created by Carl Sabanski
Digital Equatorial Sundial
Equatorial Sundial: a dial in which the dial plate is parallel to the equatorial plane and the polar-pointing gnomon is perpendicular to it.
Equatorial Plane: the plane through the Earth defined by the equator.
This particular sundial does not have a defined classification. It lies parallel to the equatorial plane but does not have a dial plate with hour lines nor a gnomon. It does have a reading line that is located on the polar axis of the sundial. The hour numbers are cut through a semi-circular ring. These numbers and any other time indicators are located in the hour planes of the times to be indicated. The sun shines through these indicators and projects them as lighted images along the reading line. Thus the term digital in the descriptor.
Figure 1 illustrates the layout of the digital equatorial sundial and the equatorial band with cut out hour markers. The hour markers are spaced at 15° intervals. Additional markers, such as circles, can be used to indicate other times. The reading line is located on the plate at the dial's centre.
Figure 1: Digital Equatorial Sundial Layout (CAD)
In order to see the hour number images throughout the year the reading line must have a minimum length determined as follows:
Line Length = tan (23.44°) x 2r
Line Length = 0.4336 x 2r
where r is the radius of the equatorial band.
The images will be at the centre of the line at the equinoxes, moving down the line until the lower limit is reached at the summer solstice and then up again until the upper limit is reached at the winter solstice. The line length should be increased to accommodate the height of the hour numbers.
As the sun moves into the hour plane of a particular time that number's image will be projected to the back of the dial. If the numbers are all the same size as shown above, they will become increasingly distorted as they move away from noon. This distortion results in the images becoming wider. Figure 2 illustrates how this occurs.
Figure 2: Hour Indicator Distortion
If the number has a width of W1 on the equatorial band its width W2 at the reading line will be:
W2 = W1 / cos HA
where HA is the hour angle.
Table 1 lists the values of W2 for various hour angles assuming that W1 has a width of one unit. To compensate for the distortion the width of the number can be reduced by cos (HA). This is shown as W1'. The calculations performed here are for an hour number image projected at the centre of the reading line, which will occur at the equinoxes.
Table 1: Distortion as a Function of Hour Angle
Figure 3 illustrates this distortion for 8 am. On the left no compensation was applied and the number projected is two times wider than on the equatorial band. On the right, when the width was reduced by half, the projected number appears normal.
Distortion of the hour number image will also occur as it moves up and down the reading line towards the solstice limits.
The half sphere shown in this sundial can be replaced by a plate that takes the place of the disc located on the pillar. The equatorial band would be attached to this plate. The numbers and other time indicators would travel across this plate until they reach the reading line at the dial's centre.
For an image complete with
shadow click here.