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
