In the complex plane, a circle of radius 1 with center at 1 (orange circle) becomes a lemniscate under the transformation
. A circle of radius greater than 1 becomes a Cassini oval. The
axis becomes the pair of lines crossing the origin at an angle of
. The vertical line passing through
becomes a hyperbola. You can follow the points on a circle of variable radius, the
axis, or the vertical line
and their images under the transformation. Two arrows go from each point as the complex square root is a two-valued function.