Editing Sundial (section)

==With fixed axial gnomon==
The most commonly observed sundials are those in which the shadow-casting style is fixed in position and aligned with the Earth's rotational axis, being oriented with [[true north]] and south, and making an angle with the horizontal equal to the geographical latitude. This axis is aligned with the [[celestial pole]]s, which is closely, but not perfectly, aligned with the [[pole star]] [[Polaris]]. For illustration, the celestial axis points vertically at the true [[North Pole]], whereas it points horizontally on the [[equator]]. The world's largest axial gnomon sundial is the mast of the [[Sundial Bridge at Turtle Bay]] in [[Redding, California ]]. A formerly world's largest gnomon is at [[Jaipur]], raised 26°55′ above horizontal, reflecting the local latitude.<ref>
{{cite web
 |title=The world's largest sundial, Jantar Mantar, Jaipur
 |date=April 2016
 |website=Border Sundials
 |url=https://www.bordersundials.co.uk/the-worlds-largest-sundial-jantar-mantar-jaipur/
 |access-date=19 December 2017 |url-status=live
 |archive-url=https://web.archive.org/web/20171222052242/https://www.bordersundials.co.uk/the-worlds-largest-sundial-jantar-mantar-jaipur/
 |archive-date=22 December 2017
}}
</ref>

On any given day, the Sun appears to rotate uniformly about this axis, at about 15° per hour, making a full circuit (360°) in 24&nbsp;hours. A linear gnomon aligned with this axis will cast a sheet of shadow (a half-plane) that, falling opposite to the Sun, likewise rotates about the celestial axis at 15° per hour. The shadow is seen by falling on a receiving surface that is usually flat, but which may be spherical, cylindrical, conical or of other shapes. If the shadow falls on a surface that is symmetrical about the celestial axis (as in an armillary sphere, or an equatorial dial), the surface-shadow likewise moves uniformly; the hour-lines on the sundial are equally spaced. However, if the receiving surface is not symmetrical (as in most horizontal sundials), the surface shadow generally moves non-uniformly and the hour-lines are not equally spaced; one exception is the Lambert dial described below.

Some types of sundials are designed with a fixed gnomon that is not aligned with the celestial poles like a vertical obelisk. Such sundials are covered below under the section, "Nodus-based sundials".

===Empirical hour-line marking===
{{see also|Schema for horizontal dials|Equation of time}}
The formulas shown in the paragraphs below allow the positions of the hour-lines to be calculated for various types of sundial. In some cases, the calculations are simple; in others they are extremely complicated. There is an alternative, simple method of finding the positions of the hour-lines which can be used for many types of sundial, and saves a lot of work in cases where the calculations are complex.<ref name="harvp|Waugh|1973|pp=106–107">{{harvp|Waugh|1973|pp=106–107}}</ref> This is an empirical procedure in which the position of the shadow of the gnomon of a real sundial is marked at hourly intervals. The [[equation of time]] must be taken into account to ensure that the positions of the hour-lines are independent of the time of year when they are marked. An easy way to do this is to set a clock or watch so it shows "sundial time"{{efn|
A clock showing sundial time always agrees with a sundial in the same locality.
}}
which is [[standard time]],{{efn|
Strictly, [[local mean time]] rather than standard time should be used. However, using standard time makes the sundial more useful, since it does not have to be corrected for time zone or longitude.}} plus the equation of time on the day in question.{{efn|The equation of time is considered to be positive when "sundial time" is ahead of "clock time", negative otherwise. See the graph shown in the section [[#Equation of time correction]], above. For example, if the equation of time is -5&nbsp;minutes and the standard time is 9:40, the sundial time is 9:35.<ref>{{harvp|Waugh|1973| p= 205}}</ref>
}}
The hour-lines on the sundial are marked to show the positions of the shadow of the style when this clock shows whole numbers of hours, and are labelled with these numbers of hours. For example, when the clock reads 5:00, the shadow of the style is marked, and labelled "5" (or "V" in [[Roman numerals]]). If the hour-lines are not all marked in a single day, the clock must be adjusted every day or two to take account of the variation of the equation of time.

===Equatorial sundials===
<!-- [[File:Precision sundial in Bütgenbach-Belgium.jpg|left|upright|thumb|Precision sundial in Bütgenbach, Belgium. (Precision = ±30 seconds){{Coord|50.4231|6.2017|type:landmark|format=dms|name=Belgium}}]]  Dup of one in galleryb-->
[[File:Tower-bridge-and-olympic-rings.jpg|upright|thumb|''Timepiece'', [[St Katharine Docks]], London (1973) an equinoctial dial by [[Wendy Taylor]]<ref>{{NHLE|num=1391106|desc=Timepiece Sculpture|grade=II|access-date=10 October 2018}}</ref>]]
[[File:beijing sundial.jpg|upright|thumb|An equatorial sundial in the [[Forbidden City]], Beijing. {{Coord|39.9157|116.3904|type:landmark|format=dms|name=Forbidden City equatorial sundial}} The gnomon points [[true north]] and its angle with horizontal equals the local [[latitude]]. Closer inspection of the [[:File:beijing sundial.jpg|full-size image]] reveals the "spider-web" of date rings and hour-lines.]]

{{anchor|equinoctial sundial}}The distinguishing characteristic of the ''equatorial dial'' (also called the ''equinoctial dial'') is the planar surface that receives the shadow, which is exactly perpendicular to the gnomon's style.<ref>{{harvp|Rohr|1996|pp=46–49}}; {{harvp|Mayall|Mayall|1994|pp= 55–56, 96–98, 138–141}}; {{harvp|Waugh|1973| pp= 29–34}}</ref> This plane is called equatorial, because it is parallel to the equator of the Earth and of the celestial sphere. If the gnomon is fixed and aligned with the Earth's rotational axis, the sun's apparent rotation about the Earth casts a uniformly rotating sheet of shadow from the gnomon; this produces a uniformly rotating line of shadow on the equatorial plane. Since the Earth rotates 360° in 24 hours, the hour-lines on an equatorial dial are all spaced 15° apart (360/24).

:<math> H_E = 15^{\circ}\times t\text{ (hours)} ~.</math>

The uniformity of their spacing makes this type of sundial easy to construct. If the dial plate material is opaque, both sides of the equatorial dial must be marked, since the shadow will be cast from below in winter and from above in summer. With translucent dial plates (e.g. glass) the hour angles need only be marked on the sun-facing side, although the hour numberings (if used) need be made on both sides of the dial, owing to the differing hour schema on the sun-facing and sun-backing sides.

Another major advantage of this dial is that equation of time (EoT) and daylight saving time (DST) corrections can be made by simply rotating the dial plate by the appropriate angle each day. This is because the hour angles are equally spaced around the dial. For this reason, an equatorial dial is often a useful choice when the dial is for public display and it is desirable to have it show the true local time to reasonable accuracy. The EoT correction is made via the relation

: <math> \text{Correction}^{\circ} = \frac{\text{EoT (minutes)} + 60 \times \Delta \text{DST (hours)}}{4} ~.</math>

Near the [[equinox]]es in spring and autumn, the sun moves on a circle that is nearly the same as the equatorial plane; hence, no clear shadow is produced on the equatorial dial at those times of year, a drawback of the design.

A ''nodus'' is sometimes added to equatorial sundials, which allows the sundial to tell the time of year. On any given day, the shadow of the nodus moves on a circle on the equatorial plane, and the radius of the circle measures the [[declination]] of the sun. The ends of the gnomon bar may be used as the nodus, or some feature along its length. An ancient variant of the equatorial sundial has only a nodus (no style) and the concentric circular hour-lines are arranged to resemble a spider-web.<ref>{{cite journal |last=Schaldach |first=K. |year=2004 |title = The arachne of the Amphiareion and the origin of gnomonics in Greece | journal = Journal for the History of Astronomy | volume = 35 | issue = 4 | pages = 435–445 | issn = 0021-8286 | doi=10.1177/002182860403500404| bibcode = 2004JHA....35..435S | s2cid = 122673452 }}</ref>

===Horizontal sundials===
{{For|a more detailed description of such a dial|London dial|Whitehurst & Son sundial (1812)}}
[[File:Garden sundial MN 2007.JPG|thumb|upright|left|Horizontal sundial in [[Minnesota]]. June 17, 2007 at 12:21. 44°51′39.3″N, 93°36′58.4″W]]
In the ''horizontal sundial'' (also called a ''garden sundial''), the plane that receives the shadow is aligned horizontally, rather than being perpendicular to the style as in the equatorial dial.<ref>{{harvp|Rohr|1996|pp=49–53}}; {{harvp|Mayall|Mayall|1994|pp= 56–99, 101–143, 138–141}}; {{harvp|Waugh|1973| pp= 35–51}}</ref> Hence, the line of shadow does not rotate uniformly on the dial face; rather, the hour lines are spaced according to the rule.<ref>{{harvp|Rohr|1996|p=52}}; {{harvp|Waugh|1973| p= 45}}</ref>

:<math>\ \tan H_H = \sin L\ \tan \left(\ 15^{\circ} \times t\ \right)\ </math>

Or in other terms:

:<math> \ H_H = \tan^{-1}\left[\ \sin L\ \tan(\ 15^{\circ} \times t\ )\ \right] </math>

where L is the sundial's geographical [[latitude]] (and the angle the gnomon makes with the dial plate), <math>\ H_H\ </math> is the angle between a given hour-line and the noon hour-line (which always points towards [[true north]]) on the plane, and {{mvar|t}} is the number of hours before or after noon. For example, the angle <math>\ H_H\ </math> of the 3&nbsp;{{sc|pm}} hour-line would equal the [[inverse trigonometric function|arctangent]] of {{nobr| [[trigonometric function|sin]] {{mvar|L}} ,}} since tan 45° = 1. When <math>\ L = 90^\circ\ </math> (at the [[North Pole]]), the horizontal sundial becomes an equatorial sundial; the style points straight up (vertically), and the horizontal plane is aligned with the equatorial plane; the hour-line formula becomes <math>\ H_H  = 15^\circ \times t\ ,</math> as for an equatorial dial. A horizontal sundial at the Earth's [[equator]], where <math>\ L = 0^\circ\ ,</math> would require a (raised) horizontal style and would be an example of a polar sundial (see below).

[[File:Kew Gardens 0502.JPG|thumb|Detail of horizontal sundial outside [[Kew Palace]] in London, United Kingdom]]
The chief advantages of the horizontal sundial are that it is easy to read, and the sunlight lights the face throughout the year. All the hour-lines intersect at the point where the gnomon's style crosses the horizontal plane. Since the style is aligned with the Earth's rotational axis, the style points [[true north]] and its angle with the horizontal equals the sundial's geographical latitude {{mvar|L}}&nbsp;. A sundial designed for one [[latitude]] can be adjusted for use at another latitude by tilting its base upwards or downwards by an angle equal to the difference in latitude. For example, a sundial designed for a latitude of 40° can be used at a latitude of 45°, if the sundial plane is tilted upwards by 5°, thus aligning the style with the Earth's rotational axis. {{Citation needed|date=August 2012}}

Many ornamental sundials are designed to be used at 45 degrees north. Some mass-produced garden sundials fail to correctly calculate the ''hourlines'' and so can never be corrected. A local standard [[time zone]] is nominally 15 degrees wide, but may be modified to follow geographic or political boundaries. A sundial can be rotated around its style (which must remain pointed at the celestial pole) to adjust to the local time zone. In most cases, a rotation in the range of 7.5° east to 23° west suffices. This will introduce error in sundials that do not have equal hour angles. To correct for [[daylight saving time]], a face needs two sets of numerals or a correction table. An informal standard is to have numerals in hot colors for summer, and in cool colors for winter.{{Citation needed|date=August 2012}} Since the hour angles are not evenly spaced, the equation of time corrections cannot be made via rotating the dial plate about the gnomon axis. These types of dials usually have an equation of time correction tabulation engraved on their pedestals or close by. Horizontal dials are commonly seen in gardens, churchyards and in public areas.

===Vertical sundials===
[[File:Houghton Hall Norfolk UK 4-face sundial.jpg|thumb|Two vertical dials at [[Houghton Hall]] [[Norfolk]] [[UK]] {{Coord|52.827469|0.657616|type:landmark|format=dms|name=Houghton Hall vertical sundials}}. The left and right dials face south and east, respectively. Both styles are parallel, their angle to the horizontal equaling the latitude. The east-facing dial is a polar dial with parallel hour-lines, the dial-face being parallel to the style.]]

In the common ''vertical dial'', the shadow-receiving plane is aligned vertically; as usual, the gnomon's style is aligned with the Earth's axis of rotation.<ref>{{harvp|Rohr|1996|pp=46–49}}; {{harvp|Mayall|Mayall|1994|pp= 557–58, 102–107, 141–143}}; {{harvp|Waugh|1973| pp= 52–99}}</ref> As in the horizontal dial, the line of shadow does not move uniformly on the face; the sundial is not ''equiangular''. If the face of the vertical dial points directly south, the angle of the hour-lines is instead described by the formula:<ref>{{harvp|Rohr|1996|p=65}}; {{harvp|Waugh|1973| p=52}}</ref>

:<math> \tan H_V = \cos L\ \tan(\ 15^{\circ} \times t\ )\ </math>

where {{mvar|L}} is the sundial's geographical [[latitude]], <math>\ H_V\ </math> is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and {{mvar|t}} is the number of hours before or after noon. For example, the angle <math>\ H_V\ </math> of the 3&nbsp;{{sc|p.m.}} hour-line would equal the [[inverse trigonometric function|arctangent]] of {{nobr| [[trigonometric function|cos]] {{mvar|L}} ,}} since {{nobr|{{math| tan 45° {{=}} 1 }} .}} The shadow moves ''counter-clockwise'' on a south-facing vertical dial, whereas it runs clockwise on horizontal and equatorial north-facing dials.

Dials with faces perpendicular to the ground and which face directly south, north, east, or west are called ''vertical direct dials''.<ref>{{harvp|Rohr|1996|pp=54–55}}; {{harvp|Waugh|1973| pp= 52–69}}</ref> It is widely believed, and stated in respectable publications, that a vertical dial cannot receive more than twelve hours of sunlight a day, no matter how many hours of daylight there are.<ref>{{harvp|Waugh|1973| p=83}}</ref> However, there is an exception. Vertical sundials in the tropics which face the nearer pole (e.g. north facing in the zone between the Equator and the Tropic of Cancer) can actually receive sunlight for more than 12&nbsp;hours from sunrise to sunset for a short period around the time of the summer solstice. For example, at latitude 20° North, on June&nbsp;21, the sun shines on a north-facing vertical wall for 13&nbsp;hours, 21&nbsp;minutes.<ref name="Sunrise">{{cite web |last=Morrissey |first=David |title=Worldwide Sunrise and Sunset map |url=http://www.sunrisesunsetmap.com/ |url-status=live |access-date=28 October 2013 |archive-url=https://web.archive.org/web/20210210004115/https://sunrisesunsetmap.com/  |archive-date=10 February 2021}}</ref> Vertical sundials which do ''not'' face directly south (in the northern hemisphere) may receive significantly less than twelve hours of sunlight per day, depending on the direction they do face, and on the time of year. For example, a vertical dial that faces due East can tell time only in the morning hours; in the afternoon, the sun does not shine on its face. Vertical dials that face due East or West are ''polar dials'', which will be described below. Vertical dials that face north are uncommon, because they tell time only during the spring and summer, and do not show the midday hours except in tropical latitudes (and even there, only around midsummer). For non-direct vertical dials – those that face in non-cardinal directions – the mathematics of arranging the style and the hour-lines becomes more complicated; it may be easier to mark the hour lines by observation, but the placement of the style, at least, must be calculated first; such dials are said to be ''declining dials''.<ref>{{harvp|Rohr|1996|pp=55–69}}; {{harvp|Mayall|Mayall|1994|p=58}}; {{harvp|Waugh|1973| pp= 74–99}}</ref>

[[File:Nové Město nad Metují sundials 2011 3.jpg|thumb|170px|"Double" sundials in [[Nové Město nad Metují]], Czech Republic; the observer is facing almost due north.]]

Vertical dials are commonly mounted on the walls of buildings, such as town-halls, [[cupola]]s and church-towers, where they are easy to see from far away. In some cases, vertical dials are placed on all four sides of a rectangular tower, providing the time throughout the day. The face may be painted on the wall, or displayed in inlaid stone; the gnomon is often a single metal bar, or a tripod of metal bars for rigidity. If the wall of the building faces ''toward'' the south, but does not face due south, the gnomon will not lie along the noon line, and the hour lines must be corrected. Since the gnomon's style must be parallel to the Earth's axis, it always "points" [[true north]] and its angle with the horizontal will equal the sundial's geographical latitude; on a direct south dial, its angle with the vertical face of the dial will equal the [[colatitude]], or 90° minus the latitude.<ref>{{harvp|Waugh|1973| p=55}}</ref>

===Polar dials===
[[File:Sundial - Melbourne Planetarium.jpg|thumb|Polar sundial at [[Scienceworks (Melbourne)|Melbourne Planetarium]]]]
[[File:Reloj de sol polar en Donramiro (Lalín, España).jpg|thumb|Monumental polar sundial in [[Lalín]] ([[Spain]])]]
In ''polar dials'', the shadow-receiving plane is aligned ''parallel'' to the gnomon-style.<ref>{{harvp|Rohr|1996|p=72}}; {{harvp|Mayall|Mayall|1994|pp= 58, 107–112}}; {{harvp|Waugh|1973| pp= 70–73}}</ref>
Thus, the shadow slides sideways over the surface, moving perpendicularly to itself as the Sun rotates about the style. As with the gnomon, the hour-lines are all aligned with the Earth's rotational axis. When the Sun's rays are nearly parallel to the plane, the shadow moves very quickly and the hour lines are spaced far apart. The direct East- and West-facing dials are examples of a polar dial. However, the face of a polar dial need not be vertical; it need only be parallel to the gnomon. Thus, a plane inclined at the angle of latitude (relative to horizontal) under the similarly inclined gnomon will be a polar dial. The perpendicular spacing {{mvar|X}} of the hour-lines in the plane is described by the formula

:<math> X = H\ \tan(\ 15^{\circ} \times t\ )\ </math>

where {{mvar|H}} is the height of the style above the plane, and {{mvar|t}} is the time (in hours) before or after the center-time for the polar dial. The center time is the time when the style's shadow falls directly down on the plane; for an East-facing dial, the center time will be 6&nbsp;{{sc|a.m.}}, for a West-facing dial, this will be 6&nbsp;{{sc|p.m.}}, and for the inclined dial described above, it will be noon. When {{mvar|t}} approaches ±6 hours away from the center time, the spacing {{mvar|X}} diverges to [[Extended real number line|+∞]]; this occurs when the Sun's rays become parallel to the plane.

===Vertical declining dials===
[[File:Verticalezonnewijzers-en.jpg|thumb|600px|center|Effect of declining on a sundial's hour-lines. A vertical dial, at a latitude of 51°&nbsp;N, designed to face due south (far left) shows all the hours from 6&nbsp;{{sc|a.m.}} to 6&nbsp;{{sc|p.m.}}, and has converging hour-lines symmetrical about the noon hour-line. By contrast, a West-facing dial (far right) is polar, with parallel hour lines, and shows only hours after noon. At the intermediate orientations of [[Boxing the compass|south-southwest, southwest, and west-southwest]], the hour lines are asymmetrical about noon, with the morning hour-lines ever more widely spaced.]]
{{Clear}}
<!-- [[File:MootHallSundial.JPG|upright|thumb|SSW facing, vertical declining sundial on Moot Hall, [[Aldeburgh]], Suffolk, England.]]  Logically better place-->
[[File:Vertical Sundial at Fatih Mosque.jpg|upright|thumb|Two sundials, a large and a small one, at [[Fatih Mosque]], [[Istanbul]] dating back to the late 16th&nbsp;century. It is on the southwest facade with an azimuth angle of 52° N.]]
A ''declining dial'' is any non-horizontal, planar dial that does not face in a cardinal direction, such as (true) [[north]], [[south]], [[east]] or [[west]].<ref>{{harvp|Rohr|1996|pp=55–69}}; {{harvp|Mayall|Mayall|1994|pp= 58–112, 101–117, 1458–146}}; {{harvp|Waugh|1973| pp= 74–99}}</ref> As usual, the gnomon's style is aligned with the Earth's rotational axis, but the hour-lines are not symmetrical about the noon hour-line. For a vertical dial, the angle <math>\ H_\text{VD}\ </math> between the noon hour-line and another hour-line is given by the formula below. Note that <math>\ H_\text{VD}\ </math> is defined positive in the clockwise sense w.r.t. the upper vertical hour angle; and that its conversion to the equivalent solar hour requires careful consideration of which quadrant of the sundial that it belongs in.<ref name="harvp|Rohr|1996|p=79">{{harvp|Rohr|1996|p=79}}</ref>

:<math> \tan H_\text{VD} = \frac{\cos L}{\ \cos D\ \cot(\ 15^{\circ} \times t\ ) - s_o\ \sin L\ \sin D\ } </math>

where <math>\ L\ </math> is the sundial's geographical [[latitude]]; {{mvar|t}} is the time before or after noon; <math>\ D\ </math> is the angle of declination from true [[south]], defined as positive when east of south; and <math>\ s_o\ </math> is a switch integer for the dial orientation. A partly south-facing dial has an <math>\ s_o\ </math> value of {{nobr|{{math| +1 }} ;}} those partly north-facing, a value of {{nobr|{{math| −1 }}.}} When such a dial faces south (<math>\ D  = 0^{\circ}\ </math>), this formula reduces to the formula given above for vertical south-facing dials, i.e.

:<math>\ \tan H_\text{V}  = \cos L\ \tan(\ 15^{\circ} \times t\ )\ </math>

When a sundial is not aligned with a cardinal direction, the substyle of its gnomon is not aligned with the noon hour-line. The angle <math>\ B\ </math> between the substyle and the noon hour-line is given by the formula<ref name="harvp|Rohr|1996|p=79"/>

:<math> \tan B = \sin D\ \cot L </math>

If a vertical sundial faces trUe south Or north (<math>\ D = 0^{\circ}\ </math> or <math>\ D  = 180^{\circ}\ ,</math> respectively), the angle <math>\ B  = 0^{\circ}\ </math> and the substyle is aligned with the noon hour-line.

The height of the gnomon, that is the angle the style makes to the plate, <math>\ G\ ,</math> is given by :
:<math>\ \sin G = \cos D\ \cos L  ~</math><ref>{{harvp|Mayall|Mayall|1994|p= 138}}</ref>

===Reclining dials===
[[File:RelSolValongo.jpg|thumb|right|Vertical reclining dial in the Southern Hemisphere, facing due north, with hyperbolic declination lines and hour lines. Ordinary vertical sundial at this latitude (between tropics) could not produce a declination line for the summer solstice. This particular sundial is located at the [[Valongo Observatory]] of the [[Federal University of Rio de Janeiro]], Brazil.]]
The sundials described above have gnomons that are aligned with the Earth's rotational axis and cast their shadow onto a plane. If the plane is neither vertical nor horizontal nor equatorial, the sundial is said to be ''reclining'' or ''inclining''.<ref>{{harvp|Rohr|1965|pp=70–81}}; {{harvp|Waugh|1973|pp=100–107}}; {{harvp|Mayall|Mayall|1994|pp=59–60, 117–122, 144–145}}</ref> Such a sundial might be located on a south-facing roof, for example. The hour-lines for such a sundial can be calculated by slightly correcting the horizontal formula above<ref>{{harvp|Rohr|1965|p=77}}; {{harvp|Waugh|1973|pp=101–103}};</ref><ref>{{cite book | first = Samuel Capt. | last = Sturmy | year = 1683 | title = The Art of Dialling | place = London, UK }}</ref>

:<math>\ \tan H_{RV} = \cos(\ L + R\ )\ \tan(\ 15^{\circ} \times t\ )\ </math>

where <math>\ R\ </math> is the desired angle of reclining relative to the local vertical, {{mvar|L}} is the sundial's geographical latitude, <math>\ H_{RV}\ </math> is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and {{mvar|t}} is the number of hours before or after noon. For example, the angle <math>\ H_{RV}\ </math> of the 3pm hour-line would equal the [[inverse trigonometric function|arctangent]] of {{nobr| {{math|[[trigonometric function|cos]]( ''L'' + ''R'' )}} ,}} since {{nobr| {{math| tan 45° {{=}} 1 }} .}} When {{nobr| {{math|''R'' {{=}} 0°}} }} (in other words, a south-facing vertical dial), we obtain the vertical dial formula above.

Some authors use a more specific nomenclature to describe the orientation of the shadow-receiving plane. If the plane's face points downwards towards the ground, it is said to be ''proclining'' or ''inclining'', whereas a dial is said to be ''reclining'' when the dial face is pointing away from the ground. Many authors also often refer to reclined, proclined and inclined sundials in general as inclined sundials. It is also common in the latter case to measure the angle of inclination relative to the horizontal plane on the sun side of the dial.
In such texts, since <math>\ I = 90^\circ + R\ ,</math> the hour angle formula will often be seen written as :

:<math> \tan H_{RV} = \sin(L + I)\ \tan(\ 15^{\circ} \times t\ )\ </math>

The angle between the gnomon style and the dial plate, B, in this type of sundial is :

:<math> B = 90^{\circ} - (L + R) </math>

or :

:<math> B = 180^{\circ} - (L + I) </math>

===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>
<!-- Description of the problem -->

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.

===Spherical sundials===
[[File:Zw stelling.jpg|thumb|170px|upright|Equatorial bow sundial in [[Hasselt]], [[Flanders]] in [[Belgium]] {{Coord|50|55|47|N|5|20|31|E|type:landmark|name=Hasselt equatorial bow sundial}}. The rays pass through the narrow slot, forming a uniformly rotating sheet of light that falls on the circular bow. The hour-lines are equally spaced; in this image, the local solar time is roughly 15:00&nbsp;hours {{nobr|( 3 {{sc|p.m.}} ).}} On September&nbsp;10, a small ball, welded into the slot casts a shadow on centre of the hour band.]]

The surface receiving the shadow need not be a plane, but can have any shape, provided that the sundial maker is willing to mark the hour-lines. If the style is aligned with the Earth's rotational axis, a spherical shape is convenient since the hour-lines are equally spaced, as they are on the equatorial dial shown here; the sundial is ''equiangular''. This is the principle behind the armillary sphere and the equatorial bow sundial.<ref>{{harvp|Rohr|1996|pp=114, 1214–125}}; {{harvp|Mayall|Mayall|1994|pp= 60, 126–129, 151–115}}; {{harvp|Waugh|1973| pp= 174–180}}</ref> However, some equiangular sundials – such as the Lambert dial described below – are based on other principles.

In the ''equatorial bow sundial'', the gnomon is a bar, slot or stretched wire parallel to the celestial axis. The face is a semicircle, corresponding to the equator of the sphere, with markings on the inner surface. This pattern, built a couple of meters wide out of temperature-invariant steel [[invar]], was used to keep the trains running on time in France before World War I.{{sfn|Rohr|1996|p=17}}

Among the most precise sundials ever made are two equatorial bows constructed of [[marble]] found in [[Yantra mandir (Jaipur)|Yantra mandir]].<ref>{{harvp|Rohr|1996|pp=118–119}}; {{harvp|Mayall|Mayall|1994|pp=215–216}}</ref> This collection of sundials and other astronomical instruments was built by Maharaja [[Jai Singh II]] at his then-new capital of [[Jaipur]], India between 1727 and 1733. The larger equatorial bow is called the ''Samrat Yantra'' (The Supreme Instrument); standing at 27&nbsp;meters, its shadow moves visibly at 1&nbsp;mm per second, or roughly a hand's breadth (6&nbsp;cm) every minute.

===Cylindrical, conical, and other non-planar sundials===
[[File:Precision sundial in Bütgenbach-Belgium.jpg|left|upright|thumb|170px|Precision sundial in Bütgenbach, Belgium.
(Precision{{space}}={{space}}±30{{space}}seconds)
{{Coord|50.4231|6.2017|type:landmark|format=dms|name=Belgium}}]]
Other non-planar surfaces may be used to receive the shadow of the gnomon.
 
As an elegant alternative, the style (which could be created by a hole or slit in the circumference) may be located on the circumference of a cylinder or sphere, rather than at its central axis of symmetry.

In that case, the hour lines are again spaced equally, but at ''twice'' the usual angle, due to the geometrical [[inscribed angle]] theorem. This is the basis of some modern sundials, but it was also used in ancient times;{{efn|
An example of such a half-cylindrical dial may be found at [[Wellesley College]] in [[Massachusetts]].<ref>{{harvp|Mayall|Mayall|1994|p=94}}</ref>
}}

In another variation of the polar-axis-aligned cylindrical, a cylindrical dial could be rendered as a helical ribbon-like surface, with a thin gnomon located either along its center or at its periphery.