MHB How Do Cylindrical and Spherical Coordinates Represent the Same Point?

[evinda]

Hello!

• Suppose that a point has cylindrical coordinates $(r, \theta, z)$ where $r$ is not zero. Describe all other cylindrical coordinates of that point.
  
• Suppose $(R, \theta_1, \phi)$ and $(R, \theta_2, \phi)$ are two representations of the same point in spherical coordinates. Is it true that $\theta_1$ and $\theta_2$ must differ by an integer multiple of $2 \pi$?

If we have the cylindrical coordinates of a point how can we find other ones so that these represent the same point?

How can we find what relation the $r, \theta$ and $z$ of the two representations have to have?

 

[Evgeny.Makarov]

Suppose that a point has cylindrical coordinates $(r, \theta, z)$ where $r$ is not zero. Describe all other cylindrical coordinates of that point.

Adding $2\pi$ to $\theta$ does not change the point.

Suppose $(R, \theta_1, \phi)$ and $(R, \theta_2, \phi)$ are two representations of the same point in spherical coordinates. Is it true that $\theta_1$ and $\theta_2$ must differ by an integer multiple of $2 \pi$?

Please remind the meaning of $\theta$ and $\phi$ in spherical coordinates. There are slightly different conventions.

 

[evinda]

Adding $2\pi$ to $\theta$ does not change the point.

What does $\theta$ represent?

Also is then $(r, \theta+ 2k \pi, z)$ the only other cylindrical coordinate of the given point?

Please remind the meaning of $\theta$ and $\phi$ in spherical coordinates. There are slightly different conventions.

The spherical coordinates of $(x,y,z)$ are defined as follows:

$$x= \rho \sin{\phi} \cos{\theta} , \ \ \ y= \rho \sin{\phi} \sin{\theta} , \ \ \ z= \rho \cos{\phi}
\\ \text{ where } \rho \geq 0, 0 \leq \theta < 2 \pi, 0 \leq \phi \leq \pi $$

But what angle do $\theta$ and $\phi$ represent?

 

[Evgeny.Makarov]

Why don't you check out definitions of spherical and cylindrical coordinates in Wikipedia? Note, however, that angles can have different names there.