Semi-in finite 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”ofthestones.Workingwithstonesthataretransparentortranslucent,acuttermakesthemostofthecolorofastonethroughcarefulchoiceof the angles between facets, depending on the refractive index of the material.At the same time, gem producers have their own goal: to use asmuch of the volume of the rough stones as possible. By applying modern methods of semi-infinite optimization to this problem, we were ableto 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 shapeand facet design without technical support. In the recent work, researchers at the Fraunhofer Institute for Industrial Mathematics inKaiserslautern, to-gether with a consortium of mechanical engineering companies and a gem producer, developed a fully automatic process forindustrialgemproductionbasedonanoptimalbalanceofvolumeyieldandidealproportionsoftheresultinggemstones.Themachinefirstmapsthe surface of the rough stone by projecting narrow bands of light onto it. Using the scan data, the optimization software chooses one of manybasic shapes (e.g., emerald, trillion, or pear; see Figure 1) and a suitable arrangement of facets (e.g., brilliant, Ceylon, or Portuguese cut; seeFigure 2), and finds an embedding of the faceted gemstone in the rough stone such that the volume yield is maximized. Once the optimal solu -tion has been found, a grinding and polishing plan isautomatically generated and transferred to a CNCmachine. Finally, with no manual intervention, thefaceted gemstone is ground and polished to a preci -sion of 10 micrometers.Parameterization of a faceted gemstone beginswith the position and orientation of the cut stone within the rough stone; other parameters describe the shape of the faceted gemstone, includ -ing height, radius, and aspect ratio. The first person to investigate the influence of different shape parameters on the appearance of the brilliantcut 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 (col -lections of articles can be found, for example, in [9] and [10]).The volume-optimization problem has been studied much less extensively. The methodsdeveloped to date concentrate on diamond cutting and assume a fixed polyhedral geometry ofthe faceted gem. Few references are available (see [8] and [15] or, for commercial publicationson problems of this type, [4, 5], [13]).The available methods are appropriate for diamond-cutting problems (where one fixed facetarrangement, the so-called round brilliant cut, predominates) and cannot be applied to the cut -ting of colored gemstones because of subtle yet very important differences between these prob -lem classes. On the one hand, the lapidary proportions are much less restrictive than the bril -liant cut proportions. On the other hand, the assumption of a fixed facet arrangement is notappropriate for the lapidary cutting problem because of the large number (several hundreds) ofpossible geometries.The requirement that several hundreds of parameterized cut variations be taken into accountprecludesthe use of fixed polyhedralgeometriesand leads to a crucial question, one that is left unanswered by the optimization methods devel -oped 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?