Hello Mike, You wrote:
>The only thought I had was that the line will not be a straight line at >higher latitudes but a curve of some sort. Your supposition that the line curves is correct. Except at the equinoxes, when it is a straight east-west line. Otherwise, it is for us North Americans an hyperbola segment, with the branches taking off asymptotically at angles somewhat north of the east- west line for the winter year-half, declination negative, and somewat south of the east-west for positive, summer declinations. My Fig. 2 in the Sept. 97 issue of the Nass "Compendium" Vol:4 No. 3, p.25, sketches the general near-winter solstice appearance for mid-latitudes. The curve is bilaterally symmetrical about the north-south line. If you can do the pegging starting in the forenoon, and ending after local culmination you can get it right by catching the shadow point at equal radii, (i.e. equal solar altitudes,) as shown best in Fig. 1, p.23, but if you use a piece of the curve solely on the same side of noon, you'll not get a true bearing. ---- Again, except at an equinox, and the error grows as the sun moves away from the equatorial plane. You can find the two solsticial date hyperbolas plotted as furniture on some horizontal dials as illustrated in dial texts. Conceptually, think of the tip of the stick as a pivot for a line drawn from the center of the sun through the pivot. Now as the sun moves in a circular arc (convex up) in the southern sky, that line of shadow produced by the point of the stick extends to describe an inverted concave up reverse-path shadow-cone mirror image of the (incomplete) cone described by the part of the line connecting the sun's path to the point. The tips of the twin cones meet at the stick or pivot point. Where the shadow cone intesects the horizontal plane, (level ground,) the conic section drawn is a portion of a hyperbola. Mathematicians like to think of cones as always in such tip to tip pairs; for their purposes, a plane that cuts them both always produces twins of what we think of as hyperbolic curves. In this case we can't see where the imagined tangent plane at the site of the observation meets the light-twin cone out there in space. ---- Pretty dramatic, hey? A story of the light heavenly cone, joined at the tip (hip?) to its dark, perverse, earthly twin, that might evilly guide you astray unless you are familiar with its potentially crooked behavior. It has cosmic implications! Post a notice to the New Agers! Sell the plot to the X-Files! Have fun dialing, Bill.
