Here is a small amendment to my previous message. Although h0 becomes negative x does not so I should have written:
if the absolute value of A exceeds 90 degrees, then h0 and thus x becomes negative. This is the reason for: If A < 90, h = x + h0 If A > 90, h = x - h0 Geoff On Thu, 18 Jul 2024 at 11:33, Geoff Thurston <thurs...@hornbeams.com> wrote: > 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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