# Is a (hyper)sphere a (hyper)plane in spherical coordinates?

1. Sep 6, 2013

### bers

Hi,
can I say that a sphere is a plane, because in spherical coordinates, I can simply express it as $<(r, \theta, \varphi)^T, (1, 0, 0)^T> = R$? It does sound too easy to me. I'm asking because I'm thinking about whether it is valid to generalize results from the John-Radon transform (over k-planes in an n-dimensional space) to curvilinear coordinates. In the book I'm reading (Analytic tomography), a k-plane is the set of points x in ℝn with $<x, u> = c$, with c real and u any unit vector in ℝn. Clearly, R is real and $(r, \theta, \varphi) = (1, 0, 0)^T$ is a unit vector in ℝn.

I believe the above is invalid because the dot product is expressed differently in spherical coordinates - is that already the answer?

Thanks
bers

2. Sep 6, 2013

### HallsofIvy

"Sphere" and "plane" are geometric objects and so completely independent of the coordinate system. They are not the same thing no matter what coordinate system you use.