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I have a circle defined by a set of $x, y$ and $z$ coordinates. The circle exists across $3$ planes. I would know how to parameterize a $2$-d circle (say in just the $x$-$y$ plane) into polar coordinates, but not in $3$-d. How do I do this? I visualized the set of $x, y$ and $z$ coordinates in Paraview so I could see the orientation of the circle; this can be seen below.

I need to paramterize this circle because I need to define the $x, y$ and $z$
points I was given as angles around the circumference of that circle.

(Note the circle of interest is the translucent white object in the centre of the cube domain)

Rick
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bjuksel
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1 Answers1

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Find from your data the center $O$ of the circle and then two points $A$, $B$ on the circle such that $OA\perp OB$. The parametric equation of the circle is then: $$ (x,y,z)=O+(A-O)\cos\phi+(B-O)\sin\phi. $$ Inserting here the coordinates of a given point of the circle you can solve for $\cos\phi$ and $\sin\phi$. Of course point $A$ corresponds to $\phi=0$.

Intelligenti pauca
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