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

  1. Sep 6, 2013 #1
    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 ℝn with [itex]<x, u> = c[/itex], with c real and u any unit vector in ℝn. Clearly, R is real and [itex](r, \theta, \varphi) = (1, 0, 0)^T[/itex] 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?

  2. jcsd
  3. Sep 6, 2013 #2


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    "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.
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