Australian Facetors' Guild Limited

I have been putting some thought into design issues with concave faceted gems.  My theory is as follows.

Imagine lines are drawn on the facet in a grid pattern and the facet is cut along a line that at some point meets the C axis.

1. The line down the centre of the concave from cutlet to girdle, is 0 tilt in both directions, so should behave like a normal facet.  Hence the usual faceting angle rules apply.
2. All lines on the facet parrallel to that line have the same angle relative to the C axis of the gem but a tangent 90 degrees to that line will deflect the light in the direction perpendicular to that tangent.  
3. For practical purposes there are an infinite number of parralel lines that can be drawn on the facet so for any given cross section of the facet perpendicular to the C axis the facet will radiate light in a parabolic fan around the facet.  
4. The arc through which a cross section of the facet perpendicular to the facet face is cut will determine the size of the fan on each cross section perpendicular to the C axis.
5. If you cut a series of cross sections through the facet perpendicular to the C axis, those beside the narrowest points  on the centre line of the facet will have a small arc and those at the widest part will have the greatest arc.  The width of the arc depends on how much  that part of the facet wraps ariond the cylindrical lap cutting the facet.
6. Because the light is fanned, its intensity drops as it spreads out in each cross section before striking another facet.
7. The sum of light from each cross section as you look down the C axis means there is more intensity along the centre of the facet than at its edges.  If the material has a slight cloudiness this will be visable as a glow at the centre of the facet.
8. This means light should seem to concentrate along the center axis of the facet and more light will reflect from facets receiving light from this light path than from other light paths.
9. With so much light bouncing about inside the stone, there must be interference patterns caused by the light waves cancelling (extinguishment, dark nodes) or reinforcing (bright nodes) each other.
10. Each colour of light is a different wavelength. So the placement of these nodes varies with the colour of the light. So concave facets should be expected to accentuate diffraction.

So it seems to me that you should best set the angle of concave facets as you would for a flat facet.

Any other thoughts or theories on this?

Regards

Gordon