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

  • Thread starter bers
  • Start date
  • #1
4
0

Main Question or Discussion Point

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 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?

Thanks
bers
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
955
"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.
 

Related Threads on Is a (hyper)sphere a (hyper)plane in spherical coordinates?

  • Last Post
Replies
4
Views
4K
Replies
1
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
3
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
4
Views
5K
  • Last Post
2
Replies
26
Views
2K
Replies
14
Views
9K
Top