created by Carl Sabanski
Direct Inclining, Reclining and Proclining Sundials
Inclining Sundial: usually applied to portable dials in which a horizontal dial, designed for a high latitude, typically 60°, may be inclined by raising its southern edge (in the N hemisphere) so that it may be used at locations with lower latitudes. The opposite arrangement is sometimes found. The term is sometimes used for near-vertical dials where the top leans away from the observer.
Proclining Sundial: a term sometimes used to describe a dial which is approximately upright but which leans toward the observer. A dial which leans forward by 10° will have an inclination of +100°. They are sometimes also called inclining dials, although that term is best reserved for dials derived from a horizontal dial.
Reclining Sundial: strictly, an approximately vertical dial which leans backwards away from the observer. However, often used as a catch-all term for any non-vertical dial. The angle is defined from the horizontal towards the observer, so a dial which leans backwards by 10° from the vertical has an inclination of +80°.
Inclination (i, I): the angle between the back of the dial plane and the horizontal for inclining or reclining dials. Equivalently, it is the angle between the zenith and the positive z co-ordinate of the dial. i = 0° implies a horizontal dial. For an inclining dial, 0 < i< ø (the latitude of the place). For a reclining dial leaning away from the observer, i < 90°; whilst i > 90° implies a proclining dial leaning forward towards the observer. Beware: this convention is not followed by all observers.
Direct inclining, reclining and proclining sundials face one of the four cardinal directions; north, south, east or west. Figure 1 illustrates the different surface orientations and how their inclination is measured. Inclining and reclining sundials are similar but have different inclination ranges. Based on the definition for inclination, the sundials can be defined as follows:
where ø is the latitude of the place.
Figure 1: Surface Inclination
Figure 2 illustrates an number of different types of sundials that face directly north or south. Note that the style of every dial points to one of the celestial poles. Starting with dial "A" a number of common sundials can be defined.
These sundials are located in the Northern hemisphere.
Figure 2: Various Sundials (CAD)
If the vertical sundial AV at latitude ø is moved to a latitude ø* that is equal to 90°- ø it then becomes a horizontal sundial. This can be done for any direct north or south inclining, reclining or proclining sundial. This process is called reduction and is illustrated in Figure 2 using dial "B". Any of these sundials can be reduced to an equivalent horizontal dial as follows:
ø* will be negative for the proclining sundials indicating that the equivalent horizontal dial is located in the Southern hemisphere. The hour lines will then be a mirror image of a sundial located at the same latitude in the northern hemisphere and the hour numbers will be reversed.
Direct facing east and west sundials are reduced to a new latitude where a vertical south declining sundial is designed. This is done as follows:
Figure 3 illustrates direct east and west reclining sundials. The noon lines are horizontal and near the bottom of both dials. The dial centre for the direct east reclining dial is at the left and for the direct west reclining dial it is at the right.
3: Direct East & West Reclining Sundials (SHADOWS)
In the case of direct east and west proclining sundials
the noon lines are horizontal and near the top of the dials. The dial
centre for the direct east proclining dial is at the right and for the
direct west reclining dial it is at the left.