Finding the Analemma by Computation. Given the latitude & longitude of the observer, the Sun's altitude & azimuth at exactly 12:00 noon Standard Time (A & az), and the height of the aperture through which the Sun's image is cast (d), the x,y coordinates for points along the analemma may be calculated using the following formulas (after Waugh, Sundials: Their Theory and Construction, pp. 21-28):

   (1)   VS = d × COT(A)
   (2)   BV = VS × COS(a)
   (3)   BS = BV × TAN(a)

   Where:

    \|/
   --o--
    /|\
   (to Sun)
        \
         \
          =
           \
            \
             C
             |\
             | \
             |  \
             |   \
           d |    \
             |     \
             |      \
             |       \
             |        \
             V---------\---B
                        \
                         \
                          \
                           S

   C  = Sun aperture (a hole ~1/4 in. diameter)
   d  = 48 in. (given height of Sun aperture)
   V  = the base, i.e., the point on the floor (or ground) vertically below aperture
   BV = the meridian
   CS = beam of sunlight passing through aperture
   S  = Sun's image ("spot" on the floor)
   ÐA = CSV (Altitude of Sun)
   Ða = BVS (azimuth of Sun or angle from the south)

Note: Solar altitude & azimuth data are obtained from the USNO Altitude and Azimuth of the Sun or Moon During One Day Web page. The values for COT(A), COS(a), and TAN(a) are obtained from Table 1.

Any point along the analemma can be located on the Cartesian Plane by an ordered pair of numbers (x,y), called the coordinates. The origin represents Point V, and in this case, also represents the center of a compass rose. The y-axis represents the meridian (BV): north is toward the top of the y-axis; south is toward the bottom. East is toward the right end of the x-axis; west is toward the left. Construct a graph of the approximate shape of an analemma (as projected on a horizontal surface) using the values for "x" & "y" in Table 2 (see orange columns). The shortest and longest and distances from the base (Point V) are inferred from the values of BV for JUN 21 and DEC 21 respectively.