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As a check for the torus segment center of mass calculation, there should be agreement in the limit where the radius of the torus goes to zero (with a non-zero correction otherwise).

# Center of mass for a circular wire segment.

As an additional check for the correctness of the result above, we should be able to compare with the center of mass of a circular wire segment, and get the same result in the limit .

For that we have

So we have

Observe that this is

which is consistent with the previous calculation for the solid torus when we let that solid diameter shrink to zero.

In particular, for of the torus, we have , and

We are a little bit up the imaginary axis as expected.

I’d initially somehow thought I’d been off by a factor of two compared to the result by The Virtuosi, without seeing a mistake in either. But that now appears not to be the case, and I just screwed up plugging in the numbers. Once again, I should go to my eight year old son when I have arithmetic problems, and restrict myself to just the calculus and algebra bits.