A simple method for the transposition of faceting angles.


By Clive Washington


One of the first problems encountered by the beginner is the conversion of published faceting designs to a form suitable for materials other than quartz. Virtually all published designs use angles which have been optimized for quartz, and since other materials have different refractive indices and critical angles, they require the facets to be cut at different angles to the original design, in order to optimize the optical performance, or avoid a 'fish-eye' stone

One of the simplest methods of doing this conversion is to add or subtract a fixed number of degrees from the design angles, following published tables of recommended crown and pavilion mains angles for different materials. Thus, for example to convert a design from quartz angles to topaz, we check the list and find that quartz pavilion mains are cut at 43 degrees and topaz at 39 degrees. So we must subtract 4 degrees from the quartz pavilion angles in the original design to obtain the topaz pavilion angles, and simply repeat this subtraction for every facet in our design; thus a set of facets cut at 57 degrees in the original quartz design must be cut at 53 degrees when using topaz.

I began to use this method when I started faceting and it rapidly became obvious that it had major shortcomings. It worked fairly well for conventional designs like the standard brilliant, where most facet angles were not too far from the mains angle, and small errors could be corrected by altering the proportions of the stone. However, if the facets cover a large angular range, or are close to the table or girdle, problems start to arise. How, for example, do you transpose the angles for one of those fancy multifaceted tables, or a bar cut, and what effect does it have on the geometry of a bead or pendeloque? Now that we have GemCad, a lot of published designs have the angles calculated quite accurately, to a tenth of a degree or better, and it seemed ridiculous that a designer should go to so much trouble to work out accurate angles, only for me to maul them around by knocking off degrees here and there. It also irked me that the procedure could not be applied uniformly to all the facets - one is obviously not going to add or subtract a transposition angle to the girdle or table angles. But a rigorous system should treat all facets alike - so what was going on?

I realized that the proper procedure was to perform a mathematical transformation on the stone that had the effect of 'stretching' it in a vertical direction alone, keeping the geometrical relation between the facets correct. It didn't take an awful lot of thought to come up with the tangent ratio method for this transform - which was probably known well before I devised it for myself in the mid 70's. This method transforms angles by scaling their tangents by the ratio of the tangents of the design and target angles - easy to do on a calculator or with a quick Excel spreadsheet, but hard to explain without formulae. I'm sure it's been described in this newsletter before, so I won't repeat it here.

However, when I got to this point the first primitive calculators were just starting to appear, scientific calculators were well outside my pocket, and the idea of a personal computer was laughable. (That was about 8 years before Bill Gates said that "640k of RAM should be enough for anyone"). A tangent ratio conversion meant half an hour's thrashing away with four-figure tables and a slide rule, and was a fairly unpleasant prospect. I wanted to find an easier way.

About this time I became interested in nomograms - graphical lines like fixed slide rules that allowed you to perform simple calculations by laying a ruler across them. I realized that it should be possible to use a nomogram for faceting angle conversion, and tried to work out the transforms for the various lines. After a bit of head-scratching, and taking into account the obvious properties of the transform (it doesn't affect angles of zero or 90 degrees, and it needs to transpose circular functions) I came up with the circular diagram in the figure. It works like this:

Suppose you want to convert from quartz pavilion angles to topaz pavilion angles. Again you look up the recommended mains angles in the tables, 43 degrees and 39 degrees. The top semicircle in the diagram is used for the original design angles, while the bottom is used for the required transformed angle (which I call the target angle). You need a ruler and a sharp pencil. Place the ruler so that it joins the design angle of 43 degrees on the top half of the diagram and the target angle of 39 degrees on the bottom half of the diagram. Draw a short line where the ruler crosses the horizontal centre line. Now to convert any other angle from the original design, all you have to do is to place the ruler so that it lies on the design angle on the top half of the diagram, and passes exactly through your mark on the centre line. The point at which the ruler crosses the bottom angular scale is the target angle you need. This is actually much easier to do than to describe. With a good ruler and a sharp pencil it is possible to work to a tenth of a degree. If you're printing the diagram from here, you should check that your printer prints a true circle by measuring across the diameter in several places. Some cheap photocopiers will also distort your geometry, but in practise it makes no difference to the transform.

tangent
The diagram above is for illustration only; a high quality version of the diagram is available for download as a Word or PDF document

word

pdf

Right click over image and select
save target as

Right click over image and select
save target as


You should remember that you will need a different centre mark for the crown conversion because the recommended crown angles are different to the pavilion ones. It pays to copy or print a few of the diagrams so that they can be disposed of when they get tatty, and it helps to use a soft pencil so that you can rub out your centre mark when you have finished, to avoid confusion with later stones. I devised a more complex version of the chart in which various centre points for different materials were marked for the crown and pavilion, but it rapidly became too cluttered and confusing. Better to make your own centre mark for the job in hand.

I find this procedure extremely quick and accurate, and can convert a design in a shorter time than it would take to start up the computer to get at a tangent ratio spreadsheet. Also not everyone has, or likes, computers or calculators, and this method is completely calculation-free, so I hope someone out there will find it useful.

One thing I would like to point out is that I have tested the procedure and it gives identical results to the tangent ratio method over its entire range. However, I have not yet proven analytically that it is exact - I'll get round to it one day, or perhaps somone else would like to have a go. Life's too short.
 
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