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

[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.

Spoiler

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

Thanks
bers

 

[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.