# Bifilar sundial

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.
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. He refers in the paper, to a previous publication on the theory of sundials in 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 orthogonally- 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.

### 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 ).
In this proof $\varphi$ (pronounced phi) is the latitude 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 $t_{\odot }$ and $\delta \,$ (pronounced t-dot and delta, respectively the known as the hour angle et declination), 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 :
    ${\begin{matrix}x_{I}&=&h_{1}{\frac {\sin t_{\odot }}{\sin \varphi \ \operatorname {tan} \delta \ +\ \cos \varphi \cos t_{\odot }}}\\\ &\ &\ \\y_{I}&=&h_{2}{\frac {-\cos \varphi \ \operatorname {tan} \delta \ +\ \sin \varphi \cos t_{\odot }}{\sin \varphi \ \operatorname {tan} \delta \ +\ \cos \varphi \cos t_{\odot }}}\end{matrix}}$
Eliminating the variable $\delta \,$ 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 $t_{\odot }$, the equation of the trace of the sun associated with the local apparent time. In its simplest form this equation is written:
    ${\frac {x_{I}}{y_{I}+h_{2}/\operatorname {tan} \varphi }}={\frac {h_{1}\sin \varphi }{h_{2}}}\ \operatorname {tan} t_{\odot }$
This relation shows that the hour traces are indeed line segments and the meeting-point of these line segments is the point $C\,$:
    $x_{C}=0\,$
    $y_{C}=-h_{2}/\operatorname {tan} \varphi$
In other words, point C is south of point O (where the wires intersect), by a distance of $h_{2}/\operatorname {tan} \varphi$, 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:
    ${\frac {x_{I}-x_{C}}{y_{I}-y_{C}}}=\operatorname {tan} t_{\odot }$
at all times, the intersection $I\,$ of the shadows on the dial plate $\Pi \,$ is such that the angle ${\widehat {OCI}}$ is equal to the hour angle $t_{\odot }$ 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.

### 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).