Bezel length for oval stones is quite an interesting problem. There

are two approaches, - one is working approximation with very little

mathematics, but steps are not easy to describe. Visual instruction

is required. So I am going to recommend my DVD “Coronet Cluster” for

this.

http://www.ganoksin.com/gnkurl/fp

Another approach is mathematical, but there are number of

complications. First, oval and ellipse are not the same thing.

Mathematically we can only deal with ellipses. So if stone is not

true ellipse, one has to determine the difference and adjust

calculations later on.

From this point on, the discussion only refers to ellipses. Let’s

take a to mean half of major axis, and b - half of minor axis. Also

assume that values were adjusted for metal thickness. If a is less

the 3b the formula is 2(Pi)*sqrt (( a^2 + b^2)/2) (Pi is 3.14). The

result is within 5% of true value.

If a is greater than 3b, more complex formula is required. Pi[ 3(a +

b) - sqrt ((3a + b) (a + 3b)) ]

There are other methods yielding exact values, but methods are quite

complex and involve infinite series. There are no reasons to get into

them here.

As anyone can see, a simple oval contains a lot of mysteries, so in

actual practice it is better to use working approximation than exact

calculations.

Practical: given ellipse of 30 X 22 - a is 15 and b is 11. Since a

is less than 3b, first formula is used

2(3.14) * sqrt (( 15^2 + 11^2) / 2 ) = 6.28 * sqrt ( 225 + 121 ) /

2) = = 6.28 * sqrt (173) = 6.28 * 13.15 = 82.6mm

There are no accounting for thickness of metal in the above example.

Also, the true value is from (82.6 - 0.413) to (82.6+0.413). Whether

one chooses higher or lower value would depend a lot on technique.

As a parting word, let me add that requirement for precision is

increasing as thickness of metal is decreasing. For thicknesses of

0.5mm and bellow, only second formula is recommended, and I have

been called to make bezels in 22k out of 0.25mm. Such bezels are used

to set important emeralds and other fragile stones. For these, exact

methods must be used. I did not describe them due to their

complexity, but on can find them in mathematical literature. Another

option is to use specialized software like Wolfram Mathematica.

Leonid Surpin

www.studioarete.com