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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