The detailed model description

To study diamond cut, a group of researchers from Moscow State University have developed specialized software called "Brill". The software employs a mathematical model that consists of the following parts:

1. Illumination sources.

The illumination model involves a set of up to 100 virtual Lambertian sources of white light. Each of the sources has its own position and angular size. To determine the fire of a diamond, 60 randomly positioned light sources are used. The angular size of these sources is small (2-10°) - to model "Chandelier" light source. To model diffuse illumination, from 2 to 5 separate sources are used, each of these having an angular size of 20-40° - light source "Office" - type. The sources are located at a semi-sphere, from which the contour of the observer is excluded. The positions of the sources are selected so that they do not illuminate the pavilion of the diamond when the latter is inclined by an angle of no more than 30°. The software is capable of modeling not only Lambertian sources, but also light sources with non-uniform directional patterns.

2. Diamond.

The diamond under study is chosen to have a standard round brilliant cut with a pointed culet and faceted girdle, with ideal symmetry and ideally polished flat facets. The computer builds up a complete 3D parametric model of diamond cut, the shape of which is determined by the following parameters:

The software allows the user to model other common fancy shapes as well. The calculations use the following fixed parameters: h=1% (at the narrow part) and q=0.82. The pattern of the crown facets is maintained so that the table and the star facets form two squares when the diamond is viewed in face-up position.

3. Ray tracing.

The calculations deal with multiple ray tracing with taking into account Fresnel reflections at facets. Partial polarization of light (as it is refracted and reflected in the diamond) is not considered. The maximum number of the multiple reflections of a single ray in round brilliant cut considered as 20. If the intensity of a ray decreases 1000 times, it is no longer traced. To simulate the facet polishing imperfection, the model includes separate light scattering coefficient for all the facets of the modeled gem. The value of the coefficient can be varied from zero to unity.

4. Observer.

The software models the angular size of the observer's pupil (the linear size of the pupil divided by the distance between the pupil and the diamond). It is taken into account that the observer's head screens some of the illumination sources. The angular size of the head can be varied from 5 to 20°. The screening of illumination sources by the observer's body is kept in mind when arranging the sources. Color computations match the standard colorimetric observer CIE 1931.

5. Absorption.

When modeling a colorless diamond, light absorption inside the crystal is neglected. For colored diamonds, the absorption is determined by a stepwise-approximated absorption spectrum of the real diamond. Color computations needed for determining the fire of the gem and the color of a separate ray match the ordinates of the composition curves of the standard colorimetric CIE 1931.

6. Coefficients.

To quantitatively analyze the modeling results, the following coefficients are introduced for estimating the light return, fire, cut quality, and cut efficiency of a diamond:

6.1. The light return coefficient LR is defined as the white fraction of the incident light returned by the diamond to the observer's eye. It is not sufficient to study any static orientation of the diamond. Therefore, the light return coefficient is averaged over a range of gem orientations. The coefficient LR_30 is calculated by averaging the LR coefficient over all the orientations when the axis of the gem inclines by no more than 30 degrees. The coefficient nLR_30 is equal to the LR_30 coefficient normalized to the light return coefficient of a diamond that matches the criteria of the Tolkowsky cut.

6.2. The fire coefficient F is defined as the mathematical color dispersion (a quantity characterizing the deviation of observed colors from white) of highlights visible on the surface of the diamond, weighed by the square root of the colored highlights area. This definition is valid only for colorless diamonds, while colored diamonds require a more complex formula. The coefficient F_30 is the mean fire coefficient F for all the directions making with the diamond axis an angle from 0 to 30 degrees. Since the perception of the fire strongly depends on the illumination conditions, an additional coefficient, MF_30, is useful to introduce. This coefficient is equal to the geometric mean of four F_30 coefficients for four different illumination conditions (two different "Office" and two different "Chandelier" illumination types). The MF_30 coefficient treats the fire as not much dependent on the illumination features. Normalizing the MF_30 coefficient to its value for a diamond that matches the criteria of the Tolkowsky cut yields the coefficient nMF_30.

6.3. The Q-factor of a cut diamond takes into account both its light return and fire. This factor is the measure of the cut quality. It is defined as the product of nLR_30 by nMF_30. In those parameter zones where the Q-factor is low, the cut should be considered as poor: it yields either small light return coefficient or weak fire. In those zones where Q is high, the cut should be further studied from the point of view of the pattern of the gem and its appearance.

6.4. If an increase in the cut quality reduces the weight of the cut diamond, the economic efficiency of such a cutting process may appear to be far too low. This is because the price of a diamond depends on both its cut quality and weight, and the latter dependence is more evident. As an experiment, the authors suggest to estimate the commercial efficiency of the cutting process with the efficiency function EFF=Q2M that takes into account both the cut quality Q and the diamond weight M.

7. Tolkowsky diamond.

In the "Diamond design" paper by M.Tolkowsky one may find the following parameters of calculated diamond cut: the crown angle - 34.5°, the pavilion angle - 40.75°, the table size - 53%. However, the author has not indicated such important parameters as the girdle thickness, the star length, the lower girdle length, and culet size (размер калетты). It is impossible to unambiguously define the cut shape without knowing the exact values of these parameters. In the present work, the "Tolkowsky cut parameters" are understood as the two angles and the table size indicated by him. The other cut parameters are the following:

• the girdle thickness -1% (at the narrow part), 2.7% (at the thick part);
• the star length is adjusted so that the table and the star facets make two squares if the diamond is viewed in the face-up position, along the main axis (for the Tolkowsky cut parameters - 0.37);
• the lower girdle length - 0.82;
• the pointed culet.