According to an analogy to quasi-Fuchsian groups, we investigate topological<br />
and combinatorial structures of Lyubich and Minsky&#039;s affine and hyperbolic<br />
3-laminations associated with the hyperbolic and parabolic quadratic maps.<br />
We begin by showing that hyperbolic rational maps in the same hyperbolic<br />
component have quasi-isometrically the same 3-laminations. This gives a good<br />
reason to regard the main cardioid of the Mandelbrot set as an analogue of the<br />
Bers slices in the quasi-Fuchsian space. Then we describe the topological and<br />
combinatorial changes of laminations associated with hyperbolic-to-parabolic<br />
degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For<br />
example, the differences between the structures of the quotient 3-laminations<br />
of Douady&#039;s rabbit, the Cauliflower, and $z \mapsto z^2$ are described.<br />
The descriptions employ a new method of tessellation inside the filled Julia<br />
set introduced in Part I that works like external rays outside the Julia set.