Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
I said it backwards. For positive declination, use NEO. For negative declination, use 180 - NEO. 48 Tu November 19th 2103 Michael Ossipoff --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
I should mention that, when posting about the trig-at-the-dial method, I assumed a positive declination. When the declination is negative, you just use NEO instead of its supplement. 48 Tu November 19th 1534 UTC --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
I've heard that dialists traditionally disregard atmospheric-refraction, when calculating sunrise an sunset times. That allows the use of spherical-trigonometry's tangent-formula, instead of the altitude-formula, a co-ordinate-transformation. But the orrery derivation of the altitude-formula seems just as easy as the derivation of spherical-trigonometry's tangent-formula. In fact, the orrery-derivations of the alt and az formulas seem, to me, easier. ...even though those formulas are larger than the tangent-formula. The tangent formula, being briefer, involves less arithmetic, but the orrery derivation of the alt and az formulas seem more naturally and easily explained. By the way, though I'd explain declination-line construction by the altitude-method, there might be advantage in calculating it by the trig-at-the-dial method. For one thing, the alt & az formulas can have subtraction, which can cause a loss of significant digits (which would only rarely matter, with today's many-digits machines). Also, if you want the measurement to be straightforward, instead of looking for the point on the hour line that's the right distance from the sub-nodus point, which isn't on the hour-lline, then you'd need to calculate the solar altitude and azimuth both. That, and the conversion to rectangular co-ordinates, and then a little work with those co-ordinates, probably amounts to a bit more arithmetic than the trig-at-the-dial method. 48 Tu Novembeer 19th 1524 UTC Michael Ossipoff --- https://lists.uni-koeln.de/mailman/listinfo/sundial
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
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
Oh alright, here's the derivation by plane trig at the dial: . .First, of course this is what gives the natural and easily-explained derivation for a Horizontal-Dial's hour-lines, and so I'll show that first: . I'll designate lines by the letter-names of their two endpoints. . When I state a sequence of three points' letter-names, I'm referring to the angle that they define. . If I want to refer to the triangle that they define, then I'll precede them with the word "triangle". . The following letters will stand for the following points: . N is the nodus. . P is the point where the shortest line from N, perpendicular to ON intersects the ground. . E is the nodus's shadow at the equinox, at the hour of interest (h/15). . D is the nodus's shadow at the declination on the date of interest (dec) at h/15. . O is the origin and intersection of the hour-lines. . Say that, at N, on the polar axis ON, there is mounted a Disk-Equatorial Dial. . NP, that dial's noon-line, has a length of OP sin lat. If that dial's hour-line for hour h/15 is extended to the ground, it extends to E. . PE = NP tan h = OP sin lat tan h. . PE/OP = tan POE (the angle on the dial for the hour-line for hour h/15). . That straighforwardly demonstrates the construction of an hour-line of a Horizontal-Dial, based on a disk-equatorial dial. . Now, for the position of D: . Regarding the triangle NEO: . OE can be gotten as OP/cos POE. . ...or as sqr( OP^2 + PE^2). . NEO = asin(ON/OE). . ...or, if you prefer, = acos(NE/OE) ...or ATAN(ON/NE) . NED = 180 - NEO, because D and O are points on the same line, on opposite sides of E. . So you have NE, NED, and END (the declination, a given quantity). . So you have ASA (angle-side-angle) . So you can solve triangle NDE for DE, by the Law of Sines. . So, measure distance DE, along the hour-line for h, to mark the point on the declination-line for the date of interest. . --- . But I'd rather give the Horizontal-Dial declination-line derivation that uses the Sun's altitude. It seems to me that the orrery derivation of the altitude-formula is easier and clearer to explain than the above trig-at-the-dial derivation for a point on the declination-line. . ...and that someone who'd like a derivation for the Horizontal-Dial's declination-lines, is going to later more likely have use or need for altitude (or terrestrial-distance) or azimuth calculations, than for other trig problems or other 3-dimensional analytic-geometry problems. . 47 Su November 17th 2140 UTC . Michael Ossipoff --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
When I listed trig, in my previous message, I was referring to the fact that there's a 3rd derivation-approach that applies plane trig at the dial itself (but of course not just on the horizontal-plane). With any one of those 3 derivation-approaches, it would be necessary to explain either some 3-dimensional analytic-geometry; or the derivation of at least the formula for altitude (or maybe azimuth too) from h and dec; or trig for the solving of triangles. The trig needed for deriving the altitude and azimuth formulas consists only of direct use of the definitions of the trig functions, whereas the trig-at-the-dial derivation involves several plane triangles, and the solution of a non-right triangle...meaning that the person you're explaining to would have to hear about more trig than that required for deriving the altitude-formula. The altitude and azimuth formulas can be directly and straighforwardly derived by applying the definitions of the trig functions to an orrery. ...for a brief and straightforward derivation that would make sense to anyone with no prior experience in the subject. On Sun, Nov 17, 2019 at 1:42 PM Michael Ossipoff wrote: > I mean, when you're choosing which declination-lines > construction-explanation to use, there's the matter of: Which of the > following is that person more likely to have occasion to use? Or which is > more likely to be of interest and use to someone interested in sundials or > astonomical matters?: > > Altitudes (or terrestrial distances) and azimuths? > > 3-dimensional analytic geometry > > trig > > 3-dimensional analytic geometry and trig are of course useful for many > things. But celestial altitudes and azimuths, and terrestrial distances, > are of frequent and direct use and interest to people interested in > sundials or astronomical matters. > > So that's why I'd give the altitude &/or azimuth explanation for > declination-line construction. > > --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
Frank-- . As you described, a Horizontal-Dials' (or any Flat-Dial's) declination-lines can be constructed, by 3-dimensional analytic geometry, as the intersection of a cone with a plane. . Here's another way: . 1. For a particular day, and at an hour shown on the dial, calclate the Sun's altitude. . 2. From that altitude can be calculated the nodus-shadow's distance of the nodus's shadow from the sub-nodus point. ...by dividing the height of the nodus by the tangent of the Sun's altitude. . 3. Measuring from the sub-nodus point, mark the point on that hour's hour-line at the above-calculated distance. . - . But maybe you'd rather just measure the distance from the hour-lines' intersection-point. In that case: . 1. Calculate both the Sun's altitude and azimuth at a particular date and time. . 2. As above, from that altitude can be calculated the nodus-shadow's distance from the sub-nodus point. That and the azimuth give you polar co-ordinates of the nodus-shadow with respect to the sub-nodus point. . 3. Convert the polar co-ordinates to rectangular co-ordinates with respect to the sub-noduc point. . 4. Add to the north-south co-ordinate the distance between the sub-nodus point and the hour-lines' intersection point. Then you have the rectangular co-ordinates of the nodus-shadow with respect to the hour-lines' intersection point. . 5. Convert the rectangular co-ordinates to polar co-ordinates, and mark the appropriate hour-line at the distance in those polar co-ordinates. . Or, alternatively: . 4. Divide the east-west co-ordinate by the north-south co-ordinate. . 5. Multiply that result by the distance between the sub-nodus point and the north edge of the dial or the construction-page. Mark that distance along that north-edge, from the dial's north-south axis. . 6. From the rectangular-co-ordinates, calculate the nodus-shadow's distance from the sub-nodus point. Measure that distance along the line to the point that you marked on the north-edge. . . So, there's the analytical-geometry solution that you described, and these various ways of doing it via calculation of the Sun's altitude, and maybe azimuth. . And those aren't the only derivations either. . Of all the derivations that I'm aware of, I prefer the altitude or altitude & azimuth, approach, because those calculations are of interest and use for various other matters, for sundials and all sorts of things. . . 47 Su November 17th 1830 UTC . Michael Ossipoff --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
Hi Frank-- Thanks for the thorough and complete explanation about conic-sections for declination-lines for flat-dials. Yes, but I want to be able to explain the actual numerical calculation of the declination-lines to someone. ...to explain the derivation of the actual formulas for the hour-lines and declination-lines. The Horizontal-Dial's hour-lines are easily and briefly explained, but I don't know of as easy an explanation for its declination-lines. I like what you said, but it would mean explaining 3-dimensional analytic geometry to someone, if you wanted two tell them exactly how the line-forumulas are derived. ...not that the other derivations are any easier. But thanks for a remarkably clear, thorough and complete description and explanation of the conic-section nature of the declination-lines for a Flat-Dial. Equatorial Dials (disk, band, or cylinder) are of course the one whose hour and declination-lines don't need explanation (well maybe a tiny bit for the declination lines--but minimal). Next is the Polar-Dial, and then maybe the Tube-Dial. But a Horizontal-Dial, while the most easily-built stationary-dial, has difficultly-explained declination-lines, if you want to tell someone in detail the derivation of the formulas. If you don't use declination-lines, or are willing to not explain them, then a Horizontal iBsut my favsorite is the Cylinder-Equatorial. . . On Sat, Nov 16, 2019 at 3:54 AM Frank King wrote: > Dear Michael, > > You ask: > > > Is there an easy explanation/derivation for the solar-declination lines > on a Horizontal-Dial? > > Yes. Here is the thought process: > > 1. You start with a plane and a point (the point must not be in the > plane) > > 2. Call the plane the 'dial plate' and call the point the 'nodus'. > > 3. Imagine a line drawn from the sun to the nodus. > > 4. Observe that, during a solar day, the line sweeps out a cone. (The > line is a generator.) > > 5. The extension of the line from the sun through the nodus sweeps out a > mirror-cone. > > 6. The common vertex of both cones is the nodus. > > 7. The common axis of both cones is polar oriented. > > 8. The intersection of the mirror cone and the plane dial plate is a > conic section. > > 9. This conic section is the required constant-declination line. > > At this stage, I have made no assumptions about the orientation of the > dial plate or the solar declination but there is an implicit assumption > that the plane and nodus are rigidly attached to the Earth. > > My nine points are best understood by considering some examples: > > EXAMPLE I – The dial plate is parallel to the Earth's equator and the > nodus is on the north side. > > [This is an equatorial dial and applies with a horizontal dial at the > north pole or a vertical direct-north-facing dial at the equator.] > > If the declination is positive, then the intersection of the mirror cone > and the dial plate is a circle whose radius increases as the declination > decreases. This circle is the constant-declination line for the assumed > declination. > > If the declination is zero, the cone and the mirror cone both degenerate > into a disc which is parallel to the dial plate so there is no > intersection. If the declination is negative, then the mirror cone is > wholly on the north side of the dial plate and there is no intersection. > > EXAMPLE II – The dial plate makes an angle of 10° to the equatorial plane. > The nodus is again on the north-side. > > [This case applies with a horizontal dial at 80°N or a vertical > direct-north-facing dial at 10°N.] > > If the declination is greater than 10°, then the sun will always be on the > north side of the dial plate and the intersection of the mirror cone and > the dial plate is an ellipse. This ellipse is the constant-declination > line for the assumed declination. > > If the declination is 10°, the ellipse becomes a parabola. If the > declination is less than 10° (but greater than −10°) then the intersection > is a hyperbola. If the declination is less than −10°, then the mirror cone > is wholly on the north side of the dial plate and there is no intersection. > > EXAMPLE III – The dial plate makes an angle of greater than 23.4° to the > equatorial plane. > > [In the northern hemisphere, this case applies with a horizontal dial > outside the arctic regions and a vertical direct-north-facing dial north of > the Tropic of Cancer.] > > Here, whatever the declination, both the cone and the mirror-cone > intersect the dial plate and the intersection of the mirror cone and the > dial plate is always a hyperbola. > > GENERAL NOTE > > Whatever the orientation of the target plane there will be some location > on the planet where this orientation is the local horizontal. The > declination lines, for that horizontal case, are precisely the declinations > required for the target plane. > > PRIVATE RANT > > Teaching geometry in schools seems to have gone out of fashion in most of > the world. In my
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
Dear Michael, You ask: > Is there an easy explanation/derivation for the solar-declination lines on a Horizontal-Dial? Yes. Here is the thought process: 1. You start with a plane and a point (the point must not be in the plane) 2. Call the plane the 'dial plate' and call the point the 'nodus'. 3. Imagine a line drawn from the sun to the nodus. 4. Observe that, during a solar day, the line sweeps out a cone. (The line is a generator.) 5. The extension of the line from the sun through the nodus sweeps out a mirror-cone. 6. The common vertex of both cones is the nodus. 7. The common axis of both cones is polar oriented. 8. The intersection of the mirror cone and the plane dial plate is a conic section. 9. This conic section is the required constant-declination line. At this stage, I have made no assumptions about the orientation of the dial plate or the solar declination but there is an implicit assumption that the plane and nodus are rigidly attached to the Earth. My nine points are best understood by considering some examples: EXAMPLE I – The dial plate is parallel to the Earth's equator and the nodus is on the north side. [This is an equatorial dial and applies with a horizontal dial at the north pole or a vertical direct-north-facing dial at the equator.] If the declination is positive, then the intersection of the mirror cone and the dial plate is a circle whose radius increases as the declination decreases. This circle is the constant-declination line for the assumed declination. If the declination is zero, the cone and the mirror cone both degenerate into a disc which is parallel to the dial plate so there is no intersection. If the declination is negative, then the mirror cone is wholly on the north side of the dial plate and there is no intersection. EXAMPLE II – The dial plate makes an angle of 10° to the equatorial plane. The nodus is again on the north-side. [This case applies with a horizontal dial at 80°N or a vertical direct-north-facing dial at 10°N.] If the declination is greater than 10°, then the sun will always be on the north side of the dial plate and the intersection of the mirror cone and the dial plate is an ellipse. This ellipse is the constant-declination line for the assumed declination. If the declination is 10°, the ellipse becomes a parabola. If the declination is less than 10° (but greater than −10°) then the intersection is a hyperbola. If the declination is less than −10°, then the mirror cone is wholly on the north side of the dial plate and there is no intersection. EXAMPLE III – The dial plate makes an angle of greater than 23.4° to the equatorial plane. [In the northern hemisphere, this case applies with a horizontal dial outside the arctic regions and a vertical direct-north-facing dial north of the Tropic of Cancer.] Here, whatever the declination, both the cone and the mirror-cone intersect the dial plate and the intersection of the mirror cone and the dial plate is always a hyperbola. GENERAL NOTE Whatever the orientation of the target plane there will be some location on the planet where this orientation is the local horizontal. The declination lines, for that horizontal case, are precisely the declinations required for the target plane. PRIVATE RANT Teaching geometry in schools seems to have gone out of fashion in most of the world. In my day, we were taught how to calculate conic sections at 16 years old. Very best wishes Frank Frank King Cambridge, U.K. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Brief explanation/derivation for Horizontal-Dial's declination-lines?
I'll check it out. I suspect that there's no way that the declination-lines for a Horizontal Dial can be anything like as brief as that for the Polar and Tube Dials, because the horizontal surface, though convenient for building the dial, lacks any of the orientation-convenience of the Polar and Tube Dials. 47 F November 15th 1714 UTC On Fri, Nov 15, 2019 at 11:33 AM Simon Wheaton Smith < illustratingshad...@gmail.com> wrote: > My book, Illustrating Time's Shadow, is now free on.. > > www.illustratingshadows.com > > And covers hDial declination lines, and several ways of drawing them. > > Simon > > > On Fri, Nov 15, 2019, 07:49 Michael Ossipoff wrote: > >> >> Is there an easy explanation/derivation for the solar-declination lines >> on a Horizontal-Dial? I mean, easiness comparable to that of the >> declination-lines for a Polar-Dial or a Tube-Dial (Circumference-Aperature >> Cylindrical-Equatorial Dial)? >> . >> The explanation for the Horizontal-Dial's hour-lines is brief enough, but >> its declination-lines seem to need a considerably longer explanation. >> . >> I ask that because, to offer or suggest a sundial or map-projection, or >> to set up a sundial for others, I prefer one that's easy to explain. It >> seems to me that people would like something better if it can be explained >> to them, preferably without their having to listen to an long explanation. >> . >> Week 47, Friday >> November 15th >> 1450 UTC >> . >> Michael Ossipoff >> --- >> https://lists.uni-koeln.de/mailman/listinfo/sundial >> >> --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Brief explanation/derivation for Horizontal-Dial's declination-lines?
Is there an easy explanation/derivation for the solar-declination lines on a Horizontal-Dial? I mean, easiness comparable to that of the declination-lines for a Polar-Dial or a Tube-Dial (Circumference-Aperature Cylindrical-Equatorial Dial)? . The explanation for the Horizontal-Dial's hour-lines is brief enough, but its declination-lines seem to need a considerably longer explanation. . I ask that because, to offer or suggest a sundial or map-projection, or to set up a sundial for others, I prefer one that's easy to explain. It seems to me that people would like something better if it can be explained to them, preferably without their having to listen to an long explanation. . Week 47, Friday November 15th 1450 UTC . Michael Ossipoff --- https://lists.uni-koeln.de/mailman/listinfo/sundial
