| ISO Overview
| COS Overview
| Difference Overview
| ISO Detail
| COS Detail
| Difference Detail |
||A Graphical Presentation of Brightness
in the Standard Round Brilliant
by Bob Keller
Old Pueblo Lapidary Club - Tucson, Arizona
A Standard Round Brilliant design file is available for GemCad/GemPrint users.
The standard round brilliant was developed by diamond cutters and is used extensively in that industry. This enduring design and its variations are widely used by commercial and hobbyist faceters for cutting many materials, and the SRB is often assigned and recommended as a first project for beginning faceters. Shown at left is my first stone, a 10mm SRB from Bolivian amethyst that I cut in an Old Pueblo Lapidary Club faceting class.
One might think that the angles used to facet a standard round brilliant would be 'standard', but in fact they can and should be changed according to what you're cutting and what characteristics you want or need to maximize. There is seemingly endless discussion and debate on the 'best' angles for quartz and of course other materials with different refractive indexes.
Brightness is one of the desiderata of gemstone design. Brightness is a significant parameter and factor in the appearance of a faceted stone. If brightness was the sole figure of merit, gemstone design would be a more straightforward affair. The fact of the matter is there are other desiderata and brightness is often traded off in the process of striking the desired balance with other requirements such as shape, yield, reflection pattern, enhancement of display of color and dispersion, and tilt performance. Some of these desiderata are inherently subjective and others are more readily modeled and quantified.
Prior to the widespread adoption of personal computers and raytracing software by gemstone designers there was a time when brightness was more subjective and elusive. During the "Dark Ages" some dedicated faceters who became interested in the effect of pavilion and crown angles on brightness attempted to get a hand on design angles by cutting a series of test stones, systematically varying the pavilion and/or crown angles and judging the results by eye. While this approach is arguably the ultimate method by virtue of looking directly in the horse's mouth, it has several significant shortcomings.
A major obstacle to the cut-and-see methodology is the number of stones that must be cut to provide a systematic set of samples. The effects of varying pavilion and crown angles are not independent of each other. Light is refracted and reflected by both the pavilion and crown of a gemstone - they work together in combination to present a unique geometry to light entering and reflecting inside a stone before it refracts back out and either returns to the viewer through the crown, or is lost by "leaking" out through the pavilion. In order to systematically test the effect of ten different angles for the crown and pavilion using cut-and-see methodology, 100 stones would have to be cut. Compounding this situation is the fact that the refractive index of the material being cut has a significant bearing on the performance envelope for brightness. So a separate set of "look-and-see" stones would have to be cut in different materials to systematically examine the effect of varying the pavilion/crown angles for each refractive index of interest.
Nowadays gemstone design and raytracing software such as the GemCad suite of design tools and utilities provide an alternate methodology to cut-and-see for modeling and systematically examining the effect of design angles and refractive index on brightness. Graphing the numerical data output by raytracing utilities such as GemRay provides a powerful means of examining the interacting effects of design angles and refractive index on brightness. The graph below provides a visual presentation and overview of the relationship and effect of design angles and refractive index on brightness of the SRB.
The graph above presents an overview of ISO brightness for the standard round brilliant over a range of angles from 15° to 75° for the pavilion mains (facet pm) in 2.5° increments and a range of 10° to 80° for the crown (facet cm) mains in 5° increments. As the value of the pavilion and crown mains angle is varied the tangent ratio function is utilized to determine the other pavilion and crown angles as the mains angles are varied so as to preserve the crown and pavilion planviews. Use the "Cycle" buttons on this and the following graphs to observe the significant effect of varying the refractive index as a variable affecting brightness.
In interpreting the above graph and several to follow, it is illuminating to understand the underlying models employed to generate the numerical data represented by the graphs. The above graph presents brightness data derived from a lighting model referred to as ISO brightness. The ISO lighting model predicates a light source consisting of an illuminating hemisphere over the crown of the stone (analogous to the dome of a planetarium) that is uniformly lit - that is to say, the light is the same in all directions from the equator of the illuminating hemisphere, which contains the plane of the girdle, to the pole of the illuminating hemisphere, which is situated directly over the table. If you were sitting in the center of the table of a gemstone illuminated with an ISO source, the "sky" you would see when looking overhead (along the z axis of the stone) is represented at left.
The brightness data represented by this graph was derived and generated for the GemCad format SRB design file provided above using the GemFrame/GemRay raytracing applications. This data set was systematically calculated for eight different refractive indexes ranging from RI = 1.43 (fluorite) to RI = 2.61 (sphalerite).
Another lighting model is referred to as COS brightness. The COS lighting model predicates a light source consisting of an illuminating hemisphere over the crown of the stone that is lit with the brightest area at the pole directly over the crown of the stone graduating to black at the equator according to a cosine function. If you were sitting in the center of the table of a gemstone illuminated with a COS source, the "sky" you would see when looking overhead is represented at left. This COS Overview Graph presents COS brightness for the SRB over the same range of angles as the ISO overview graph presented above.
Let's now take a closer look at the behavior of ISO brightness over a narrower range of possible design angles with a more resolute ISO Detail Graph presenting 2501 data points corresponding to angles ranging from 30° to 50° pavilion mains in .5° increments and 20° to 50° in .5° increments. This data set was systematically recalculated for eight different refractive indexes ranging from RI = 1.43 (fluorite) to RI = 2.61 (sphalerite), each of which can be viewed by using the 'Cycle' buttons on the graph in like manner as the preceding overview graphs.
And of course here is a corresponding closer look at the behavior of COS brightness over a narrower range of possible design angles with a more resolute COS Detail Graph presenting the same 2501 data points corresponding to angles ranging from 30° to 50° pavilion mains in .5° increments and 20° to 50° in .5° increments. This data set was also systematically recalculated for eight different refractive indexes ranging from RI = 1.43 (fluorite) to RI = 2.61 (sphalerite), each of which can be viewed by using the 'Cycle' buttons on the graph in like manner as the preceding graphs.
Finally, let's take a look at a graphical presentation of the *differences* between the ISO and COS brightnesses values calculated by GemRay for these same ranges of angles and refractive indexes with a corresponding ISO-COS Difference Overview Graph and a corresponding ISO-COS Difference Detail Graph.
'Best' and 'Brightest' are not necessarily the same thing. If you want to see something really bright, just turn on your MAG light and stare directly into it for 30 seconds or so. Bright perhaps, but all in all, not very interesting! Gemstone design would be much simpler if brightness was the sole figure of merit for gemstones. Enhancement and display of color, dispersion, scintillation, tilt performance, and of course shape of the rough combined with a desire for high yield with precious materials all come into play. In the matter of brightness, gemstone design is somewhat a matter of wanting to have your cake and eat it too, in that other desiderata are often paid for with brightness. However, gemstones suffer if they are too dark overall due to too much brightness being traded off, or just plain leaked out. I like to think of brightness as what you've got left to have fun with after paying the bills.
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