Visualizing construction of a certain function Consider a map $f$ defined on $\mathbb{R}^3$ with the following properties:\
1) $f$ fixes the poles $(0,0,\pm1)$.\
2) $f$ is symmetric in the plane $\{x=0\}$ and the plane $\{y=0\}$.\
3) $F$ is folded in the $z-$direction, that is, there is $z\in{(0,1)}$ such that 
$F_3(0,0,z)<F_3(0,0,-z),$ where $F=(F_1,F_2,F_3)$ denotes harmonic extension of $f$ given by Poisson integral.
This construction appeared in a paper where the author proves that harmonic extension of a homeomorphism may not be injective in $\mathbb{R}^3$. This construction is called "TENNIS BALL" construction by the author. In the construction it comes out that this function drags the northern hemisphere towards south pole and southern hemisphere towards north pole.
Can anyoe help me with how to visualize this construction. I am not able to see why this construction is called tennis ball construction.

You didn't provide a link to the article, so it's just a guess. There is a line on a tennis ball that divides its surface in two parts. Those parts can be considered as deformations of hemispheres. Two "tongs" are dragged to the North pole and the two others to the South pole.
