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)
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 $\phi=0$.