# Change from cylindrical to rectangular coordinates and transformations

I have been given an exercise to convert switch coordinated from cylindrical to rectangular ones. This task is easy but one of them is a strange looking. The point in cylindrical coordinates is $(0,45,10)$. This corresponds to $r=0$. What is this point? Is it not the origin? But why then the angle and z-coordinate. I have one more question that is how to describe the geometric meaning of the following transformation in cylindrical coordinates: $(r,\theta,z)$ to $(-r,\theta-\pi/4,z)$

$-r$ makes the whole problem here.

In the cylindrical coordinate system $r$ is the distance from the z-axis, not the distance from the origin. The point $(0,45,10)$ has a $z$ coordinate of 10. The 45 is irrelevant.
For your second question I believe it makes no sense since $0 \le r \lt \infty$. If it was a typo and was supposed to be $r$, rather than $-r$, then the $\theta - \pi/4$ with constant $z$ would represent a clockwise rotation of $\pi/4$ about the z axis.