- SUNDIAL CALCULATIONS -

      To derive these 365 average data or date points for the gnomon, I used the Ephemeris report data for the position of the sun as observed at noon from a fixed reference point on Earth from the dates March 1, 1997 through February 28, 1999.

      The 730 days were then paired (March 1, 1997 with March 1, 1998, etc.) and a weighted average was then used to calculate the sun's azimuth and altitude position for 365 "average" days of the year. The data for the days closest to February 28, 1998 and March 1, 1998 were weighted highest, and the data for the days closest to March 1, 1997 and February 28, 1999 were weighted lowest. Thus, the average position (Pos_Avg) for the sun on a given pair of days was calculated by:

     PosAvg = (dA - 1) / 364 x PosA + (730 - dB) / 364 x PosB ; where:

     dA = days 1 through 365 (March 1, 1997 through February 28, 1998)
     dB = dA + 365 = days 366 through 730 (March 1, 1998 through February 28, 1999)
     PosA = the sun's azimuth and altitude position on the dates between days 1 and 365.
     PosB = the sun's azimuth and altitude position on the dates between days 366 and 730.

     PosAvg = (1 - 1) / 364 x PosA + (730 - 366) / 364 x PosB ; where

     PosA = the sun's azimuth and altitude position on day 1, March 1, 1997; and 
     PosB = the sun's azimuth and altitude position on day 366, March 1, 1998. 

     PosAvg = (365 - 1) / 364 x PosA + (730 - 730) / 364 x PosB ; where

     PosA = the sun's azimuth and altitude position on day 365, February 28, 1998; and
     PosB = the sun's azimuth and altitude position on day 730, February 28, 1999. 

Selecting the Observer's Position-

      The design of this sundial is such that the date is revealed by moving the edge of your finger above the figure-8 cutout and date markings on the analemmic gnomon... when the edge of your finger's shadow also intersects with the timeline on the dial below, then that edge of your finger's shadow also indicates the correct date mark on the gnomon. If I were to calculate date construction points for the gnomon from data for the sun's apparent position exactly at noon above Greenwich (0° Long.), then if the sundial was located in London, at precisely noon, the shadow intersecting the dial's timeline would be cast from the date mark indicating the exact start of that day. At 6:00pm in London the sundial would indicate that the day is only 1/4 day old. At 6:00am the next morning in London the sundial would still be indicating only 3/4 of the way through the prior day. The date indicated by the sundial would always be 1/2 day slow. Thus, the most desirable location for noting the sun's noon position for calculation of the 365 date construction points for the gnomon would be with the observer positioned 180° around the Earth from the desired sundial location. Then at noon at the sundial's location, the sundial would indicate that the day (date) is exactly half over. This is equivalent to mathematically advancing the date points by 1/2 day from the observed noon position of the sun at the sundial location, so that at local noon, the date is indicated as being 1/2 day complete.

      The accuracy for this sundial could be improved by designing two analemmic gnomon plates. One could be created for use year-round in Asia, and one for use year-round in the U.S. The U.S. analemmic gnomon would be created by plotting the position of the sun at noon from 90° E., in Asia. If this sundial was placed in Greenwich, at 6:00am it would indicate the exact start of the day, and at 6:00pm the sundial would indicate the day is 1/2 over. In the U.S. from 6:00 am to 6:00pm the sundial would indicate that the day is between 1/4 and 3/4 day old, and at noon it would indicate that the day is exactly halfway over.

      The Asian analemmic gnomon would be created by plotting the position of the sun at noon from 90° W., in the U.S. Thus at 6:00am on the International Date Line the sundial would indicate the exact start of the day, and at 6:00pm the sundial would indicate the day is 1/2 over. In Asia from 6:00 am to 6:00pm the sundial would indicate that the day is between 1/4 and 3/4 day old, and at noon it would indicate that the day is exactly halfway over.

Analemmic Gnomon Design-

     Angle ai = (180° - Azmi) , and is measured in the equatorial plane.
     Angle bi = Alti ; a positive value is north of the equatorial plane.
     r = the desired radius of the sundial.
     Azmi and Alti are the apparent Azimuth and Altitude of the sun at noon on day i.
     xi and yi are (x,y) date coordinates from the center of the gnomon plate, calculated by:

		(xi,yi) = (r×sin(ai), r×cos(ai)×tan(bi))

      When reading the sundial, the gnomon plate is always swiveled about the NCP axis to directly face the sun, casting the largest possible figure-8 shadow onto the dial ring below. In the above diagram, because the sun at noon is off by Angle a_i from the direct overhead position, the gnomon plate is also rotated by Angle a_i to directly face the sun. Thus:

		xi = r×sin(ai) ; where:

     r = the radius of the sundial, measured from the equatorial dial to the front of the gnomon plate.
     Angle ai = (180° - Azmi) , and is measured in the equatorial plane.
     Azmi is the apparent Azimuth of the sun at noon on day i.
     x0 is the horizontal coordinate at the center of the gnomon plate, where Azm0 = 180°.
     xi is the date coordinate's horizontal distance from the center of the gnomon plate.

      Although the gnomon plate can swivel about the NCP axis to directly face the sun, it is always perpendicular to the equatorial plane. Because it swivels about the NCP axis to directly face the sun, the actual distance from the equatorial dial to the front of the gnomon plate shortens from r, to r×cos(a_i). Thus:

		yi = r×cos(ai)×tan(bi) ; where:

	r×cos(ai) = the distance from the equatorial dial to the front of the rotated gnomon plate.
	Angle bi = Alti ; a positive value is north of the equatorial plane.
	Alti is the apparent Altitude of the sun at noon on day i.
	y0 is the vertical coordinate at the center of the gnomon plate, where Alt0 = 0°.
	yi is the date coordinate's vertical distance from the center of the gnomon plate.

      Slightly off topic, but can our perception really be deceived by shadows so easily? Yes, it can.

Ephemeris Data and Date Coordinates-

Last Updated: May 4, 2010
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