# Optimal Utilization of Colored Gemstones

The production of colored gemstones consists of several steps starting with the mining of the raw material and ending with the sale of the precious stone. Up to some years ago, all steps in between had to be done manually.Our software allows to evaluate the cuts and the clarity of the gemstones before the stones are cut.

Stones generated by this way are more beautiful and have a larger revenue. At the same time this helps to bring back the production, which moved to low-wage countries.

#### From a mathematical point of view the following challenges exist:

1. Optimal Dissection of the Raw Material
2. Embedding of a Gemstone in an Unmachined Part
3. Optimal Faceting of Colored Gemstones

## Optimal Dissection of the Raw Material

The basic raw material for the production of colored gemstone is mined. However, the obtained material cannot be processed directly to precious stones. Rather, defects like cracks and inclusions must be removed. This usually happens by dissecting the raw material into several parts in such a way, that the defects are as close to the boundary of the parts as possible. The unmachined parts are then the basis for the actual cutting. The dissection is currently done by manual sawing.

On the one hand we want to calculate partitionings in such a way, that as few as possible material is to be removed from the stone when further processing. On the other hand we like to confer existing water jet technology to the dissection of gemstone raw material and to automate the resulting process.

From the mechanical point of view the challenges are to derive physical models of the water jet and the handling of the raw material. From the mathematical point of view the sticking point is the modeling of the so called »developable surfaces«, which describe the cut resulting from the water jet, and the development of appropriate methods for solving the arising optimization problems.

## Embedding of a Gemstone in an Unmachined Part

The challenge is the physical modeling of the cutting and polishing process, which is a precondition for an appropriate controlling of the robotics. Also the mechanical implementation are to be derived in elaborate experiments. With this we can reproduce the abilities of the manual grinder with machines and exceed their precision in a clearly observable way.

When crafting manually the part is preformed, in a second step, the individual facets are cut and, finally, polished. In this project we develop the first automation of this production process. Here, high-precision cutting robots generate the facets based on optimal facetings and polish them afterwards. A preceding preforming is not necessary anymore.

## Optimal Faceting of Colored Gemstones

To achieve an automation of the gemstone production, we have to be able to deal with many different facetings. There is a multitude of basic shapes (pear, emerald, round, rect, trillion, …) and several styles of cuts (portuguese, ceylon, …).

In this project we found parametrizations of these facetings and derived algorithms, which yield appropriate values for these parameters. Hereby, appropriate means two things: First, as few as possible material of the unmachined part should be lost. Second, a beholder should regard the resulting precious stone as beautiful.

An essential challenge is to put the beauty into a mathematical form. The shape, the regularity of the faceting and the brilliance, i. e. the way the light is reflected by the stone, are the main factors to be considered.

In close cooperation with our partner Paul Wild OHG we succeeded in representing the relevant knowledge of the grinders within our software. From this we get a computer application which is not only able to calculate solutions with maximal volume yield, but also allows the user to select those parameters which depend on taste and fashion like the choice of the shape and the cutting style.

## Video: Optimal utilization of gemstones

(only available in German)