Change from cylindrical to rectangular coordinates and transformations

Question

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.

Answer 1

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.

Answer 2

I reckon your second question can be addressed by considering -r to be 180 degrees, pi radians, from the stated direction.

Consider (r, 3 pi /4, z) transformed by (-r, theta, z). The result could be considered (r, pi / 4, z). It's poor form though. The radius should be a zero, or positive quantity.