Semi - infinite Optimization Meets Industry : A Deterministic Approach to Gemstone Cutting

For five centuries artisans have been cutting faceted gemstones, creating jewels that sparkle with internally reflected light, showing off the “fire” of the stones. Working with stones that are transparent or translucent, a cutter makes the most of the color of a stone through careful choice of the angles between facets, depending on the refractive index of the material. At the same time, gem producers have their own goal: to use as much of the volume of the rough stones as possible. By applying modern methods of semi-infinite optimization to this problem, we were able to improve the volume yield significantly while guaranteeing optimal optical properties of the faceted gemstones. Experienced cutters of such stones—rubies, sapphires, tourmalines, and others—have always done their work manually, de-ciding on shape and facet design without technical support. In the recent work, researchers at the Fraunhofer Institute for Industrial Mathematics in Kaiserslautern, to-gether with a consortium of mechanical engineering companies and a gem producer, developed a fully automatic process for industrial gem production based on an optimal balance of volume yield and ideal proportions of the resulting gemstones. The machine first maps the surface of the rough stone by projecting narrow bands of light onto it. Using the scan data, the optimization software chooses one of many basic shapes (e.g., emerald, trillion, or pear; see Figure 1) and a suitable arrangement of facets (e.g., brilliant, Ceylon, or Portuguese cut; see Figure 2), and finds an embedding of the faceted gemstone in the rough stone such that the volume yield is maximized. Once the optimal solution has been found, a grinding and polishing plan is automatically generated and transferred to a CNC machine. Finally, with no manual intervention, the faceted gemstone is ground and polished to a precision of 10 micrometers. Parameterization of a faceted gemstone begins with the position and orientation of the cut stone within the rough stone; other parameters describe the shape of the faceted gemstone, including height, radius, and aspect ratio. The first person to investigate the influence of different shape parameters on the appearance of the brilliant cut was Marcel Tolkowski, at the beginning of the 20th century [14]. The optimal proportions he calculated (called the Tolkowski Ideal Cut) have long served as a reference for the quality of a brilliant cut. Recently, numerous groups have studied the optics of faceted gemstones (collections of articles can be found, for example, in [9] and [10]). The volume-optimization problem has been studied much less extensively. The methods developed to date concentrate on diamond cutting and assume a fixed polyhedral geometry of the faceted gem. Few references are available (see [8] and [15] or, for commercial publications on problems of this type, [4, 5], [13]). The available methods are appropriate for diamond-cutting problems (where one fixed facet arrangement, the so-called round brilliant cut, predominates) and cannot be applied to the cutting of colored gemstones because of subtle yet very important differences between these problem classes. On the one hand, the lapidary proportions are much less restrictive than the brilliant cut proportions. On the other hand, the assumption of a fixed facet arrangement is not appropriate for the lapidary cutting problem because of the large number (several hundreds) of possible geometries. The requirement that several hundreds of parameterized cut variations be taken into account precludes the use of fixed polyhedral geometries and leads to a crucial question, one that is left unanswered by the optimization methods developed so far but that needs to be answered before we can tackle the lapidary cutting problem: If the polyhedral description of a faceted gemstone cannot be used during optimization, what, then, do we optimize?