MHB Geometric Meaning of Cylindrical & Spherical Mappings

[mathmari]

Hey!

What is the geometric meaning of the following mappings, that are written in cylindrical coordinates?? (Wondering)

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?? (Wondering)

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

 

[Fallen Angel]

Hi mathmari,

For example, the first is a symmetry with respect to the plane $z=0$.

For the others, take a point in the space, $(x,y,z)$, where you know $x=rcos\theta, \ y=r\sin \theta, \ z=z$ (and the corresponding in spherical coordinates), calculate its image and try to relate both points.

 

[mathmari]

I see... Thank you very much! (Smile)