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 <f...@cl.cam.ac.uk> 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 day, we were taught how to calculate conic sections at > 16 years old. > > Very best wishes > Frank > > Frank King > Cambridge, U.K. >
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