Hi John, Here is a try:
John Carmichael wrote: << Snip >> > THE PROBLEM: The plotting techniques which use tabulated angles or > computation produce timeline plotting angles in degrees which the dialist > must mark onto the dial plate using a protractor. These angles will be as > precise as the number of decimal places used in the calculations. However, > even though one takes great care to obtain precise timeline angles, this > amount of precision is useless if one's protractor isn't equally precise. > The graphical plotting method also requires an accurate protractor, of course. <<Snip >> I assume that you are at the "Layout Stage" here and want to precisely locate the hour and other marks on the actual surface of the Dial. For "Celeste" at: http://sciencenorth.on.ca/AboutSN/polaris/index.html the radius of the dial circle is 20/pi meters, about 6.366 m (just under 21 feet). When laying out the marks I found it easiest to measure the length along the arc of the circle from the reference at north to the various marks, using a flexible cloth surveyor's tape. The center of each brass 15 minute marker is positioned to within .5 mm of where it "should be". There was some "slop" in the positioning when we placed the terrazzo matrix into the welded brass framework which holds the Roman numerals and the quarter-hour marks. I calculated the angles between each of the hour and quarter-hour marks and North. The length of the arc along the circumference of the dial for each angle is given by: arc length = (radius) x (angle in radians), the units being the ones to measure radius. or arc length=(radius)x(angle in degrees)x(pi/180) In the case of Celeste: The dial circle has a circumference, C, of 40 meters. (Another subject, but the reason for this is that both Celeste and Terra are at one millionth scale of Earth, and use the classical definition of the metre, namely, 1/10,000,000 of the distance between a pole and the equator.) Celeste's circumference of dial circle, C = 40 meters Radius, R, of dial circle = C / 2pi Celeste's Radius, R = 40 / 2pi meters = 20/pi meters L, length of Celeste's arc = (20/pi) x (pi/180) = 1/9 meters/degree ~ .11111 meters/degree This is 111.1 mm or about 4.37 inches of arc length per degree of subtending angle. You can easily mark to the nearest millimeter, so you can easily mark to the nearest hundredth of a degree. In fact you can probably get down to a half a minute of angle in precision of marking. You can probably guess why I used a pipe for a 20 foot gnomon. (Yes, it is much easier for the eye to interpolate to the centre of two fuzzy shadow edges to make a reading then to debate where the actual shadow is on one side. The symmetry avoids all of the penumbral and diffraction difficulties.The error due to non-linearity of the dial scale is negligible. Besides I needed some rigidity.) Commentary on A, B and C: > SOLUTION A: << Snip >> > I'm thinking that during the construction phase of a very > large sundial, I could make a temporary giant protractor located just > outside the hourline radius. This would be a fairly simple thing to do using > plane geometry. It would be easier to turn your angles using a theodolite at the centre of the dial than to make a giant protractor. I used one to check the time marks before the terrazzo was poured and the hairline was on the brass in every case. Just remember to set the theodolite lower than usual, because you'll have to tilt the 'scope down to mark your angles. And you'll need a helper to hold the pencil point on the marks. > SOLUTION B: If the unit square method is used (Waugh, pg. 40-43), then no > protractor is needed. One only needs a good long measuring tape for laying > out the lines. ( The limits of precision would again depend on the number of > decimal places used.) I don't have Waugh's book. I assume he converts to rectangular coordinates to do the layout. Great solution if you have to transcribe curved or non-radial lines, but a lot of work at the layout stage. Your computer program can usually give you your x-y coordinates as a matter of course - just punch in points on the curves and note the coordinates off of the cursor coordinates. Sometimes it is easier to grid it as finely as you need and then interpolate to give you plot points. > SOLUTION C: What if I built the gnomon first and use its shadow to tell me > the position of the time lines???? << Snip >> Guess who put up a temporary gnomon for a few sunny days before placing terrazzo! It worked superbly. I wouldn't build a big one without doing some "field checking"! Saves lot of potential embarrassment too. Big dials usually have a lot of volunteer (field supervisors) because they are usually very visible in a public place and these folk can usually be conscripted to hold cardboard and pinholes to sharpen the moving shadow and to call "mark" when the second hand sweeps through 12 on the electromechanical time manufacturer worn on the wrist. They can share the excitement of finding out that the thing really works! That is, they find that their wristwatch, after suitable synchronization is just as good as the spinning earth to keep time! I do hope that you do build a big one John. It is a great experience. Cheers, t > Thanks, > > John Carmichael
