In foliations on the open 3-ball by complete surfaces, Inaba and Masuda give an example for $\Bbb B^2$ at the end of page $2.$ I would like to understand this example first, before I keep progressing through the paper:
A complete closed uni-leaf foliation on $\Bbb B^2= \lbrace (x,y)\in \Bbb R^2 | x^2+y^2<1 \rbrace $ can easily be constructed as follows. Begin with the standard foliation on $\mathcal H$ on $\Bbb B^2$ defined by $dy=0.$ Then obviously all leaves of $\mathcal H$ are diffeomorphic to the real line and closed in $\Bbb B^2.$ Let $h$ be a diffeomorphism of $\Bbb B^2$ defined by $h(r,\theta)=\bigg (r,\theta + \tan \frac{\pi r^2}{2}\bigg),$ where $(r,\theta)$ are the polar coordinates. Then, $h$ sends any leaf $l$ of $\mathcal H$ to a complete curve $\mathcal H(l)$ in $\Bbb B^2,$ because each end of $h(l)$ spirals asymptotically on $\partial \Bbb B^2.$ Hence $h(\mathcal H)$ is a foliation we have desired. Note that, since $h$ is real analytic $(C^{\omega}),$ so is $h(\mathcal H).$
I understand the first sentence. The second sentence I understand to mean that this standard foliation, $\mathcal H$ is partitioning the ball by open line segments with zero slope. I understand that all leaves of $\mathcal H$ are diffeomorphic to the real line, but I'm not completely sure why they are closed in $\Bbb B^2.$ Lastly, in the fifth sentence, I'd like to get some intuition for this statement. When they say "each end of $h(l)$ spirals asymtotically on $\partial \Bbb B^2$" can I think of this as the two ends of the open line segments being transformed so that they spiral around the ball getting closer and closer to the boundary? Why does this create a complete curve $\mathcal H(l)$ in $\Bbb B^2?$