As is well-known, lines and circles are converted into lines and circles by circle inversion, or by any Möbius transformation for that matter. What bothers me is what happens in the Pappus's classical construction of a chain of circles inscribed into a region between two tangent circles (so-called Archimedes's arbelos).
One can apply a strategic circle inversion that converts this region into a strip between two parallel lines (see the first image on Mathhelp's Pappus Chain and below). Naturally, the inscribed circles are inverted into a vertical stack of circles inscribed into the strip, and their centers lie on its midline. The trouble is that the centers of the original Pappus chain circles lie on an ellipse (this is easy to show using its focal property, see e.g. Wikipedia's Pappus Chain). Since the inversion is involutive it would seem that it inverted a line into an ellipse?!
I am probably missing something very simple but I am not sure what. Is it that circle centers are not inverted into circle centers? Where does this ellipse go then?
Your guess is correct. Circle centers aren’t mapped onto circle centers.
For simplicity, consider inversion in the unit circle. Let $c$ be the radial distance of the center of a circle of radius $r$ to be inverted. The ray through this point intersects the circle at radial distances of $c\pm r$. These two points are inverted to $(c\pm r)^{-1}$. Their midpoint—the center of the circle’s image—is at ${c\over c^2-r^2}$, but the image of the circle’s center is at $\frac1 c$.
For the Pappus chain, the ellipse on which the centers lie has the centers of the outer and inner circles that generate the chain as its foci. The inversion circle is orthogonal to the first circle of the chain (the circle that completes the arbelos). Taking, for example, an outer diameter of $7$ and inner diameter of $4$, the central ellipse has semi-axis lengths $\frac{11}4$ and $\sqrt7$. The inversion circle has a radius of $2\sqrt7$, so the inversion formula is $\mathbf r'=28{\mathbf r \over \|\mathbf r\|^2}$. With some simplification help from Mathematica, the inversion of the central ellipse is $$x = {1232 \over 233+ 9\cos t}, y={448 \sqrt{7} \tan \frac t2 \over 233+ 9\cos t},$$ the red curve in the following illustration.
This curve asymptotically approaches the vertical centerline of the inverted circle stack, so as you get farther along the chain, the inverted circle centers do almost lie on a straight line, but none of them actually lie on it. With Mathematica’s help again, the parameter $t$ can be eliminated to get the implicit equation $$2\arctan{11y \over 4\sqrt7 x} = \arccos{1232-233x\over9x},$$ but I don’t find that particularly illuminating.