Complex Transformations of the Circle

This applet investigates some complex transformations of the circle, drawn on the complex plane, including sine , cosine, invertion, conjugation and the Joukowski transformation giving the "Joukowski Aerofoil".

Author and programmer: Ron Barrow

UK University Mathematics - Complex Analysis


Instructions below    See also:    Complex numbers 1    Complex nth roots

[Applet - Some Complex Transformations of the Circle]

How to Use this Applet

This applet is an Argand diagram, and helps you visualize the effects of some common complex transformations of the circle, or arcs of the circle.
You can change various parameters by clicking and dragging with the mouse.
The circle (drag the dot at its centre)
The circle's radius (drag the the slider on the left)
The arc length (drag the slider on the bottom)
The start angle, in radians, of the arc drawn (drag the slider on the right)

By combining these actions, you can put the image circle (or any arc of it) in different positions and sizes.
Different transformations can be chosen using the drop-down menu at the top of the screen. Clicking the "Reset" button redraws the full circle at 0, with radius = 1 (i.e. the circle |z| = 1).
You are now in a position to investigate this tricky but fascinating topic. Enjoy it!