Sunny Day! I have been trying to understand the consequences of shadow tapering on a monumental sundial a little better. To help visualize this I have attached an image I modified from Wikipedia. From previous emails I come up with the following.
If the shadow casting object is cylindrical the resulting shadow is theoretically as shown in the image. The diameter of the sun is about 32' as was mentioned. The umbra will disappear at a distance "ud" equal to: ud = r / tan (16/60) = 214.9 x r where "r" is the radius (D/2) of the cylinder. This is the maximum theoretical distance and is not reality. The umbra is not likely to be visible at that distance just as the penumbra is not visible as shown in the image. That is why a gnomon diameter equal to 1/105th of the distance from the origin to the point where the readings will occur is recommended. Now at any point "P" that is a distance "d" from the cylinder the umbra has tapered by: t = 2 x tan (16/60) x d x r = 0.009308 x d x r and the width of the umbra is D - t. If the object is a vertical cylinder the section that casts its shadow at point "P" varies throughout the year as the sun's elevation changes. But as the cylinder is vertical "d" will remain constant a so will the taper value "t". This will also remain the same at different latitudes. If the cylinder is sloped the section that casts its shadow at point "P" still varies throughout the year but now so does "d". As a result the taper value "t" will vary. The width of the umbra at point "P" will vary throughout the year. However, as the gnomon is cylindrical and it is the centre of the umbra that is of interest, this change in the taper value does not matter. Now, if the wide gnomon is not a cylinder but a triangle then the style is an edge. The shadow cast by this edge will also taper. At any given point on the dial plate this taper will vary throughout the year because it originates from different positions on the style. If the dial plate is not designed to give constant values of "d" in the reading area the taper will also vary from line to line in the reading area. The taper will also be affected by the latitude. If I have this straight then when the hour lines are adjusted for taper this will not be a definitive adjustment. On a smaller sundial the taper variation will be small because of the small changes in "d". On a monumental sundial, this variation may be significant depending on the layout of the dial plate.. If a gnomon goes to a point that acts as a nodus the result of tapering will be noticed very quickly I would think. This would be especially true at large values of "d". If this is wrong please let me know. Happy Dialling! Carl Sabanski www.mysundial.ca
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