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Re: spherical right triangles, napiers circle
From: William Allen
Date: 2004 Jan 28, 15:06 -0800
From: William Allen
Date: 2004 Jan 28, 15:06 -0800
I know of at least two ways to prove the cosine rule (one shown in Nautical Astronomy by Smart and the other in the Kerns, Kell book I mentioned) and there are many different ways of expressing it. Working through the proofs helps understand the different ways it can be expressed. Regards, Bill Allen -----Original Message----- From: Navigation Mailing List [mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Patrick Stanistreet Sent: Wednesday, January 28, 2004 2:34 PM To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM Subject: Re: spherical right triangles, napiers circle Thanks for the link to the online book. I downloaded the entire book as a single zip file only 685 K. Just use the url without the chapter information. In my original email I wasn't too sure what needed to be included in the question. The link provided http://home.t-online.de/home/h.umland/Chapter10.pdf does have the pertinent information regarding napiers circle and the the two rules. The Congleton book I am using has the same information. The reason I am interested in a derivation or a proof of Napiers rules is that the cosine rule for spherical triangles is shown in the Congleton book to follow. Perhaps there are other ways to derive the cosine rule in which case I would still like to read through the proofs. Also I would guess that the derivations of Napiers rules and the circle are historically interesting. Ex: These notes are from page 92-94 of Congletons book Given an oblique sph. triangle with 3 points ABC. A left bottom, B top middle and C right bottom Now form 2 rt. sph. triangles by dropping a perpendicular from B to segment AC. The point of intersection to be called D. The segment from B to D is called y and from A to D is called x and from D to C is called b - x. In triangle DBC by napiers rule Eq1: sin(co - a) = cos(b - x)cos(y) Eq2: cos(a) = cos(b - x)cos(y) using the trig identity for the difference of two angles Eq2 becomes cos(a) = cos(y)( cos(b)cos(x) + sin(b)sin(x) ) Eq3: cos(a) = cos(y)cos(b)cos(x) + cos(y)sin(b)sin(x) Using triangle ABD and napiers rules we can solve for x and y in terms of c and A Eq4: sin(y) = sin(c)sin(A) Eq5: sin(x) = tan(y)cot(A) Eq6: cos(c) = cos(x)cos(y) Using Equations 4,5 and 6 in Equation 3 we get the cosine rule. cos(a) = cos(b)cos(c) + sin(b)sin(c)cos(A) I like the proof but it bugs me about using Napiers rules since I dont have any proof for them. Noyce, Bill wrote: > George Huxtable asks: > > >>Patrick does well to ask where those rules are derived from. >>I haven't seen them before, and wonder if they are true. >>Is Patrick really quoting from the book word-for-word? >>Has he left something out? I'm not very conversant with "Napier's > > rules". > > Yes, there's something left out. Napier's "parts" are typically > arranged in a circle, with the right angle omitted. The parts > are the two sides adjacent to the right angle, and the *complements* > of the other angles and other side. See, for example, the nice > tuorial at http://home.t-online.de/home/h.umland/Chapter10.pdf > > So for George's example, if we consider the right angle at > Lat=0, Lon=0 to be "the" right angle to be omitted, then > the parts are, in order: The 90d length of the arc from > there to the pole, 0d for the 90d angle at the pole, 0d for > the 90d arc back to the equator, 0d for the angle there, > and 90d for the arc along the equator. With these fixes the > formulas test out fine. > > I don't recall having seen a derivation of these rules. I > suspect that given time I could work them out myself. The > critical thing to notice is that an angle on the surface of > the sphere is simply the angle between two planes defining > the great circles, which can be measured in a plane normal > to both of them. Similarly, the "length" of an arc on the > sphere is the angle between its endpoints at the center of > the sphere. > > -- Bill > >