NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Douglas Denny
Date: 2010 Sep 1, 13:56 -0700
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
----------------------------------------------------------------