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I'm currently enrolled in a course dealing with Isogeometric Analaysis (Splines, NURBS, ...) and in one of our exercises, we are asked to numerically evaluate an integral using methods we have implemented. This works fine for me, but I actually don't know what I'm computing. Maybe someone could give an interpretation of the formula for me.

\begin{align} I &= \frac{1}{4 \pi} \int_{C1} \int_{C2} \frac{(dC1 \times dC2)\cdot (C2-C1)}{||C2-C1||^3} \\ &=\frac{1}{4 \pi} \int_0^1 \int_0^1 \frac{(C1'(\xi) \times (C2'(\eta))\cdot (C2(\eta)-C1(\xi))}{||C2(\eta)-C1(\xi)||^3} d\xi d\eta \end{align} The curves $C1$ and $C2$ are both unit circles. One inside the $xy$ plane centered at $(0,0,0)$ and the other in the $xz$ plane, centered at $(1,0,0)$. Here is a picture: C1 and C2

Maybe someone could give me a hint how to interpret the integral (which hopefully has a value that converges towards 1).

Thank you!

Thomas
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  • The short answer -- I don't know. I assume you have parametric equations for the two circles. I'd suggest you draw the vectors $C2(\eta)-C1(\xi)$ and $C1'(\xi) \times C2'(\eta)$ for a few different values of $\xi$ and $\eta$. Another (more important) tip: quite a few of the claimed benefits of isogeometric analysis are a bit far-fetched (to put it politely). Caveat emptor. – bubba Dec 02 '13 at 07:49
  • You are computing the magnetic force between two current-carrying wires. See Ampère's Force Law – David H Dec 02 '13 at 08:17

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A partial answer: the numerator is a scalar triple product. It represents the volume of the parallelipiped with sides $C1'(\xi)$, $C2'(\eta)$ and $C2(η)−C1(ξ)$. Beyond that, I'm as puzzled as you are, I'm afraid.

bubba
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