Square Inversion of a Family of Lines

Square inversion is a little-known analogue of circle inversion. This paper works out what happens to families of lines under inversion with respect to the unit square, where the mapping is defined piecewise depending on the region. Horizontal lines map to parabolic arcs joined with horizontal segments, diagonal lines map to closed contours made of two parabolic pieces. The results are elementary but the shapes are unexpected and visually striking.