Table 1. Altitude & Azimuth of the Sun (at 10:00 a.m. EST)
Washington, D.C., ~39°N (38°53'N), ~77°W (77°02'W)

Month	Day	Altitude (A)	VCS = 90 - A	COT(A) = TAN(ÐVCS)	Azimuth (az)	a = az - 180	COS(a)	TAN(a)
JAN	1	20.9	69.1	2.6187	147.7	-32.3	0.8453	-0.6322
	15	21.9	68.1	2.4876	145.4	-34.6	0.8231	-0.6899
FEB	1	25.0	65.0	2.1445	142.3	-37.7	0.7912	-0.7729
	15	28.7	61.3	1.8265	139.7	-40.3	0.7627	-0.8481
MAR	1	33.3	56.7	1.5224	137.0	-43.0	0.7314	-0.9325
	15	38.3	51.7	1.2662	134.2	-45.8	0.6972	-1.0283
	21	40.5	49.5	1.1708	132.9	-47.1	0.6807	-1.0761
APR	1	44.5	45.5	1.0176	130.5	-49.5	0.6494	-1.1708
	15	49.2	40.8	0.8632	126.9	-53.1	0.6004	-1.3319
MAY	1	53.7	36.3	0.7346	122.2	-57.8	0.5329	-1.5879
	15	56.5	33.5	0.6619	117.7	-62.3	0.4648	-1.9047
JUN	1	58.3	31.7	0.6176	112.9	-67.1	0.3891	-2.3673
	15	58.6	31.4	0.6104	110.3	-69.7	0.3469	-2.7034
	21	58.5	31.5	0.6128	109.7	-70.3	0.3371	-2.7929
Month	Day	Altitude (A)	ÐVCS = 90 - A	COT(A) = TAN(ÐVCS)	Azimuth (az)	a = az - 180	COS(a)	TAN(a)
JUL	1	57.9	32.1	0.6273	109.7	-70.3	0.3371	-2.7929
	15	56.5	33.5	0.6619	111.4	-68.6	0.3649	-2.5517
AUG	1	54.2	35.8	0.7212	116.0	-64.0	0.4384	-2.0503
	15	51.7	38.3	0.7898	121.2	-58.8	0.5180	-1.6512
SEP	1	48.0	42.0	0.9004	128.3	-51.7	0.6198	-1.2662
	15	44.5	45.5	1.0176	134.1	-45.9	0.6959	-1.0319
	23	42.3	47.7	1.0989	137.3	-42.7	0.7349	-0.9228
OCT	1	39.9	50.1	1.1959	140.2	-39.8	0.7683	-0.8332
	15	35.7	54.3	1.3916	144.5	-35.5	0.8141	-0.7133
NOV	1	30.7	59.3	1.6842	148.2	-31.8	0.8499	-0.6200
	15	26.9	63.1	1.9711	149.9	-30.1	0.8651	-0.5797
DEC	1	23.4	66.6	2.3109	150.4	-29.6	0.8695	-0.5680
	15	21.5	68.5	2.5386	149.7	-30.3	0.8634	-0.5844
	21	21.1	68.9	2.5916	149.1	-30.9	0.8581	-0.5985

Finding the Analemma by Computation. Given the latitude & longitude of the observer, the Sun's altitude & azimuth at exactly 10:00 a.m. 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.