tan delta)/(sin theta tan delta
+cos theta cos t) (11)
where: theta = latitude; t = solar hour angle; delta = solar
declination; g1 is height of the thread along the y-axis (= 1/cos theta
for a conventional bifilar dial); g2 is the height of the thread along
the x-axis (= tan theta in the usual case).
I then put the equations in a spreadsheet and calculated the x and y
coordinates for a number of hour angles during the day. And repeated
the exercise for different solar declinations. I put the resulting
coordinates into a statistics software package and plotted the results.
(see the attached .gif which is c. 7 kb in size). (I hope the cryptic
legend is sufficiently clear for the purposes of my question)
At first, I was quite pleased because the resulting family of curves was
roughly what I anticipated, from my conceptualisation of the bifilar
dial as a sort of dial cast by a vertical gnomon. But then arose my
puzzlement. Although the lines though the hour marks converge to a
point (as they should)...the angles _between_ hours are not equal.
Hence my puzzlement. Needless to add, I would be most grateful for an
indication of what am I doing wrong!
warmest wishes,
Peter
--
Peter Mayer
Politics Department
Adelaide University, AUSTRALIA 5005
Ph : +61 8 8303 5606
Fax : +61 8 8303 3446
e-mail: [EMAIL PROTECTED]
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