Well, here's another question. I've been thinking (again) about what a proper shadow would be for a bowstring equatorial or a hemisperium or hemicyclium. I made up a dumb little spreadsheet which is at the URL below to thrash about with the numbers some. Parts of that spreadsheet are shown below. From the output, you can see how I've been thinking, but how _should_ I be thinking about this problem? What _is_ a proper shadow?
Thanks, John Bercovitz http://www-eng.lbl.gov/~bercov/math/Sundials/UPE.xls Umbrae, Penumbrae and Equatorial dials dramatis personae d - diameter of rod or sphere D - distance of center of rod or sphere from where you read the shadow A - angular subtense of the sun in radians (32' = .0093 radians) U - umbra width or diameter (for rod or sphere respectively) = d - .0093D P - linear or radial width of penumbra = .0093D S - highly fictitious "sensitivity" = D/U Angle is small enough that small-angle approximation applies d D A U P S 0.125 6 0.0093 0.0692 0.0558 86.70520231 0.250 12 0.0093 0.1384 0.1116 86.70520231 0.250 6 0.0093 0.1942 0.0558 30.89598352 0.500 12 0.0093 0.3884 0.1116 30.89598352 When D/d is large, is it too hard to find where the umbra is within the shadow? When D/d is small, is it too hard to find where the center of the umbra is? If the shadow-caster is a sphere instead of a rod, must the umbra be relatively larger? -
