Hi,
can I say that a sphere is a plane, because in spherical coordinates, I can simply express it as [itex]<(r, \theta, \varphi)^T, (1, 0, 0)^T> = R[/itex]? 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 ℝ^{n} with [itex]<x, u> = c[/itex], with c real and u any unit vector in ℝ^{n}. Clearly, R is real and [itex](r, \theta, \varphi) = (1, 0, 0)^T[/itex] is a unit vector in ℝ^{n}.

Spoiler

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

"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.