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
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