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how plot this region $S=[1,2] \times[0, \dfrac{\pi}{2}] \times[0, \dfrac{\pi}{2}]$ in spherical coordinates?

Under the commands command,

{{ParametricPlot3D[FromSphericalCoordinates[{{{1,[{0, Pi},p/(2 Pi)], p}, {[Placeholder]}}}] // Evaluate, {p, 0, 2 Pi}]},{}}

but without success

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    $\begingroup$ This is basically one octant of a spherical shell, no? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Apr 2, 2018 at 11:31
  • $\begingroup$ I believe so. But as I'm starting to use mathematica now, I'm curious to try to plot the regions, and I could not $\endgroup$
    – Ilovemath
    Commented Apr 2, 2018 at 11:59
  • $\begingroup$ If all you want is that spherical shell section, there are much easier methods. For the general case: maybe try ParametricRegion[]? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Apr 2, 2018 at 12:49

2 Answers 2

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For Mathematica 10 or later, the built-in function TransformedRegion sketches the image of a region under a 2D or 3D transformation. For simplicity, let me just write $r, t, p$ for the spherical coordinates $\rho, \theta, \phi$. We want to sketch the image of the region $$S=\{(r,t,p):1\le r\le 2,\ 0\le t\le\pi/2,\ 0\le p\le\pi/2\}$$ under the transformation $$x=r\sin p\cos t, z=r\sin p\sin t, y=r\cos p.$$ In Mathematica:

S = ImplicitRegion[1<=r<=2&&0 <= t <= Pi/2 && 0 <= p <= Pi/2, {r, t, p}];
f[r_, t_, p_] := {r Sin[p] Cos[t], r Sin[p] Sin[t], r Cos[p]};
R = TransformedRegion[S, f];
Region[R, Axes -> True, AxesLabel -> {x, y, z}]

enter image description here

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  • $\begingroup$ Usually it's $x=r\sin\theta\cos\phi$, $y=r\sin\theta\sin\phi$, $z=r\cos\theta$. $\endgroup$
    – Roman
    Commented Jun 4, 2022 at 19:36
  • $\begingroup$ Although I agree with @Roman, you code looks fine to me and certainly is original. The plot looks strange, however. $\endgroup$
    – bbgodfrey
    Commented Jun 4, 2022 at 19:47
  • $\begingroup$ The equivalent R = ParametricRegion[{{Cos[t] r Sin[p], r Sin[t] Sin[p], Cos[p] r}, 1 <= r <= 2 && 0 <= t <= Pi/2 && 0 <= p <= Pi/2}, {r, t, p}] is simpler. $\endgroup$
    – bbgodfrey
    Commented Jun 4, 2022 at 20:16
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Well, in this case it is very easy. No need to use ParametricPlot3D, as RegionPlot3D can do the job:

RegionPlot3D[1 < x^2 + y^2 + z^2 < 2^2, {x, 0, 2}, {y, 0, 2}, {z, 0, 2}, 
Mesh -> None]

enter image description here

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