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===Reclining-declining dials===
Some sundials both decline and recline, in that their shadow-receiving plane is not oriented with a cardinal direction (such as [[true North]] or [[true South]]) and is neither horizontal nor vertical nor equatorial. For example, such a sundial might be found on a roof that was not oriented in a cardinal direction. The formulae describing the spacing of the hour-lines on such dials are rather more complicated than those for simpler dials. In fact it is only in the last decade that agreement has been found on the correct hour angle formula for this type of dial using either the methods of rotation matrices; or by making a 3D model of the reclined-declined plane and its vertical declined counterpart plane, extracting the geometrical relationships between the hour angle components on both these planes and then reducing the trigonometric algebra.{{sfn|''Sundial Design Using Matrices.'' H. Brandmaier, NASS Compendium, Vol 12, No. 1, pp.16-23, Mar 2005.}} {{sfn|''[http://dls-website.com/documents/SundialDesignConsiderations.pdf Sundial Design Considerations.]'' Donald L. Snyder, NASS Compendium, Vol 22, No. 1, Mar 2015.}} Previous formulae given by Rohr and Mayall are not correct. {{efn| This is most probably due to the difficulty in making a reliable three dimensional drawing of the reclined-declined situation as well as the ease of making errors in the geometrical relationships and trig algebra. They had proposed that
:<math>
H_\text{RD} = H_\text{RD1} + H_\text{RD2}
</math>
:<math>
H_\text{RD1} = \tan D \cos R ,
</math>
:<math>
H_\text{RD2} = \frac{\cos R \cos D \sin L + \sin R \cos L - \cos R \sin D \cot(15^{\circ} \times t)}{\sin D \sin L + \cos D \cot(15^{\circ} \times t)}
</math>
where L is the sundial's geographical [[latitude]], ''t'' is the time before or after noon, and R and D are the angles of inclination and declination, respectively.
and that the angle B between the substyle and the noon-line is given by<ref>Rohr (1965), p. 78.</ref>
:<math>
\tan B = \sin R \sin D \frac{\tan L \cos R + \sin R \cos D}{\cos R - \tan L \cos D \sin R}.
</math>
}}
The angle <math> H_\text{RD} </math> between the noon hour-line and another hour-line is given by the formula below. Note that <math> H_\text{RD} </math> advances anticlockwise with respect to the zero hour angle for those dials that are partly south-facing and clockwise for those that are north-facing.
:<math>
\tan H_\text{RD} = \frac{\cos R \cos L - \sin R \sin L \cos D - s_o \sin R \sin D \cot(15^{\circ} \times t)}{\cos D \cot(15^{\circ} \times t) - s_o \sin D \sin L }
</math>
within the parameter ranges : <math> D < D_c </math> and <math> -90^{\circ} < R < (90^{\circ} - L) </math>.
Or, if preferring to use inclination angle, <math> I </math>, rather than the reclination, <math> R </math>, where <math> I = (90^{\circ} + R) </math> :
:<math> \tan H_\text{RD} = \frac{\sin I \cos L + \cos I \sin L \cos D + s_o \cos I \sin D \cot(15^{\circ} \times t)}{\cos D \cot(15^{\circ} \times t) - s_o \sin D \sin L }
</math>
within the parameter ranges : <math> D < D_c </math> and <math> 0^{\circ} < I < (180^{\circ} - L) </math>.
Here <math>L</math> is the sundial's geographical [[latitude]]; <math>s_o</math> is the orientation switch integer; ''t'' is the time in hours before or after noon; and <math> R </math> and <math> D </math> are the angles of reclination and declination, respectively.
Note that <math> R </math> is measured with reference to the vertical. It is positive when the dial leans back towards the horizon behind the dial and negative when the dial leans forward to the horizon on the sun's side. Declination angle <math> D </math> is defined as positive when moving east of true south.
Dials facing fully or partly south have <math>s_o</math> = +1, while those partly or fully north-facing have an <math>s_o</math> value of -1.
Since the above expression gives the hour angle as an arctan function, due consideration must be given to which quadrant of the sundial each hour belongs to before assigning the correct hour angle.
Unlike the simpler vertical declining sundial, this type of dial does not always show hour angles on its sunside face for all declinations between east and west. When a northern hemisphere partly south-facing dial reclines back (i.e. away from the sun) from the vertical, the gnomon will become co-planar with the dial plate at declinations less than due east or due west. Likewise for southern hemisphere dials that are partly north-facing.
Were these dials reclining forward, the range of declination would actually exceed due east and due west.
In a similar way, northern hemisphere dials that are partly north-facing and southern hemisphere dials that are south-facing, and which lean forward toward their upward pointing gnomons, will have a similar restriction on the range of declination that is possible for a given reclination value.
The critical declination <math> D_c </math> is a geometrical constraint which depends on the value of both the dial's reclination and its latitude :
:<math>
\cos D_c = \tan R \tan L = - \tan L \cot I
</math>
As with the vertical declined dial, the gnomon's substyle is not aligned with the noon hour-line. The general formula for the angle <math> B </math>, between the substyle and the noon-line is given by :
:<math>
\tan B = \frac {\sin D}{\sin R \cos D + \cos R \tan L} = \frac {\sin D}{\cos I \cos D - \sin I \tan L}
</math>
The angle <math> G </math>, between the style and the plate is given by :
:<math> \sin G = \cos L \cos D \cos R - \sin L \sin R = - \cos L \cos D \sin I + \sin L \cos I
</math>
Note that for <math> G = 0^{\circ}</math>, i.e. when the gnomon is coplanar with the dial plate, we have :
:<math> \cos D = \tan L \tan R = - \tan L \cot I </math>
i.e. when <math> D = D_c </math>, the critical declination value.
====Empirical method====
Because of the complexity of the
===Spherical sundials===
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