What is this logarithm doing in Wikipedia's formula for the unbounded Möbius surface?

Question

According to Wikipedia, the cartesian coordinates of a Möbius strip of width $1$, with its medial circle of radius $1$ in the $(x,y)$ plane, is given parametrically by $$x=(1+\tfrac12v\,\cos\tfrac12u)\cos u,\qquad y=(1+\tfrac12v\,\cos\tfrac12u)\sin u,\qquad z=\tfrac12v\sin\tfrac12 u,$$where $0\leqslant u<2\pi$ and $-1\leqslant v\leqslant1.$ That's fine; but then the article goes on to give the equation for the unbounded extension of the strip in cylindrical polar coordinates as $$\log r\sin\tfrac12\theta=z\,\cos\tfrac12\theta.$$Here I presume that the parameter $u$ in the cartesian formulae plays the same role as $\theta$ in the unbounded equation, and that $z$ has the same meaning in both cases. The problem I have is that, after eliminating $v$ (and ignoring its bounds), and using $r^2=x^2+y^2$, the parametric equations yield $$r=1+z\,\cot\tfrac12\theta\quad(0<\theta<2\pi),$$with $r$ indefinite and $z=0$ at $\theta=0$; here negative $r$ is allowed so that the whole generator line for each $\theta$ can be be obtained. I can't see, in particular, how the logarithm arises in the Wikipedia formula.