What is the geometric meaning of the following mappings, that are written in cylindrical coordinates?
The mappings are: $$(r, \theta, z) \rightarrow(r, \theta , -z) \\ (r, \theta , z) \rightarrow (r, \theta +\pi , -z)$$
And what is the geometric meaning of the following mappings, that are written in spherical coordinates?
The mappings are: $$(\rho , \theta , \phi) \rightarrow (\rho , \theta +\pi , \phi) \\ (\rho , \theta , \phi) \rightarrow (\rho , \theta , \pi-\phi)$$
On
(a) the mapping $$(r, \theta, z) \to (r,\theta, -z)$$ is the reflection on the plane $z = 0$ or the $xy$ plane.
(b) the mapping $$(r, \theta, z) \to (r,\pi + \theta, -z)$$ is the reflection on the plane $z = 0$ or the $xy$ plane followed by half rotation about the $z$-axis.
(c) the mapping $$(\rho, \theta, \phi) \to (r,\pi+\theta, \phi)$$ is the half rotation about the $z$-axis
(d) the mapping $$(\rho, \theta, \phi) \to (r,\theta, \pi-\phi)$$ is the reflection on the $xy$ plane.
It looks to me like the question includes two pairs of mappings. Nonetheless:
Hint One concrete way to see this is to write each of the coordinate triples involved in rectangular coordinates using the usual transformation rules. (Alternately, one can meditate a bit on the geometric meaning of each of the coordinates $r, \theta, z, \rho, \phi$. In fact, I think it would be instructive to do this concretely as above, and then "meditatively".)