Hyperbolas, Lemniscates, and Cassini Ovals Generated by w = Sqrt(z)

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.