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 shadow is thrown on a dial-plate marked out like a simple equatorial sundial.
The proof.
The first wire formula_2 is orientated north-south at a constant distance formula_3 from the dial plate formula_4<br>
The second wire formula_5 is orientated east-west at a constant distance formula_6 from the dial plate formula_4 (thus formula_5 is orthogonal to formula_2 which lies on the plane of the meridian ).
In this proof formula_10 (pronounced phi) is the latitude of the dial plate.
Respectively, formula_11 and formula_12 are the vertical projections of wires formula_2 and formula_5 on the dial plate formula_4.
Point formula_16 is the point on the dial plate directly under the two wires' intersection.<br> 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 formula_17 et formula_18 (pronounced t-dot and delta, respectively the known as the hour angle et declination), the co-ordinates formula_19 and formula_20 of point formula_21, the intersection on the two shadows on the dial-plate formula_4 have values of :
Eliminating the variable formula_18 in the two preceding equations, one obtains a new equation defined for formula_19 and formula_20 which gives, as a function of the latitude formula_10 and the solar hour angle solaire formula_17, the equation of the trace of the sun associated with the local apparent time. In its simplest form this equation is written:
This relation shows that the hour traces are indeed line segments and the meeting-point of these line segments is the point formula_30:
In other words, point C is south of point O (where the wires intersect), by a distance of codice_3, where formula_10 is the latitude.
If one arranges the two wire heights formula_6 and formula_3 such :
then the equation for the hour lines can be simply written as:
at all times, the intersection formula_21 of the shadows on the dial plate formula_4 is such that the angle formula_41 is equal to the hour angle formula_17 of the sun so thus represents solar time.
So provided the sundial respects the la condition formula_1 the trace of the sun corresponds to the hour-angle shown by lines (rays) centred on the point formula_30 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 (formula_45) above the midday line. That east-west wire is placed at a height of 7.826 (formula_46) 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 (formula_46) 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.