In the date at the bottom of my post, I mistakenly said "48 M, November 18th", but the correct Greenwich date of course was and is:
48 Tu November 19th Michael Ossipoff On Mon, Nov 18, 2019 at 7:42 PM Michael Ossipoff <[email protected]> wrote: > First, an omission in my post about the trig-at-the-dial derivation: > . > I should have said that, by the definition of the cosine: > . > NE = NP/cos h. > . > ------------------ > . > Yes, the thing is that all of the declination-line calculations and > explanations that we've discussed here require telling the person about > some other mathematical topic or method that must be applied. > . > So It's just a question of which one. > . > Dialists are familiar with the calculation of altitude and azimuth, and > often with co-ordinate transformations in general. > . > So, understandably, some dialists would find it more convenient to use > what they already use for varius other things. > . > Which would be more useful for sundials in general? > . > If the declination-lines derivation that you explain to someone is the one > that uses altitude, then the person to whom you're explaining it knows the > altitude-formula, and knows where it comes from, how it's derived. > . > What else is it used for? > . > 1. Babylonian and co-Italian Hours: > . > Well, the altitude-formula is the basis of sunrise and sunset > calculations, and so that person will also know where the Babylonian and > co-Italian hour-lines come from, how they're calculated, and from where > comes the formula by which they're calculated > . > 2. Altitude-Dials: > . > Altitude dials are the most easily-built portable dials. And they're the > most easily-used of the easily-built portable dials. > . > And the altitude-formula is their basis. > . > 3. Reclining-Declining Dials and Co-ordinate Transformations: > . > Of course the formulas for alt and az from h and dec are the general > formulas for spherical co-ordinate transformations. > . > And of course one use of co-ordinate transformations is the constructeeion > of Flat-Dials on any surface in any orientation. ...including > Reclining-Declining Dials. > . > So I'd say that the person you're explaining declination-line construction > to gets a lot of other sundial-useful appications with the altitude formula > alone, and moreso with the altitude and azimuth formulas. Of course the > azimuth-formula's orrery derivation is very similar to that of the > altitude-formula. Explain one, and nothing in the other will be new to the > person. > . > (...other than the easily-explained matter of the quadrant of the > azimuth-answer, depending on the signs of the numerator and denominator in > the formula.) > . > [quote] > You are right that people are more familiar > with altitude and azimuth than they are with > three-dimensional coordinates BUT... > . > You wanted an explanation that was easy to > understand and when you say: > . > > For a particular day, and at an hour shown > > on the dial, calculate the Sun's altitude. > . > I think: Hey, he has introduced a whole > lot of things I don't need to know about > when considering declination lines... > . > If I want to draw the declination lines > then I don't need to know about the day, > or the hour or the altitude or (and you > haven't said this) the latitude. > [/quote] > . > For any stationary sundial, you DO need to know its latitude, regardless > of what method you use for the declination-lines. > . > The day? What's actually needed in the methods that I described isn't > really the day. It's the declination. And that's needed for any method of > drawing a declination-line. To draw a declination-line, you need to know > the declination for which you're drawing the line. > . > Yes, in my discussion, I spoke of the day. But that conversational > reference to the day wasn't intended to imply that the day was an > additional independent-variable needed in addition to the declination. > . > Of course if you want to mark the declination-lines with their > correspoinding dates, then you need to know them. ...regardless of which > declination-line method you use. > . > The hour? Do you need to know the hour? You bet you do! > . > ...just as, with your method, when you've written the equation of that > conic-section, from the interection of the cone with the plane, and you're > plotting the curve--you need to know x in order to calculate y. > . > With the analytic-geometry method, or the altitude-method, or the > trig-at-the-dial method...with any of the methods we've discussed, it > ultimately comes to calculation of a distance from an independent-variabe. > ...such as h or x. > . > The altitude? You don't need to know that, though its calculation is part > of one of the methods. Its calcuation uses quantities that you need > regardless of which method you're using. > . > ------------------ > . > Yes, analytic geometry can construct the hour-lines on any plane, and, > likewise, everyting about a Flat-Dial can be appied to any flat-surface in > any orientation, via a co-ordinate transformation. > . > For a Vertical-Declining Dial, much can be done without > co-ordinate-transformation. > . > ------------------- > . > The analytic-geometry declination-line method might make less use of > trig-functions, and might use less CPU-time. As an explanation, it requires > introducing the person to 3-dimensional analytic-geometry, > . > As is often the case in other matters too, the various methods have their > own advantages. > . > I was just telling some advangtages of the altitude method, or the alt & > az method. > . > Undeniably 3-dimensional analytic-geometry is fun. I once had need for it, > because it's a good way to show why, on a Steregraphic map, any circle on > the globe maps to a circle on the map. > . > Regarding which declination-construction derivation-explanation one uses: > It's a matter of of which math-topic you introduce someone to, and I > emphasize the matter how useful it will be in other sundial topics. > > I don't mean any criticism of other construction-explanation. Of course > there can be any of various considerations, suggesting any of the various > methods. > . > 48 M > November 18th > 0037 UTC > . > Michael Ossipoff >
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