Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17
*Dear Mr Gianni Ferrari,* ** * Greetings, happy new year and Merry Christmas to you and all members.* ** * As I recall, the azimuth of a body and its time of transit are not affected by refraction. Please kindly do comment.* ** * Also, I would like to have a good and applied article on astronomical corrections (Refraction, Parallax, Dip of Horizon and Semi-Diameter.) I wonder if any respectful member could help.* ** *Best regards,* ** *Mashallah Ali-Ahyaie* On Dec 26, 2007 2:30 PM, <[EMAIL PROTECTED]> wrote: > Send sundial mailing list submissions to >sundial@uni-koeln.de > > To subscribe or unsubscribe via the World Wide Web, visit >https://lists.uni-koeln.de/mailman/listinfo/sundial > or, via email, send a message with subject or body 'help' to >[EMAIL PROTECTED] > > You can reach the person managing the list at >[EMAIL PROTECTED] > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of sundial digest..." > > > Today's Topics: > > 1. R: Re: Azimuth of Sunrise - Sunset ([EMAIL PROTECTED]) > > > -- > > Message: 1 > Date: Tue, 25 Dec 2007 22:58:32 +0100 (GMT+01:00) > From: "[EMAIL PROTECTED]" <[EMAIL PROTECTED]> > Subject: R: Re: Azimuth of Sunrise - Sunset > To: > Message-ID: <[EMAIL PROTECTED]> > Content-Type: text/plain;charset="UTF-8" > > > And the air refraction ? > > Because of refraction, the sun is already > below the horizon when we observe > the upper limb kiss the horizon. In > this moment the altitude of the Sun?s center is about > ?(34+14) = -48? > (the refraction at 0? degree is about 34?) > The Azimuth of the point > where we begin to see the Sun (limb) is then different from the > theoretical and geometrical value, also if the differences are small at > mean latitudes. > > With Lat = 45 on Winter Solstice, the Azimut of the > Sun?s limb (when it appears) is about 56.8 degree ; the Azimuth of the > Sun?s center 56.4? and the theoretical Azimuth 55.8?. > The differences > increase when Latitude increases. > With Lat.= 66?. the Sun?s limb > appears with an Azimuth about 7 degree more than the theoretical value > (18.8 and 11.9?) > > My best wishes for a sunny New Year ! > Gianni Ferrari > > > > > -- > > ___ > sundial mailing list > sundial@uni-koeln.de > https://lists.uni-koeln.de/mailman/listinfo/sundial > > > End of sundial Digest, Vol 24, Issue 17 > *** > -- Mashallah Ali-Ahyaie --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Azimuth of Sunrise - Sunset Proof
Dear Roger, You chide us :-) > ... few have been doing their homework, proving that > Cos Phi = Sin Lat / Cos Dec The truth is that Geoff Thurston supplied the answer (well almost the answer) on Christmas Eve so I thought we could all have a rest! Geoff and you have both referred to the same crucial spherical triangle. He calls it the PZX triangle and you call it the Navigation Triangle. I am happy with either term. It may be that some readers are not familiar with this triangle so let's just rehearse it... The three vertices are: P the celestial pole Z the Zenith (vertically above the observer) X the sun (or other celestial body) The angles at these vertices are: at P H" (180 - the hour-angle of the sun) at Z A (the azimuth of the sun) at X p (what you call the path of the sun) [I measure hour-angle from midnight and the azimuth clockwise round from due north.] The three sides of the triangle are: opposite P h' (90 - altitude of the sun) opposite Z dec' (90 - declination of the sun) opposite X phi' (90 - observer's latitude) [Alas, I always follow Meeus and use phi for latitude!] The first cosine rule can be applied to each vertex in turn: cos(h') = cos(phi').cos(dec') + sin(phi').sin(dec').cos(H") cos(dec') = cos(h').cos(phi') + sin(h').sin(phi').cos(A) cos(phi') = cos(dec').cos(h') + sin(dec').sin(h').cos(p) [Note how the arguments change cyclically: h', dec', phi'] At sunrise and sunset the altitude h=0 so h'=90 and cos(h')=0 and sin(h')=1. Noting also that cos(H") = -cos(H) we can simplify all three: 0 = cos(phi').cos(dec') - sin(phi').sin(dec').cos(H) cos(dec') = sin(phi').cos(A) cos(phi') = sin(dec').cos(p) Noting now that cos(90-x)=sin(x) and sin(90-x)=cos(x) gives: 0 = sin(phi).sin(dec) - cos(phi).cos(dec).cos(H) sin(dec) = cos(phi).cos(A) sin(phi) = cos(dec).cos(p) Your three results then follow: cos(H) = tan(phi).tan(dec) cos(A) = sin(dec)/cos(phi) cos(p) = sin(phi)/cos(dec) I don't suppose many people will read this far but I shall use this message for reference purposes myself!! You are wise to steer clear of Clairaut's Law which comes up in the big subject of differential geometry where you also need to know about the calculus of variation. I thought such matters were fine for those who study relativity but not needed in the diallists' toolkit. Then some client came up with an absurd request... Oh, and I like the hat. Best wishes Frank --- https://lists.uni-koeln.de/mailman/listinfo/sundial
R: Does Refraction Affect Azimuth?
Dear Mr Mashallah Ali-Ahyaie, it' true! The azimuth of a body and its time of transit are not affected by refraction, that is they don't change even if the refraction changes or it is not present. On the contrary the same thing is not true if we consider the time and the azimuth of the Sun when it is at a given altitude above (or under) the horizon. At dawn if the refraction is present we see that the Sun reaches a given altitude some time before the instant in which we would see it without refraction, and therefore with an Azimuth greater in absolute value. I attach a simple sketch to explain the phenomenon Best wishes Gianni Ferrari azimuth alba.pdf Description: Adobe PDF document --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17
Mashallah, I believe you are correct about azimuth and time of transit being unaffected by refraction. "Astronomical Algorithms" by Jean Meeus has a good section on calculating effects of refraction and the sun's semi-diameter. Also a valuable section about how to correct for the effects of Nutation. I'm not so sure about the other topics you mention. I would guess parallax of the sun from the perspective of the earth's diameter is covered in surveying books, or perhaps one of the astronomical trigonometry books by Smart. -Bill Gottesman - Original Message - From: Mashallah Ali-Ahyaie To: sundial@uni-koeln.de Sent: Thursday, December 27, 2007 4:24 AM Subject: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17 Dear Mr Gianni Ferrari, Greetings, happy new year and Merry Christmas to you and all members. As I recall, the azimuth of a body and its time of transit are not affected by refraction. Please kindly do comment. Also, I would like to have a good and applied article on astronomical corrections (Refraction, Parallax, Dip of Horizon and Semi-Diameter.) I wonder if any respectful member could help. Best regards, Mashallah Ali-Ahyaie On Dec 26, 2007 2:30 PM, <[EMAIL PROTECTED]> wrote: Send sundial mailing list submissions to sundial@uni-koeln.de To subscribe or unsubscribe via the World Wide Web, visit https://lists.uni-koeln.de/mailman/listinfo/sundial or, via email, send a message with subject or body 'help' to [EMAIL PROTECTED] You can reach the person managing the list at [EMAIL PROTECTED] When replying, please edit your Subject line so it is more specific than "Re: Contents of sundial digest..." Today's Topics: 1. R: Re: Azimuth of Sunrise - Sunset ( [EMAIL PROTECTED]) -- Message: 1 Date: Tue, 25 Dec 2007 22:58:32 +0100 (GMT+01:00) From: " [EMAIL PROTECTED]" <[EMAIL PROTECTED]> Subject: R: Re: Azimuth of Sunrise - Sunset To: Message-ID: <[EMAIL PROTECTED]> Content-Type: text/plain;charset="UTF-8" And the air refraction ? Because of refraction, the sun is already below the horizon when we observe the upper limb kiss the horizon. In this moment the altitude of the Sun?s center is about ?(34+14) = -48? (the refraction at 0? degree is about 34?) The Azimuth of the point where we begin to see the Sun (limb) is then different from the theoretical and geometrical value, also if the differences are small at mean latitudes. With Lat = 45 on Winter Solstice, the Azimut of the Sun?s limb (when it appears) is about 56.8 degree ; the Azimuth of the Sun?s center 56.4? and the theoretical Azimuth 55.8?. The differences increase when Latitude increases. With Lat.= 66?. the Sun?s limb appears with an Azimuth about 7 degree more than the theoretical value (18.8 and 11.9?) My best wishes for a sunny New Year ! Gianni Ferrari -- ___ sundial mailing list sundial@uni-koeln.de https://lists.uni-koeln.de/mailman/listinfo/sundial End of sundial Digest, Vol 24, Issue 17 *** -- Mashallah Ali-Ahyaie -- --- https://lists.uni-koeln.de/mailman/listinfo/sundial --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17
Hi Bill, If I recall your comments at the NASS meeting correctly, while azimuth might be unaffected by refraction, because the "real time" of sunset is later due to refraction, the azimuth at a later time needs to be calculated. That is, for us at North latitudes the sunset is further North than a normal calculation of sunset would give. Assume no objects on the horizon. Did I miss something? Warren - Original Message - From: Bill Gottesman To: Mashallah Ali-Ahyaie ; sundial@uni-koeln.de Sent: Thursday, December 27, 2007 11:12 AM Subject: Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24,Issue 17 Mashallah, I believe you are correct about azimuth and time of transit being unaffected by refraction. "Astronomical Algorithms" by Jean Meeus has a good section on calculating effects of refraction and the sun's semi-diameter. Also a valuable section about how to correct for the effects of Nutation. I'm not so sure about the other topics you mention. I would guess parallax of the sun from the perspective of the earth's diameter is covered in surveying books, or perhaps one of the astronomical trigonometry books by Smart. -Bill Gottesman - Original Message - From: Mashallah Ali-Ahyaie To: sundial@uni-koeln.de Sent: Thursday, December 27, 2007 4:24 AM Subject: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17 Dear Mr Gianni Ferrari, Greetings, happy new year and Merry Christmas to you and all members. As I recall, the azimuth of a body and its time of transit are not affected by refraction. Please kindly do comment. Also, I would like to have a good and applied article on astronomical corrections (Refraction, Parallax, Dip of Horizon and Semi-Diameter.) I wonder if any respectful member could help. Best regards, Mashallah Ali-Ahyaie On Dec 26, 2007 2:30 PM, <[EMAIL PROTECTED]> wrote: Send sundial mailing list submissions to sundial@uni-koeln.de To subscribe or unsubscribe via the World Wide Web, visit https://lists.uni-koeln.de/mailman/listinfo/sundial or, via email, send a message with subject or body 'help' to [EMAIL PROTECTED] You can reach the person managing the list at [EMAIL PROTECTED] When replying, please edit your Subject line so it is more specific than "Re: Contents of sundial digest..." Today's Topics: 1. R: Re: Azimuth of Sunrise - Sunset ( [EMAIL PROTECTED]) -- Message: 1 Date: Tue, 25 Dec 2007 22:58:32 +0100 (GMT+01:00) From: " [EMAIL PROTECTED]" <[EMAIL PROTECTED]> Subject: R: Re: Azimuth of Sunrise - Sunset To: Message-ID: <[EMAIL PROTECTED]> Content-Type: text/plain;charset="UTF-8" And the air refraction ? Because of refraction, the sun is already below the horizon when we observe the upper limb kiss the horizon. In this moment the altitude of the Sun?s center is about ?(34+14) = -48? (the refraction at 0? degree is about 34?) The Azimuth of the point where we begin to see the Sun (limb) is then different from the theoretical and geometrical value, also if the differences are small at mean latitudes. With Lat = 45 on Winter Solstice, the Azimut of the Sun?s limb (when it appears) is about 56.8 degree ; the Azimuth of the Sun?s center 56.4? and the theoretical Azimuth 55.8?. The differences increase when Latitude increases. With Lat.= 66?. the Sun?s limb appears with an Azimuth about 7 degree more than the theoretical value (18.8 and 11.9?) My best wishes for a sunny New Year ! Gianni Ferrari -- ___ sundial mailing list sundial@uni-koeln.de https://lists.uni-koeln.de/mailman/listinfo/sundial End of sundial Digest, Vol 24, Issue 17 *** -- Mashallah Ali-Ahyaie --- https://lists.uni-koeln.de/mailman/listinfo/sundial -- --- https://lists.uni-koeln.de/mailman/listinfo/sundial --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17
Hello Mashallah, My reference on this topic is "The American Navigator" H.O. Pub No. 9, originally by Nathaniel Bowditch and the altitude correction tables inside the front cover of the Nautical Almanac. These cover the topic very well from a navigators' point of view. There are additional refraction corrections for temperature, temperature gradient and atmospheric pressure. Height of eye and dip short due to a false horizon are also important. The Newgrange video showed the rising sun was visible from the top of the mound significantly before lighting the window. These effects are quite pronounced in Canada at higher latitudes and colder temperatures. An astronomer friend in Alberta bought a farm and built his observatory based on the lights of Calgary being far enough away to be below the horizon. This was based on observations on a summer night. On clear cold winter nights he was surprised and disappointed to see the lights of the distant city come into direct view due to increased refraction. The experience of Tony Moss and the first light on the Longyearbyen sundial at 78º 13' North are relevant here. The date of the first return of the sun can vary do to refraction changing with the weather conditions. John Dunn's photo at http://www.arcticlight.com/ shows the path of the midnight sun quite well. I was once on a ski tour* with John. We were amazed at his comfort waking around in the snow in underwear and flip flops when we were bundled up in down parkas. His metabolic rate was quite a bit higher than ours and quite suitable for arctic explorer. Regards, Roger Bailey *Not the Baffin Island traverse! - Original Message - From: Mashallah Ali-Ahyaie To: sundial@uni-koeln.de Sent: Thursday, December 27, 2007 1:24 AM Subject: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17 Dear Mr Gianni Ferrari, Greetings, happy new year and Merry Christmas to you and all members. As I recall, the azimuth of a body and its time of transit are not affected by refraction. Please kindly do comment. Also, I would like to have a good and applied article on astronomical corrections (Refraction, Parallax, Dip of Horizon and Semi-Diameter.) I wonder if any respectful member could help. Best regards, Mashallah Ali-Ahyaie --- https://lists.uni-koeln.de/mailman/listinfo/sundial
To Open the URL, Password Is Required
*Dear Mr Gianni Ferrari,* ** * Greetings, thanks for your explanation. I couldn't open the following URL, for your attached file, since password is required:* * https://lists.uni-koeln.de/mailman/private/sundial/attachments/20071227/b2a54e25/attachment-0001.pdf * <https://lists.uni-koeln.de/mailman/private/sundial/attachments/20071227/b2a54e25/attachment-0001.pdf> *Best regards,* ** *Mashallah Ali-Ahyaie* --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17
Warren, It is good to hear from you. And now I know at least one person was awake during my talk. I think you are exactly right. Did I misinterpret Mashallah's question? By "Time of Transit," I took that to mean when the sun or a star passes the north-south meridian overhead, and not the sunrise or sunset. Frankly, I forgot that this question was asked in the context of the solstice sunrise at Newfane. Refraction does not affect azimuth at any given time, but it affects the azimuth of sunset because it also affects the time of (apparent) sunset. -Bill - Original Message - From: Warren Thom To: Bill Gottesman ; Mashallah Ali-Ahyaie ; sundial@uni-koeln.de Sent: Thursday, December 27, 2007 12:47 PM Subject: Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24,Issue 17 Hi Bill, If I recall your comments at the NASS meeting correctly, while azimuth might be unaffected by refraction, because the "real time" of sunset is later due to refraction, the azimuth at a later time needs to be calculated. That is, for us at North latitudes the sunset is further North than a normal calculation of sunset would give. Assume no objects on the horizon. Did I miss something? Warren - Original Message - From: Bill Gottesman To: Mashallah Ali-Ahyaie ; sundial@uni-koeln.de Sent: Thursday, December 27, 2007 11:12 AM Subject: Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24,Issue 17 Mashallah, I believe you are correct about azimuth and time of transit being unaffected by refraction. "Astronomical Algorithms" by Jean Meeus has a good section on calculating effects of refraction and the sun's semi-diameter. Also a valuable section about how to correct for the effects of Nutation. I'm not so sure about the other topics you mention. I would guess parallax of the sun from the perspective of the earth's diameter is covered in surveying books, or perhaps one of the astronomical trigonometry books by Smart. -Bill Gottesman - Original Message - From: Mashallah Ali-Ahyaie To: sundial@uni-koeln.de Sent: Thursday, December 27, 2007 4:24 AM Subject: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17 Dear Mr Gianni Ferrari, Greetings, happy new year and Merry Christmas to you and all members. As I recall, the azimuth of a body and its time of transit are not affected by refraction. Please kindly do comment. Also, I would like to have a good and applied article on astronomical corrections (Refraction, Parallax, Dip of Horizon and Semi-Diameter.) I wonder if any respectful member could help. Best regards, Mashallah Ali-Ahyaie On Dec 26, 2007 2:30 PM, <[EMAIL PROTECTED]> wrote: Send sundial mailing list submissions to sundial@uni-koeln.de To subscribe or unsubscribe via the World Wide Web, visit https://lists.uni-koeln.de/mailman/listinfo/sundial or, via email, send a message with subject or body 'help' to [EMAIL PROTECTED] You can reach the person managing the list at [EMAIL PROTECTED] When replying, please edit your Subject line so it is more specific than "Re: Contents of sundial digest..." Today's Topics: 1. R: Re: Azimuth of Sunrise - Sunset ( [EMAIL PROTECTED]) -- Message: 1 Date: Tue, 25 Dec 2007 22:58:32 +0100 (GMT+01:00) From: " [EMAIL PROTECTED]" <[EMAIL PROTECTED]> Subject: R: Re: Azimuth of Sunrise - Sunset To: Message-ID: <[EMAIL PROTECTED]> Content-Type: text/plain;charset="UTF-8" And the air refraction ? Because of refraction, the sun is already below the horizon when we observe the upper limb kiss the horizon. In this moment the altitude of the Sun?s center is about ?(34+14) = -48? (the refraction at 0? degree is about 34?) The Azimuth of the point where we begin to see the Sun (limb) is then different from the theoretical and geometrical value, also if the differences are small at mean latitudes. With Lat = 45 on Winter Solstice, the Azimut of the Sun?s limb (when it appears) is about 56.8 degree ; the Azimuth of the Sun?s center 56.4? and the theoretical Azimuth 55.8?. The differences increase when Latitude increases. With Lat.= 66?. the Sun?s limb appears with an Azimuth about 7 degree more than the theoretical value (18.8 and 11.9?) My best wishes for a sunny New Year ! Gianni Ferrari -- ___ sundial mailing list
Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24,
There is a Java applet showing the effects of refraction and flattening at: http://www.jgiesen.de/refract/index.html If it came from this list originally, I apologize in advance. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17
As Gianni Ferrari pointed out, the simplified equations solve for the theoretical time, direction and path of sunset when the altitude is zero. Reality is different. You can always use the full spherical triangle sin sin sin cos cos cos equations with an input of say 49' to account for the average values of semidiameter and refraction. Gianni gave a couple of examples. I worked through a couple of others in an article I am writing on designing a sunset sundial from scratch. In that case, the last crack of sunset was 5 minutes later at Lat 37º in mid November. Frank King gave the full equations for the solution of the triangle. They look formidable but are actually quite solvable with a normal scientific calculator. Frank gave the true Cosine form for the sides which are CoLat, CoDec and CoAlt. I prefer to substitute Cos (90-x) = sin x as outlined below. Your known inputs are the sides: Declination, Dec, Latitude Lat and Altitude Alt and you solve for final term, the desired contained angle in the triangle, time t, direction Az, or path Phi. Time: Sin Alt= Sin Lat x Sin Dec + Cos Lat x Cos Dec x Cos t Direction: Sin Dec = Sin Lat x Sin Alt + Cos Lat x Cos Dec x Cos Az Path: Sin Lat = Sin Alt x Sin Dec +Cos Alt x Cos Dec x Cos Phi >From these equations it is easy to see what disappears when the Altitude is >zero. to give the simplified forms. See "Sunset Phenomenon" a 1999 NASS presentation, publication #17 at my website. www.walkingshadow.info . Regards, Roger Bailey - Original Message - From: Warren Thom To: Bill Gottesman ; Mashallah Ali-Ahyaie ; sundial@uni-koeln.de Sent: Thursday, December 27, 2007 9:47 AM Subject: Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24,Issue 17 Hi Bill, If I recall your comments at the NASS meeting correctly, while azimuth might be unaffected by refraction, because the "real time" of sunset is later due to refraction, the azimuth at a later time needs to be calculated. That is, for us at North latitudes the sunset is further North than a normal calculation of sunset would give. Assume no objects on the horizon. Did I miss something? Warren - Original Message - From: Bill Gottesman To: Mashallah Ali-Ahyaie ; sundial@uni-koeln.de Sent: Thursday, December 27, 2007 11:12 AM Subject: Re: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24,Issue 17 Mashallah, I believe you are correct about azimuth and time of transit being unaffected by refraction. "Astronomical Algorithms" by Jean Meeus has a good section on calculating effects of refraction and the sun's semi-diameter. Also a valuable section about how to correct for the effects of Nutation. I'm not so sure about the other topics you mention. I would guess parallax of the sun from the perspective of the earth's diameter is covered in surveying books, or perhaps one of the astronomical trigonometry books by Smart. -Bill Gottesman - Original Message - From: Mashallah Ali-Ahyaie To: sundial@uni-koeln.de Sent: Thursday, December 27, 2007 4:24 AM Subject: Does Refraction Affect Azimuth? / Re: sundial Digest, Vol 24, Issue 17 Dear Mr Gianni Ferrari, Greetings, happy new year and Merry Christmas to you and all members. As I recall, the azimuth of a body and its time of transit are not affected by refraction. Please kindly do comment. Also, I would like to have a good and applied article on astronomical corrections (Refraction, Parallax, Dip of Horizon and Semi-Diameter.) I wonder if any respectful member could help. Best regards, Mashallah Ali-Ahyaie --- https://lists.uni-koeln.de/mailman/listinfo/sundial
How About the Moon? / Re: R: To Open the URL, Password Is Required
*Dear Mr Gianni Ferrari,* ** * Greetings, thank you very much for your explanation as follows and the enclosure:* * > > Dear Mr Mashallah Ali-Ahyaie, > it' true! > The azimuth of a body and its > time of transit are not affected by refraction, that is they don't > change even if the refraction changes or it is not present. > On the > contrary the same thing is not true if we consider the time and the > azimuth of the Sun when it is at a given altitude above (or under) the > horizon. > At dawn if the refraction is present we see that the Sun > reaches a given altitude some time before the instant in which we > would see it without refraction, and therefore with an Azimuth > greater in absolute value. > I attach a simple sketch to explain the > phenomenon > Best wishes > Gianni Ferrari > * *By the same token, this explanation will also apply for the moon rise and moon set, while calculating its azimuth. Am I right?* ** *Best regards,* ** *Mashallah Ali-Ahyaie* On Dec 28, 2007 2:44 AM, [EMAIL PROTECTED] <[EMAIL PROTECTED]> wrote: > > > Here is the same drawing in JPG > Best wishes > Gianni Ferrari --- https://lists.uni-koeln.de/mailman/listinfo/sundial
