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--- Begin Message ---Thank you Geoff! I can clearly see the significance of the 90° threshold for azimuth in connection with the cosine function of the first equation. I also want to thank Hervé for pointing out that, when south is 0°, azimuth can be measured as - to the east and + to the west. I typically think in terms of 360 positive degrees. I'll have to program these equations now and see if I can calculate results that match others' values. Now that I understand the instructions, I'm ready to move forward. Thanks for everyone's help, John > On Jul 18, 2024, at 12:03 PM, Geoff Thurston <thurs...@hornbeams.com> wrote: > > 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 > <mailto: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 >> <mailto: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 <mailto:johngood...@mac.com>> >>> To: Sundial List <sundial@uni-koeln.de <mailto: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 >>>> <mailto: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 >>>> <mailto:sundial@uni-koeln.de>@uni-koeln.de <mailto:sundial@uni-koeln.de>> >>>> À: "Sundial List" <sundial <mailto:sundial@uni-koeln.de>@uni-koeln.de >>>> <mailto: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. 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