# The geometric meaning of certain mappings written in cylindrical or spherical coordinates

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: $$(\rho , \theta , \phi) \rightarrow (\rho , \theta +\pi , \phi) \\ (\rho , \theta , \phi) \rightarrow (\rho , \theta , \pi-\phi)$$

• That first looks like two mappings. Are you asking about each? Same for the second question. – Thomas Andrews Mar 6 '15 at 0:51
• Yes, I am asking for each... I edited my post, and added the plural... @ThomasAndrews – Mary Star Mar 6 '15 at 0:52

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".)

• Could you explain to me further what I have to do?? – Mary Star Mar 6 '15 at 0:55
• I'm off to teach, but if someone hasn't written a more explicit answer by the end of the day, I'll come back and expand my own to show how to use rectangular coordinates to understand one of the maps geometrically. – Travis Mar 6 '15 at 0:59

(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.

• Could you explain to me the cases $(b)$ and $(c)$ about the rotations?? How do we know that it is a rotation and that the rotation is about the $z-$axis?? – Mary Star Mar 6 '15 at 2:19
• @MaryStar, you can think of the $\theta$ coordinate as the amount of rotation about the $z$-axis needed to bring the $x$-axis to the projection of the point onto the $xy$-plane. – abel Mar 6 '15 at 2:22
• I got stuck right now... Could you explain it further to me?? – Mary Star Mar 6 '15 at 2:33
• you need to plot some points $(r \theta, 0)$ on the $xy$ plane. see what the coordinates $r$ and $\theta$ mean. then plot pairs of points $(r, \theta), (r, \pi + theta)$ and see how they are related. – abel Mar 6 '15 at 2:38
• Ok, I will do that... I have also an other question... At $(a)$ do we have to prove that= it is the reflection on the plane $z=0$ or canwe just say that?? @abel – Mary Star Mar 6 '15 at 2:58

Is the geometric figure for the first one the following @abel ?? EDIT1: EDIT2: • yes, very nice diagram. – abel Mar 6 '15 at 3:16
• For the second case how will be $\theta +\pi$ at the figure?? @abel – Mary Star Mar 6 '15 at 3:30
• the green half should be rotated $180^\circ$ about the $z$-axis. – abel Mar 6 '15 at 3:35
• Is it as I have drawn in the second picture?? @abel – Mary Star Mar 6 '15 at 3:40
• yes. that is correct. pictures are very nice. how do you make them and put them here. – abel Mar 6 '15 at 3:43