Hi John, I have just been studying Jim's book and I have to admit that I was initially confused by the diagrams. However, after redrawing them for myself, I am convinced by Jim's argument and equations provided that azimuth is measured from the southern meridian. Referring to his equations:
tan h0 = cos A / tan ϕ sin x = cos h0. sin d / sin ϕ >From the first equation, we can see that h0 takes the same value for positive and negative azimuths which is what we would expect. However, if the absolute value of A exceeds 90 degrees, then h0 and thus x become negative. This is the reason for: If A < 90, h = x + h0 If A > 90, h = x - h0 Best wishes, Geoff On Wed, 17 Jul 2024 at 19:11, John Goodman via sundial <sundial@uni-koeln.de> wrote: > Diese Nachricht wurde eingewickelt um DMARC-kompatibel zu sein. Die > eigentliche Nachricht steht dadurch in einem Anhang. > > This message was wrapped to be DMARC compliant. The actual message > text is therefore in an attachment. > > > ---------- Forwarded message ---------- > From: John Goodman <johngood...@mac.com> > To: Sundial List <sundial@uni-koeln.de> > Cc: > Bcc: > Date: Wed, 17 Jul 2024 15:11:55 -0400 > Subject: Re: Azimuth and altitude > Thank you for explaining. I can see how sine functions are symmetric > around 90 degrees but I still don't see how 90 degrees is an axis of > symmetry for measuring azimuth. > > If South is treated as 0 degrees instead of 180 degrees, then, as you > describe it, East could be -90 degrees and West would be +90 degrees. The > symmetry around South would then be Az < 0° or > 0° but still not the Az < > 90° or > 90° that was specified in Morrison's equations. > > 1. Calculate the sun's altitude when the declination is zero: tan h0 = > cos A / tan phi > > 2. Calculate an auxiliary angle x from: sin x = (cos h0 sin d) / sin phi > > 3. Calculate sun's altitude, h, for the azimuth and declination from: > > If A < 90, h = x + h0 > If A > 90, h = x - h0 > > NOTE: > > h = altitude > A = azimuth > d = solar declination > phi = latitude > > > On Jul 12, 2024, at 4:40 AM, Hervé Guillemet <guillemet.he...@free.fr> > wrote: > > Hi John, > > If I understand your point, I think that the confusion comes from the fact > that for gnomonistthe origin of azimuth is South, and not North like it > is for navigators (salesmen, pilots). In such case East is -90°, West is > +90°, North is +/- 180°. > With this origin (South = 0°), there is a symetry in the trigonometric > circle : sin (pi - A) = sinA. As an example sin 120° = sin 60° = > 0,866... For the same sine value, you have 2 angles. This is why you need > to determine whether Az is > 90° or < 90° > > I hope that it answers your point. > Best regards Hervé Guillemet > > ------------------------------ > *De: *"John Goodman via sundial" <sundial <sundial@uni-koeln.de> > @uni-koeln.de <sundial@uni-koeln.de>> > *À: *"Sundial List" <sundial <sundial@uni-koeln.de>@uni-koeln.de > <sundial@uni-koeln.de>> > *Envoyé: *Jeudi 11 Juillet 2024 20:03:10 > *Objet: *Azimuth and altitude > > Diese Nachricht wurde eingewickelt um DMARC-kompatibel zu sein. Die > eigentliche Nachricht steht dadurch in einem Anhang. > > This message was wrapped to be DMARC compliant. The actual message > text is therefore in an attachment. > --------------------------------------------------- > https <https://lists.uni-koeln.de/mailman/listinfo/sundial>:// > <https://lists.uni-koeln.de/mailman/listinfo/sundial>lists > <https://lists.uni-koeln.de/mailman/listinfo/sundial>.uni-koeln.de/ > <https://lists.uni-koeln.de/mailman/listinfo/sundial>mailman > <https://lists.uni-koeln.de/mailman/listinfo/sundial>/ > <https://lists.uni-koeln.de/mailman/listinfo/sundial>listinfo > <https://lists.uni-koeln.de/mailman/listinfo/sundial>/ > <https://lists.uni-koeln.de/mailman/listinfo/sundial>sundial > <https://lists.uni-koeln.de/mailman/listinfo/sundial> > > > --------------------------------------------------- > https://lists.uni-koeln.de/mailman/listinfo/sundial > >
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