I am trying to make sense of the scale factors in cylindrical coordinates and their geometrical meaning. To start with something simpler, begin with Cartesian coordinates: $$h_x=h_y=h_z=1$$ One can say that the meaning of this is: the scale of coordinate $x$ is the same as the scales of coordinates $y$ and $z$. That is, all three coordinates have equal unit length. Does this make sense?
Now, cylindrical coordinates - $$h_r=h_z=1$$ Then, $r$ and $z$ have the same scale and unit of length. Last, $$h_{\theta}=r$$ What would be the geometric interpretation of this? Why $r$? What would be a good "rule of thumb" to explain this last scale factor.