what is the polar coordinate parametrisation of a plane?

Question

The equation of a plane I am working with is $ \sigma= a+ub+vc $ with $b$ perpendicular to $c$. I am unsure about how to parametrize this line using the generalized polar coordinates.

Answer 1

I'm assuming your $(a,b,c)$ corresponds to the standard Euclidean triplet $(x,y,z).$ Also, by "generalized polar coordinates", I'm assuming you mean spherical coordinates $(r,\theta,\phi)$. Let me know if any of these assumptions are wrong.

$(x,y,z)$ transforms into $(r,\theta,\phi)$ by the relations (found here)

$$x=r\cos\theta \sin\phi$$ $$y=r\sin\theta\sin\phi$$ $$z=r\cos\phi,$$ such that a plane $$ax+by+cz+d=0$$ becomes $$a\cos\theta \sin\phi+b\sin\theta\sin\phi+c\cos\phi+\frac{d}{r}=0.$$