Fred Sawyer
Tue, 31 Aug 2010 07:25:22 -0700
Frank, > > The key paragraph in your note is: > > > "I use this order because both rotations are > about axes IN THE PLANE being used. This > keeps rotation matrices simple and so on." > > There is no real need to do one before the other - unless you are concerned > about making the matrix manipulations simple. Historically, this was not an > issue - and in particular it was not a worry that rotations are not > commutative. The rotations can be defined so that it doesn't matter which > order they are done in - and that is perhaps why it's a question that did > not come up (I don't believe) in historical texts. The inclination is a > rotation about a horizontal line in the dial plane and the declination is a > rotation about a vertical line (whether in the plane or not). If you use > these definitions, you can incline and then decline or you can decline and > then incline - you'll get the same plane. > > In trying to treat the math as matrix manipulation, you'd like the rotation > about the vertical line to be in the plane of the dial - so that's why you > choose the order you do - but it's simply a matter of convenience. > Inclination is the angle the line of greatest slope makes with the > horizontal plane. Declination is the angle the horizontal line in the dial > plane makes with the horizontal meridian. (There may be slight ambiguity if > you begin with a horizontal plane - but this goes away if you draw > east-west/north-south axes that will become the line of greatest slope and > the always horizontal line). > > The history of inclination / declination is indeed messy. Things got to be > so weird that you even have someone like John Wells who labels a dial as > horizontal that you and I would today call a vertical direct south dial. > Even today people use inclination and reclination in different ways. I tend > to think of inclination as the angle above the horizontal, and reclination > (and proclination) as the angle from a vertical plane. > > Fred Sawyer > > > > > > > On Tue, Aug 31, 2010 at 2:37 AM, Frank King <frank.k...@cl.cam.ac.uk>wrote: > >> Dear All, >> >> An intriguing question has been raised on the >> Italian Sundial list by Fabio Savian. Slightly >> re-expressed he is asking... >> >> When specifying the orientation of a plane >> dial, is your order: >> >> A) Declination then Inclination >> >> B) Inclination then Declination ? >> >> This is important because rotations are not >> commutative: R1.R2 does not equal R2.R1. >> >> My own answer is firmly (A) and (slightly >> simplified and avoiding using either word) >> I think as follows: >> >> 1. Start with a vertical plane facing >> due north (sic). >> >> 2. Rotate about a vertical axis until >> it has the correct azimuth. >> >> 3. Rotate about a horizontal axis until >> the tilt is right. >> >> I use this order because both rotations are >> about axes IN THE PLANE being used. This >> keeps rotation matrices simple and so on. >> >> An alternative is: >> >> 1. Start with a vertical plane facing >> due north. >> >> 2. Rotate about a horizontal axis until >> the tilt is right. >> >> 3. Rotate about a vertical axis until >> the azimuth is right. >> >> This is fine but the second rotation is not >> about an axis in the plane being used. >> >> Here I quote a delightful observation by >> John Davis in the BSS Glossary referring >> to a definition of inclination: >> >> Beware: this convention is not followed >> by all authors. >> >> He could be referring to a host of other >> definitions :-) >> >> OK. Which way do you all do things: >> >> Dec then Inc or Inc then Dec ? >> >> There is no right answer but it would be >> interesting to hear who does what! >> >> Frank King >> Cambridge, U.K. >> >> --------------------------------------------------- >> https://lists.uni-koeln.de/mailman/listinfo/sundial >> >> >
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