Bifilar sundial
   HOME

TheInfoList



OR:

A bifilar dial is a type of
sundial A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat ...
invented by the German mathematician Hugo Michnik in 1922. It has two non-touching threads parallel to the dial. Usually the second thread is
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
-(perpendicular) to the first. The intersection of the two threads' shadows gives the
local apparent time Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial ti ...
. When the threads have the correctly calculated separation, the hour-lines on the horizontal surface are uniformly drawn. The angle between successive hour-lines is constant. The hour-lines for successive hours are 15 degrees apart.


History

The bifilar dial was invented in April 1922 by the mathematician and maths teacher, Hugo Michnik, from
Beuthen Bytom (Polish pronunciation: ; Silesian: ''Bytōm, Bytōń'', german: Beuthen O.S.) is a city in Upper Silesia, in southern Poland. Located in the Silesian Voivodeship of Poland, the city is 7 km northwest of Katowice, the regional capita ...
,
Upper Silesia Upper Silesia ( pl, Górny Śląsk; szl, Gůrny Ślůnsk, Gōrny Ślōnsk; cs, Horní Slezsko; german: Oberschlesien; Silesian German: ; la, Silesia Superior) is the southeastern part of the historical and geographical region of Silesia, located ...
. He studied the horizontal dial- starting on a conventional XYZ cartesian framework and building up a general projection which he states was an exceptional case of a Steiner transformation. He related the trace of the sun to
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s and the angle on the dial-plate to the
hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the '' meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle'' ...
and the calculation of
local apparent time Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial ti ...
, using conventional hours and the historic
Italian Italian(s) may refer to: * Anything of, from, or related to the people of Italy over the centuries ** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom ** Italian language, a Romance language *** Regional Ita ...
and Babylonian hours. He refers in the paper, to a previous publication on the theory of sundials in 1914.Beiträge zur Theorie der Sonnenuhren, Leipzig, 1914 His method has been applied to vertical near-declinant dials, and a more general declining-reclining dial. Work has been subsequently done by Dominique Collin.


Horizontal bifilar dial

This was the dial that Hugo Michnik invented and studied. By simplifying the general example so: :* the wires cross
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
ly- one running north-south and the other east-west :* The east west wire passes under the north south dial, so the h_2 = h_1 \sin\varphi\quad (latitude) the shadow is thrown on a dial-plate marked out like a simple
equatorial sundial A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat ...
.


The proof

The first wire f_1\, is orientated north-south at a constant distance h_1\, from the dial plate \Pi\,
The second wire f_2\, is orientated east-west at a constant distance h_2\, from the dial plate \Pi\, (thus f_2\, is orthogonal to f_1\, which lies on the plane of the
meridian Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...
). In this proof \varphi (pronounced phi) is the
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
of the dial plate. Respectively, (\mathcal D_1) and (\mathcal D_2) are the vertical projections of wires f_1\, and f_2\, on the dial plate \Pi\,. Point O\, is the point on the dial plate directly under the two wires' intersection.
That point is the origin of the X,Y co-ordinate system referred to below. The X-axis is the east–west line passing through the origin. The Y-axis is the north–south line passing through the origin. The positive Y direction is northward. One can show that if the position of the sun is known and determined by the
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
t_\odot et \delta\, (pronounced t-dot and delta, respectively the known as the
hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the '' meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle'' ...
et
declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the ...
), the co-ordinates x_I\, and y_I\, of point I\,, the intersection on the two shadows on the dial-plate \Pi\, have values of : :\begin x_I &=& h_1 \frac \\ \ &\ & \ \\ y_I &=& h_2 \frac \end Eliminating the variable \delta\, in the two preceding equations, one obtains a new equation defined for x_I\, and y_I\, which gives, as a function of the latitude \varphi and the solar hour angle solaire t_\odot, the equation of the trace of the sun associated with the local apparent time. In its simplest form this equation is written: :\frac = \frac\ \operatornamet_\odot This relation shows that the hour traces are indeed
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s and the meeting-point of these line segments is the point C\,: :x_C = 0\, :y_C = -h_2 / \operatorname\varphi In other words, point C is south of point O (where the wires intersect), by a distance of h_2 / \operatorname\varphi, where \varphi is the latitude. ;Special case If one arranges the two wire heights h_2\, and h_1\, such : : h_2 = h_1 \sin\varphi\quad then the equation for the hour lines can be simply written as: :\frac = \operatornamet_\odot at all times, the intersection I\, of the shadows on the dial plate \Pi\, is such that the angle \widehat is equal to the
hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the '' meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle'' ...
t_\odot of the sun so thus represents solar time. So provided the sundial respects the la condition h_2 = h_1 \sin\varphi\quad the trace of the sun corresponds to the hour-angle shown by lines (rays) centred on the point C\, and the 13 rays that correspond to the hours 6:00, 7:00, 8:00, 9:00... 15:00, 16:00, 17:00, 18:00 are regularly spaced at a constant angle of 15°, about point C. by the French mathematician Dominique Collin.


A practical example

A London dial is the name given to dials set for 51° 30' N. A simple London bifilar dial has a dial plate with 13 line segments drawn outward from a centre-point C, with each hour's line drawn 15° clockwise from the previous hour's line. The midday line is aligned towards the north. The north–south wire is 10 cm (h_1) above the midday line. That east-west wire is placed at a height of 7.826 (h_2) centimeters- equivalent to 10 cm x sin(51° 30'). This passes through C. The east–west wire crosses the north–south wire 6.275 cm north of the centre-point C- that being the equivalent of - 7.826 (h_2) divided by tan (51° 30').


Reclining-declining bifilar sundials

Whether a sundial is a bifilar, or whether it's the familiar flat-dial with a straight style (like the usual horizontal and vertical-declining sundials), making it reclining, vertical-declining, or reclining-declining is exactly the same. The declining or reclining-declining mounting is achieved in exactly the same manner, whether the dial is bifilar, or the usual straight-style flat dial. For any flat-dial, whether bifilar, or ordinary straight-style, the north celestial pole has a certain altitude, measured from the plane of the dial. # Effective latitude:If that dial-plane is horizontal, then it's a horizontal dial (bifilar, or straight-style). Then, of course the north celestial pole's altitude, measured from the dial-plane, is the latitude of the location. Well then, if the flat-dial is reclined, declined, or reclined-&-declined, everything is the same as if it the dial were horizontal, with the celestial pole's altitude, measured with respect to the dial-plane, treated as the latitude. # Dial-North:Likewise, the north celestial pole's longitude, measured with respect to the plane of the dial, with respect to the downward direction (or the direction that a marble would roll, if the dial is reclining) on the dial-face, is the direction that is treated as north, when drawing the hour-lines. I'll call that direction "dial-north". # Equatorial Longitude (hour-angle) of dial-north:It's necessary to find the equatorial longitude of the dial-north direction (drawn on the dial). In the case of the horizontal dial, of course that's an hour-angle of zero, the south meridian. That determines what time ("dial-north time") is represented by the dial-north line. Other times, before and after that, can then have their lines drawn according to their differences from dial-north time—in the same way as they' be drawn on a horizontal dial-face according to their differences from 12 noon (true solar time).


References

;Footnotes ;Notes ;Bibliography * *


External links

{{Time measurement and standards Sundials