basic procedure part 2

Here is a worked example for a bifilar sundial with equiangular hour lines on an arbitrary plane.
The starting data for this dial are:

Location of the substyle on the pole style sundial

These constants tell us that the dial face corresponds to a horizontal plane in a southern latitude of 46.71 degrees and a longitude of 36.04 degrees westerly of ours (translation rule).
The hour lines run anti-clockwise (the style height is negative).
The pattern for apparent solar time on this dial is pictured above.
Of particular interest are the location of the substyle and the value for the style height.

Instead of a pole style, we place a gnomon of height g perpendicularly to the sundial, at a distance of g / tan(|style height|) from the hour line centre.
In our example we choose g = 20 mm, making this distance 20 / tan(44.97) = 20.02 mm.
The top of the gnomon is an indicator on the now imaginary pole style, and the sundial will function as before.
Now, we place two wires in the same height equal to g = 20 mm, through the gnomon top and parallel to the dial face. We remove the gnomon.
One wire should be over the substyle, the other at right angles to it.
The wires intersect and the sundial will function as before, but it is still not the wanted bifilar dial.

Still the same sundial

Now we lower the wire which is at right angles to the substyle, to a height of g2 = g1 . SIN(|style height|).
This value is 20 . SIN(44.97) = 14.13 mm.
At any point in time, the shadow of the unchanged wire at g1 will be in the same place, but that of the lowered wire g2 will now have moved by a factor equal to SIN(|style height|) = SIN(44.97) = 0.7067.
This will compress the sundial pattern in the direction of the substyle by that factor.
A photocopier could do this.
After this procedure, the sundial will look like the figure below.
The hour lines need to be extended somewhat.

Transformation by a factor in the indicated direction
Hour lines are now equiangular.

The transformation just applied to the apparent solar time hour lines is equally applicable to all other sorts of lines on a sundial.
Here is the starting sundial again, expanded with three date lines, an equation of time curve around the XII-hour line and the horizon.
You could add various other lines if you wish.
The indicator on the (imaginary) pole style is once more replaced by a gnomon of height g = 20 mm, and that, in turn, by two wires at equal heights g = 20 mm.

The starting sundial

We lower the wire at right angles to the substyle again by the factor SIN(|style height|) = SIN(44.97) = 0.7067, which compresses the pattern by that factor.
This creates the desired bifilar sundial with equiangular hour lines.
Notice that after this procedure, the horizon line is no longer horizontal.

Transformed into bifilar sundial
with equiangular hour lines

To construct a bifilar sundial, use an ordinary sundial, with all the desired lines, as a basis.
Through the point where the indicator for all these lines would be, place a wire in the direction of the substyle and let its height = g1.
At right angles, place a wire of height g2 = g1 . SIN(|style height|).
Compress the entire sundial pattern in the direction of the substyle with the factor SIN(|style height|).
This will create the bifilar sundial with hour lines for local suntime all spaced fifteen degrees.
As you can see, there are no difficult equations and calculations at all, as long as you know how to construct the starting sundial.

There are many more degrees of freedom in the construction of bifilar sundials.
The height factor for the second wire may be different.
Depending on value, the pattern is either compressed (0 < factor < 1) or enlarged (factor > 1).
But the hour lines will not be equiangular anymore.

The wires do not need to be parallel to the dial face, nor do they have to be at right angles to each other.
They do not have to be over the substyle, and they may even be curved.

However, all this would be outside the scope of these pages, which mean to present only the basics from 1923.