-------- Original Message -------- Subject: Re: Locus ofintersections in bifilar dial Date: Thu, 10 Jun 2004 12:52:47 -0400 From: Stephen Madden <[EMAIL PROTECTED]> To: Peter Mayer <[EMAIL PROTECTED]> References: <[EMAIL PROTECTED]> Hi Peter, I have just gone on vacation so I don't have my notes with me, but I seem to recall that if we eliminate the time from the x,y equations, we obtain equations for a family of hyperbolas for the daily paths of the intersection. The parameters of the hyperbolas depend on the declination. The intersection point lies on one branch of the hyperbola family until the equinoxes, and then it jumps to the other branch. The degenerate hyperbola at the equinoxes is a straight line. It will be another two weeks before I can check my notes, so I hope that someone else in the group can chime in on this. Steve Madden Peter Mayer wrote: > Hi, > > 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 > > > ------------------------------------------------------------------------ > -
