Unit 6 : Equatorial Sundials : Variations : Horizontal and Vertical Sundials : Tech Notes : Unit Questionniare

# Tech Notes

The following section is included for those who want to understand some of the mathematical reasoning behind the various sundial designs.  This section is optional and your grade will not be affected by material in this section.

Horizontal Sundial with a Vertical Gnomon

A horizontal sundial with a vertical gnomon would measure the azimuth of the sun.  The sundial could be reasonably accurate at the equinoxes, but it would then be up to one hour off in midwinter or midsummer, as you can see from this graph, constructed for Ames, Iowa (latitude 42N).

Horizontal Sundial with a Slanted Gnomon

In constructing the horizontal sundial with the slanted gnomon, we made a sundial that measures the hour angle of the sun rather than the azimuth of the sun (as contrasted to the sundial above).  This slanted gnomon feature makes it much more accurate, since solar time is defined as "the hour angle of the sun plus 12 hours."

The formulae used to generate the graph below are:

cos(altitude) sin (azimuth) = cos(dec) sin(hour angle)

cos(altitude) cos (azimuth) = - cos(latitude) sin(dec) + sin(latitude) cos(dec) cos(hour angle)

where the hour angle of the sun = (time - 12 hours)x15 deg / hour

It is easiest to find the altitude first, and then the azimuth.  The altitude formula is presented next.

Shepherd's Dial

The shepherd's dial and similar dials that measure the length of a shadow cast by the sun are measuring mostly the altitude of the sun.  These dials always need to be corrected for the seasons to be usable, as they could be off by more than 2 hours at the Ames, Iowa latitude.

The formula used to generate the graph is:

sin(altitude) = sin(latitude) sin(dec) + cos(latitude) cos(dec) cos (hour angle)

The hour angle is defined above in the last graph discussion.

In principle, if you know the altitude and azimuth at all times of day, then a little geometry is all you need to construct any kind of sundial at all.  That doesn't mean, however, that it is always easy.