Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
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 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
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
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
