For those who are interested in building sundials a more extensive treatment is given here.
The angles calculated by this program are accurate for building a sundial. The governing equations are derived from the geometry of the earth-sun positions, and the local position of the dial placed on the earth. Since the earth rotates 360 degrees in 24 hours, an angle on the earth can be measured in either units of degrees or time. One hour of time is equal to 15 degrees of arc. If we define the origin of our time angle to be 12:00 noon, then at 1:00 we have turned through 15 degrees. Similarly at 11:00 we are 1 hour before noon which is again 15 degrees of arc. So, if we want to calculate the position of a certain hour line, convert that hour to degrees, and call it "timeang". For example, the 3:00 hour line is identical to 3.0 * 15 = 45 degrees. Also, given the latitude of the observer, call the latitude "lat". Then the angle between the gnomon, which lies on the true north-south line, and the hour line is given by
tan(angle) = tan(timeang) * sin(lat)
The equation above is good for finding the local apparent time almost anywhere on the earth. However, if you want to find the angles for your local mean time you need to know where your standard meridian is. If you are located east of your meridian, you see the sun reach its zenith at an earlier time than a person standing at the meridian would. Therefore, you must compensate by shifting the noon line, and all other lines, over a few degrees to make up the difference. Call the difference between the meridian longitude and your longitude "deltalon". Now convert deltalon to time. This is how much earlier (east of the meridian) or later (west of the meridian) you see the sun appear at a certain point in the sky. For example, if you are at longitude 92 degrees, you are 2 degrees west of the central time zone meridian, which corresponds to 2/15 * 60 = 8 minutes of time. If we now calculate the angle of the hour line for 11:52 A.M. in our apparent time, we would draw that line at the calculated angle, but we would label it 12:00. This would coincide us with a sundial at the meridian whose lines were calculated for its local apparent time.
The gnomon, as mentioned before, must lie along the true north-south line of the earth. This is not necessarily magnetic north as given by a compass. True north-south lies along a line of longitude. A simple gnomon is just a right triangle whose hypotenuse makes an angle with the dial plate equal to the latitude of the observer. Click the button below to show how a gnomon is situated. Note that the apex rests at the intersection of the hour lines.
Additional lines may be added to the dial plate for different purposes. The position of the shadow of the tip of the gnomon can be calculated if we know the declension of the sun for a given day. These values are well known and tabulated in many reference books on astronomy. Knowing the declension, more lines may be added such that the shadow points out the altitude of the sun, or traces out a curve for any day of the year. The latter is especially useful for determining the solstices and equinoxes.
Because our orbit about the sun is not a perfect circle, a discrepancy between our apparent time and the mean time can exist. This discrepancy is called the equation of time. A sundial adjusted to show local mean time will be fast or slow as compared to an accurate watch. The difference can be as much as 16 minutes, but since this is also well known, tables exist to correct for the equation of time and a chart is included on the next card to diagram it. Since a sundial shows standard time, always remember to correct for daylight savings.
Other types of sundials exist which can be placed on walls, roofs, and reflected to ceilings. There are a number of books detailing sundials which are available from retail or the library. This introduction may get some people interested to try it themselves. Enjoy!
