Here are some images, code and description showing some parametrisations :
http://commons.wikimedia.org/wiki/File:Lemniscates5.png
circle to boundary parametrisation
http://commons.wikimedia.org/wiki/File:Jung200.png
Newton method :
http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Component_by_Newton_method.png
circle to component ( or its part) :
http://commons.wikimedia.org/w/index.php?title=File:Mandelbrot_set_Components.jpg

