I've never done it, but a quick Google search revealed that Bezier curves are implemented in http://java.sun.com/j2se/1.5.0/docs/api/java/awt/geom/QuadCurve2D.Double.html
Then, you can getPathIterator() from this curve and with that, according to what documentation says, you get the "coordinates of shape boundaries", which, i suppose, is what you are looking for.