# Bifilar sundial

A bifilar dial showing the two wires
Stainless steel bifilar sundial in Italy

A bifilar dial is a type of sundial 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-(perpendicular) to the first. [1] The intersection of the two threads' shadows gives the local apparent time.

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, Upper Silesia. 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 sections and the angle on the dial-plate to the hour angle and the calculation of local apparent time, using conventional hours and the historic Italian and Babylonian hours. [1] He refers in the paper, to a previous publication on the theory of sundials in 1914.[2]

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.[3]

## Horizontal bifilar dial

This was the dial that Hugo Michnik invented and studied. By simplifying the general example so:

• the wires cross orthogonally- one running north-south and the other east-west
• The east west wire passes under the north south dial, so the ${\displaystyle h_{2}=h_{1}\sin \varphi \quad }$ (latitude)

the shadow is thrown on a dial-plate marked out like a simple equatorial sundial.

### The proof

The first wire ${\displaystyle f_{1}\,}$ is orientated north-south at a constant distance ${\displaystyle h_{1}\,}$ from the dial plate ${\displaystyle \Pi \,}$
The second wire ${\displaystyle f_{2}\,}$ is orientated east-west at a constant distance ${\displaystyle h_{2}\,}$ from the dial plate ${\displaystyle \Pi \,}$ (thus ${\displaystyle f_{2}\,}$ is orthogonal to ${\displaystyle f_{1}\,}$ which lies on the plane of the meridian ).

In this proof ${\displaystyle \varphi }$ (pronounced phi) is the latitude of the dial plate.

Respectively, ${\displaystyle ({\mathcal {D}}_{1})}$ and ${\displaystyle ({\mathcal {D}}_{2})}$ are the vertical projections of wires ${\displaystyle f_{1}\,}$ and ${\displaystyle f_{2}\,}$ on the dial plate ${\displaystyle \Pi \,}$.

Point ${\displaystyle O\,}$ 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 ${\displaystyle t_{\odot }}$ et ${\displaystyle \delta \,}$ (pronounced t-dot and delta, respectively the known as the hour angle et declination), the co-ordinates ${\displaystyle x_{I}\,}$ and ${\displaystyle y_{I}\,}$ of point ${\displaystyle I\,}$, the intersection on the two shadows on the dial-plate ${\displaystyle \Pi \,}$ have values of :

${\displaystyle {\begin{matrix}x_{I}&=&h_{1}{\frac {\sin t_{\odot }}{\sin \varphi \ \operatorname {tan} \delta \ +\ \cos \varphi \cos t_{\odot }}}\\\ &\ &\ \\y_{I}&=&h_{2}{\frac {-\cos \varphi \ \operatorname {tan} \delta \ +\ \sin \varphi \cos t_{\odot }}{\sin \varphi \ \operatorname {tan} \delta \ +\ \cos \varphi \cos t_{\odot }}}\end{matrix}}}$

Eliminating the variable ${\displaystyle \delta \,}$ in the two preceding equations, one obtains a new equation defined for ${\displaystyle x_{I}\,}$ and ${\displaystyle y_{I}\,}$ which gives, as a function of the latitude ${\displaystyle \varphi }$ and the solar hour angle solaire ${\displaystyle t_{\odot }}$, the equation of the trace of the sun associated with the local apparent time. In its simplest form this equation is written:

${\displaystyle {\frac {x_{I}}{y_{I}+h_{2}/\operatorname {tan} \varphi }}={\frac {h_{1}\sin \varphi }{h_{2}}}\ \operatorname {tan} t_{\odot }}$

This relation shows that the hour traces are indeed line segments and the meeting-point of these line segments is the point ${\displaystyle C\,}$:

${\displaystyle x_{C}=0\,}$
${\displaystyle y_{C}=-h_{2}/\operatorname {tan} \varphi }$

In other words, point C is south of point O (where the wires intersect), by a distance of ${\displaystyle h_{2}/\operatorname {tan} \varphi }$, where ${\displaystyle \varphi }$ is the latitude.[1]

Special case

If one arranges the two wire heights ${\displaystyle h_{2}\,}$ and ${\displaystyle h_{1}\,}$ such :

 ${\displaystyle h_{2}=h_{1}\sin \varphi \quad }$

then the equation for the hour lines can be simply written as:

${\displaystyle {\frac {x_{I}-x_{C}}{y_{I}-y_{C}}}=\operatorname {tan} t_{\odot }}$

at all times, the intersection ${\displaystyle I\,}$ of the shadows on the dial plate ${\displaystyle \Pi \,}$ is such that the angle ${\displaystyle {\widehat {OCI}}}$ is equal to the hour angle ${\displaystyle t_{\odot }}$ of the sun so thus represents solar time.

So provided the sundial respects the la condition ${\displaystyle h_{2}=h_{1}\sin \varphi \quad }$ the trace of the sun corresponds to the hour-angle shown by lines (rays) centred on the point ${\displaystyle C\,}$ 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. [a][1]

### 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 (${\displaystyle h_{1}}$) above the midday line. That east-west wire is placed at a height of 7.826 (${\displaystyle h_{2}}$) centimeters- equivalent to 10cm 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 (${\displaystyle h_{2}}$) 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.

1. 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.
2. 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".
3. 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
1. ^ This property was named the homogenity of hour lines ' French: homogénéité des lignes horaires by the French mathematician Dominique Collin.
Notes
1. ^ a b c d
2. ^ Beiträge zur Theorie der Sonnenuhren, Leipzig, 1914
3. ^
Bibliography
• Michnik, H (1922). "Title: Theorie einer Bifilar-Sonnenuhr". Astronomische Nachrichten (in German). 217 (5190): 81–90. Bibcode:1922AN....217...81M. doi:10.1002/asna.19222170602. Retrieved 17 December 2013.
• Collin, Domenique (2000). "Théorie sur le cadran solaire bifilaire vertical déclinant" (in French). F 62000 Calais. Retrieved 8 February 2015.