# More with fibers of functions

I posted earlier on a way of visualizing the fibers of certain maps from high dimensions to low dimensions.  Specifically, if the range can be embedded in the domain so that f is the identity of the image of the range, then we can draw the inverse image at each point.  I had some images of functions whose inverse image was a torus, but had trouble making these sorts of images for maps $f: \Omega \subset \mathbb{R}^3 \to \mathbb{R}^2$, so that the inverse image of a point is a line.  Well, no more!  Here are two images, one is the projection of a cube onto a square, and the other is somewhat more complicated, and is the string hyperboloid map.  See the previous post for more details on these specific maps, but I just thought these were nice images!

Fibers of the projection map from the cube to the square.

Fibers of the "twisted cylinder", which are again straight lines.

1. In the first case the fibers are equidistant; are they in the second? Two sets A, B are equidistant if for every $a\in A$ there is $b\in B$ such that $d(a,b)=dist(A,B)$; and vice versa.