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Re: Impossible lunar example
From: George Huxtable
Date: 2010 Sep 1, 22:22 +0100
From: George Huxtable
Date: 2010 Sep 1, 22:22 +0100
It may be worth clearing this matter up. If Douglas had been following this thread, he would realise that the test for impossibility, that nobody has so far questioned, and applies to both plane triangles and spherical triangles, apparent lunar triangles, and cleared lunar triangles, is as follows- No side of any such triangle can be longer than the sum of the other two. If Douglas tries sketching out a few triangles it will become obvious to him that this must be the case. And the question is not about whether the trig process "works", when starting with an "impossible" apparent triangle, to produce some sort of result. It does. The question is whether that result, of another "impossible" triangle, has any physical meaning. George. contact George Huxtable, at george@hux.me.uk or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. ----- Original Message ----- From: "Douglas Denny"To: Sent: Wednesday, September 01, 2010 9:56 PM Subject: [NavList] Re: Impossible lunar example I admit to not following this thread so might be talking out of turn: but do not understand why you call this triangle impossible? It is not an impossible spherical triangle at all. The limiting case for a spherical triangle is where one side is 180 degrees in which case it occupies a hemisphere. So: each side or angle of a spherical triangle is < 180 the sum of the three sides is between 0 and 360 the sum of the three angles is between 180 and 540 the area of of any sph. triangle must be less than 2.pi.R^2 By co-incidence I have just recently programmed my HP50 for "clearing" Lunar distances with the standard formula which I used as a basis from Cotter's 'History of Nautical Astronomy' p.209 cos D = sin S sin M + (cos d - sin s sin m )(cos S cos M)/(cos s cos m) D=lunar distance; S = true altitude star; M = true altitude of Moon; s = apparent alt star; m = apparent altitude Moon; d = apparent lunar distance. Putting your figures into my calculator for clearing a lunar distance (it includes refraction correction) gives: For the example of the "impossible" triangle LD = 103 ZDmoon = 71 ZDsun = 19 Cleared distance = 103.061 degrees. Douglas Denny. Chichester. England. ====================== Original Message:- One way to explain why clearing the lunar distance on the impossible triangle does not fail and proceeds as, George put it, "without meeting along the way a sin or cos greater than 1, or a square root of -1" is that it is operationally equivalent to clearing the lunar distance on a particular "possible" triangle. To see this, consider clearing the lunar distance using the cosine formula of spherical trigonometry in the form cos(LD) = cos(ZDmoon)*cos(ZDsun) + sin(ZDmoon)*sin(ZDsun)*cos(Z) LD = lunar distance ZDmoon = Moon's zenithal distance ZDsun = Sun's zenithal distance Z = Sun and Moon azimuth difference For the example of the impossible triangle LD = 103 ZDmoon = 71 ZDsun = 19 Note however that cos(ZDmoon)*cos(ZDsun) = cos(180 - ZDmoon)*cos(180 - ZDsun) sin(ZDmoon)*sin(ZDsun) = sin(180 - ZDmoon)*sin(180 - ZDsun) And hence clearing the lunar distance on the impossible triangle is operationally equivalent to clearing the lunar distance on a triangle with sides LD = 103 ZDmoon = 180 - 71 = 109 ZDsun = 180 - 19 = 161 which is an entirely "possible" triangle. Of course with ZD's > 90 this is not one that would arise in practice but the operations performed in the course of lunar distance clearing are all trigonometrically valid and geometrically meaningful even though a positive correction (refraction + dip etc.) applied to the ZD's of the impossible triangle corresponds to a negative correction applied to the sides of its "possible" cousin, Robin Stuart ---------------------------------------------------------------- NavList message boards and member settings: www.fer3.com/NavList Members may optionally receive posts by email. To cancel email delivery, send a message to NoMail[at]fer3.com ----------------------------------------------------------------