by Bob Keller, Old Pueblo Lapidary Club
A mathematical procedure known as tangent ratio conversion or translation is frequently employed by faceters to optically optimize or tweek a gemstone design’s pavilion and/or crown angles to work with materials of differing refractive indexes (critical angles) and degrees of color saturation. If you feel a need to bone up, Refractive Index and Critical Angle is an online primer explaining these fundamentals as pertaining to gemstone materials and design.
Another common application for tangent ratio conversion is modifying the elevation angles of a design’s crown to increase its height when cutting a stone from a precious material. If the rough is proportioned to allow it, increasing the height of the crown also increases its volume, and hence the carat weight of the finished stone. Tangent ratio conversion can also be employed to lower the crown height as the rough dictates. Sometimes a faceter will discover after cutting a stone’s pavilion there is just not quite enough material left after he’s transferred it to cut the crown at the planned upon design angles. Other reasons to lower the crown are recovering from a cutting error or the discovery of a flaw in the stone at a late stage, or possibly to repair a worn or damaged stone. Problems of this nature can sometimes be eliminated or at least minimized by recutting the crown at shallower angles.
If you should want or need to change the design angles given for the pavilion or crown of a gemstone design you have an application for tangent ratio conversion. The concept of gemstone planviews underlies the application and utility of tangent ratio conversion. Tangent ratio conversion is a procedure or algorithm for increasing or decreasing the elevation angles on the pavilion and/or crown on a design without altering the planviews.
If you are familiar with the Cartesian coordinate system, basically what’s going on when you do a tangent ratio conversion is you are scaling the z ordinates of the facets (by changing their elevation angles), making them deeper or shallower, while leaving their x and y ordinates unaffected. All of the angles on a pavilion or crown must be scaled proportionately in order to preserve its original planview. It can be mathematically demonstrated that the ratio of the tangents (a trigonometric function) between each of the original angles and the corresponding converted angles is constant when the planview is preserved. That elegantly simple but important relationship constitutes the algorithmic basis for performing tangent ratio conversions.
You might want to change the angles on both the crown and the pavilion when you are converting the angles on a design to cut it in materials of different refractive indexes and degrees of color saturation. Or you might not… If you do want to lower or raise the crown angles, that normally gets done as a separate and independent conversion from the pavilion angles. Algorithm wise, tangent ratio conversion is blind as to whether the angles you input are on the pavilion or the crown (or something else altogether). So in the circumstance where you wanted to convert both a crown and pavilion using the same tangent ratio you there is no mathematical reason you can’t calculate them together. However, in practice this circumstance is usually not the case.
The process of doing a tangent ratio conversion is one of finding the tangents for the original reference angle and the new reference angle, and then calculating the ratio of these tangents. Because this ratio is constant for all of the other angles when the planview is maintained, each of the other new angles can be found given their original values and the tangent ratio constant. If you are a Windows user, there is a handy calculator utility bundled with Windows with a scientific mode that can be used to perform tangent ratio conversions. If you have Windows installed on your system in C:Windows you may be able to use this link to run it, or else open your Start/Programs/Accessories folder and click on the “Calculator”, or do a Start/Run and enter “calc.exe”., or just “calc”. Below is a (nonfunctional) screen graphic of the Windows calculator utility. I find this virtual calculator so generally handy I keep a shortcut to it in my Windows desktop’s system tray on the task bar.
When you first run the Windows calculator you may need to set its “View” pulldown menu options to select the scientific mode. To perform tangent ratio conversions with this calculator as discussed below, make sure the mode control widgets are set to “Dec” (Decimal display) and “Degrees” as illustrated in this screen shot graphic. The “Inv” (Inverse) mode control widget should be initially unchecked until that mode is needed in the course of the calculations. The “Hyp” (Hyperbolic) mode control widget should be left unchecked throughout the conversion procedure.
Let’s work through an example tangent ratio conversion with the Windows calculator (or your own nonvirtual scientific calculator) to illustrate the tangent ratio process. Suppose you find a design to your liking in an old Lapidary Journal that you’d like to cut in quartz. It has five sets or tiers of facets on the pavilion, P1=39°, P2=42.3° P3=43.9°, P4=68° and G=90°. You know that quartz has an RI=1.54 with a corresponding critical angle of 40.49° so you want to convert all of the other angles as required to preserve the planview of the pavilion when P1 is changed from 39° to 42° to make the shallowest pavilion angle a degree and a half above the critical angle for quartz.
You can do a preliminary check on the configuration of the calculator by entering the value “45” via its virtual keypad and then clicking the [tan] button. Assuming the mode widgets are set as described, the calculator will return “1”, which is the tangent of 45°.
To solve for the other angles, the first step is to find the tangents of the original and desired angles for P1, which serves as the reference facet in this example.
(1) Find the tangent of the original angle for the reference facet. For this example enter 39 on the Windows calculator via its virtual keypad and then click the [tan] button. The calculator will return 0.809784033195007148036991374235771. Temporarily save this value in the calculator’s memory by clicking on the [MS] (memory store) button.
(2) Find the tangent of the desired angle for the reference facet. For this example enter 42 and then click the [tan] button. The calculator will return 0.900404044297839945120477203885372.
(3) Find the ratio of the tangents of the desired and original angles for the reference facet (tangent of the desired angle divided by the tangent of the original angle). Click the [/] (divide) button and then click the [MR] (memory recall) button. The calculator will return 1.11190639403606300833966851251044, which is theratio of the tangents aka the tangent ratio (0.900404044297839945120477203885372 / 0.809784033195007148036991374235771) expressed in decimal form. This value is a constant used to proportionately scale the other original angles in the following steps. Save the tangent ratio in the calculator’s memory by clicking on the [MS] (memory store) button.
Now that you’ve found and stored the operant tangent ratio, you’re ready to convert the original angles given for the balance of facets. To do this:
(4a) Find the tangent of original angle.
(4b) Multiply the tangent of the original angle by the tangent ratio stored in step (3).
(4c) Find the angle having a tangent equal to the product of the tangent of the original angle multiplied by the tangent ratio. This is the converted angle.
To do this with the Windows calculator for P2 in our example which is given as 42.3° on the original design, enter 42.3 via its virtual keypad and then click the [tan] button. The calculator will return 0.909929988177737465792254001732893. Click the [*] (multiply) button followed by the [MR] button. The calculator will return 1.01175697197998550941993171742263. Click on the “Inv” mode widget so that the box beside it is enabled (checked). Click the [tan] button. The calculator will return 45.3348402490362345016173397566199°, which is the converted angle for P2.
Repeat steps (4a) through (4c) for the other original facet angles that you need to convert. In our example P3 converts from 43.9° to 46.9371057498344816960634311809375° and P4 converts from 68° to 70.0307015406712909436051079680664°.
Of course you can round off the values of the converted angles to the nearest tenth or hundredth of a degree for the purpose of recording and cutting them, but it is best to carry through the intermediate values used in the course of calculating the converted angles as high precision numbers. Fortunately calculators and computers make grinding out these kind of calculations a relatively painless task.
The above procedure is summarized by this formula:
angle x_{new} = tan^{1}{tan(angle x_{original}) * [(tan(angle ref_{new}) / tan(angle ref_{original})]}
Work this equation “from the inside parentheses out” on the right side of the equality to convert the original angles.
You may read or get “practical” advice to the effect that if you want to convert a design from say 39° to 42° pavilion mains as in our example, just add the difference (3°) to all of the other pavilion facets and you will be “close enough”. This is more or less true. It is more true when the spread of angles is relatively low, as in an SRB pavilion where the mains and break angles are typically cut within several degrees of each other. In such cases the angular difference between the results of a tangent ratio conversion and adding a constant difference like 3° is so small that it is beyond the resolution of the vernier or indicator on our faceting machines.
However, as you can see in its graph below the tangent function is not linear. As the spread between the lowest and highest pavilion or crown angles increases as is typical with more complex designs, the practice of adding a constant difference to the angles breaks down and the results diverge from a tangent ratio conversion. This can cause noticeable deviations in the planview and may also have ramifications for meet points. The table presented adjacent to the graph illustrates a case of this divergence.
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Here’s several properties worthy of general note about the tangent function applied to angles in the range of 0° to 90°: The tangent of 0° is 0. As the angle increases between 0° and 90° the value of its tangent increases, but not in a linear manner. The tangent of angles between 0° and 45° is less than one and the tangent of angles greater than 45° and less than 90° is greater than 1. The tangent of 45° is, you guessed it, 1. Note that as the angle approaches 90 degrees the value of its tangent increases without bound. tan(89°)~=57.29, tan(89.5°)~=114.59, tan(89.99°)~=5,729.58, tan(89.999°)~=57,295.78… Thus the tangent of 90° is mathematically undefined. (Try finding the tangent of 90° with the Windows calculator.)
As far as the math is concerned, you can employ any particular angle (besides 0° or 90°) from the original set of angles given in the design instructions for a pavilion or crown as your reference angle. However, most of the time you will probably be most concerned with pavilion or crown mains facet angles and employ those for the reference angles when you are doing tangent ratio conversions.
You might want to change the angles on the crown as well as the pavilion when you are converting the angles on a design to cut it in materials of different refractive indexes. Or you might not… If you do want to lower or raise the crown angles, that gets done as a separate and independent conversion from changes you might also be making to the pavilion angles. Algorithm wise, the conversion process is blind as to whether the angles you input are on the pavilion or crown (or something else altogether). So in the circumstance where you wanted to convert both a crown and pavilion using the same tangent ratio you could do this in a single pass with the converter. However, in practice this circumstance is usually not the case.
The good news is once you’ve worked through a couple of tangent ratio conversions on the Windows calculator or a hand held you will probably come to appreciate it as a straightforward process, but one that can get a little tedious (and error prone) if the design has a lot of different facet angles that need to be calculated for the conversion. The really good news is that Internet servers can be custom programmed to perform tangent ratio conversions too…
Summary of Some Key Fundamental Concepts
 Tangent ratio conversion is a mathematical procedure for changing the elevation angles of the facets on the pavilion or crown of a gemstone design without altering their respective plan views. Tangent ratio conversion can be used to both lower or raise angles as desired.
 Tangent ratio conversions are usually independently applied to a design’s pavilion or crown. An original set of angles for the design must be known to perform a tangent ratio conversion. This information is typically obtained from published cutting instructions for the design.
 Crown or pavilion main facets are typically used as the reference facets to determine the tangent ratio constant for the conversion, but other facets can be used. The original angle for the reference facet is a “given” – specified in the design’s diagram or cutting instructions. The new angle for the reference facet is selected and specified at the faceter’s discretion. Once a new angle for the reference facet is specified to replace the original angle, the new angles for the balance of its facets are dictated and determined by the tangent ratio.
 This formula can be used to perform tangent ratio conversions:
angle x_{new} = tan^{1}{tan(angle x_{original}) * [(tan(angle ref_{new}) / tan(angle ref_{original})]}  Original facet angles specified at 0° or 90° (tables and girdles) remain at 0° or 90° and do not need to be converted using the tangent ratio process.
 In some cases the difference in calculated values is sufficiently small the same practical results are obtained by adding or subtracting the difference between the original and new angles for the reference facet to convert the balance of the facet angles on a pavilion or crown design. However, as the range or spread of angles increases as is typical of more complex designs, this quick and dirty technique breaks down as a “close enough” approximation due to the nonlinearity of the mathematical relationship between facet elevation angles and their corresponding planviews. The safest and most conservative approach is always do conversions using the tangent ratio method and then round off the results according to the resolution of your machine’s protractor and vernier. You can do a tangent ratio conversion much faster than you can recut a stone, especially so when you use a custom programmed calculator like the Online Tangent Ratio Converter. Enjoy!