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Here are some images, code and description showing some parametrisations :

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