Editing Sundial (section)

===Declining-reclining dials/ Declining-inclining 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.

There are various solution approaches, including some using the methods of rotation matrices, and some 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.{{sfnm|1a1=Brandmaier|1y=2005|1pp=16{{ndash}}23 |1loc=Vol. 12, Issue 1|2a1=Snyder|2y=2015|2loc=Vol. 22, Issue 1}}

One system of formulas for Reclining-Declining sundials: (as stated by Fennewick)<ref name=Fennewick>{{cite web |last=Fennerwick |first=Armyan |title=the Netherlands, Revision of Chapter&nbsp;5 of ''Sundials'' by René R.J. Rohr, New York 1996, declining inclined dials part&nbsp;D Declining and inclined dials by mathematics using a new figure |location=Netherlands |website=demon.nl |url=http://lester.demon.nl/mywww/rohr/ |access-date=1 May 2015 |url-status=live |archive-url=https://web.archive.org/web/20140818142829/http://lester.demon.nl/mywww/rohr/  |archive-date=18 August 2014}}</ref>
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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 counterclockwise 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; {{mvar|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 = +1\ ,</math> while those partly or fully north-facing have an <math>\ s_o = -1 ~.</math>
Since the above expression gives the hour angle as an arctangent 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.<ref name=Fennewick/>

====Empirical method====
Because of the complexity of the above calculations, using them for the practical purpose of designing a dial of this type is difficult and prone to error. It has been suggested that it is better to locate the hour lines empirically, marking the positions of the shadow of a style on a real sundial at hourly intervals as shown by a clock and adding/deducting that day's equation of time adjustment.<ref name="harvp|Waugh|1973|pp=106–107"/> See [[#Empirical hour-line marking|Empirical hour-line marking]], above.