I've been playing in my garden, staking out the path of the shadow of a corner of the barn. I found the point directly below my makeshift gnomon by hanging a rock on a string and found the shortest shadow by measuring with a string from this point. I compared the direction of the shortest shadow against North from a compass (corrected for the local deviation). I compared the time of the shortest shadow against Greenwich noon (corrected for my longitude and the Equation of time). Both showed good agreement. Cool. And old hat for most of you.
Now I want to verify my latitude (and the current declination of the sun). I missed the equinox by several days and am too impatient to wait for the next one (much less to wait for both the solstices). I could also measure a few points and do a bunch of calculations, but that lacks the elegance of the stake and string methods. I thought of a way to do it given the asymptotes of the hyperbolic shadow path, but I had difficulties finding the asymptotes accurately because of buildings and hills and trees on the horizon. I figured out another way it could be done without morning and evening measurements, but it is very complicated, involving 3-D constructions. *** Does anybody know a relatively simple method for finding the latitude from observations of the sun over the course of several hours without recourse to tables and calculations? *** Ideally, from only the position and height of the shadow-casting point and three shadows one could determine North, the latitude, and the declination. In the course of these considerations, I also wished for an easy method to construct a hyperbola. You all know how to draw an ellipse with a loop of string around two pegs. There is another method to make an ellipse involving a stick or string of length "a" with a mark at distance "b" from one end. If one end is moved along one axis and the mark is moved along the perpendicular axis, then the other end traces out an ellipse. I have found a few ways to construct hyperbolae that are mathematically correct but not especially practical. Is there any way to construct a hyperbola which is of similar elegance and practicality to the methods for ellipses? Is there an easy way, given a hyperbola, to find its axes, asymptotes, or foci? Thanks for your help. Art Carlson
