  A theoretical development is carried to establish fundamental results about
rank-initial embeddings and automorphisms of countable non-standard models of
set theory, with a keen eye for their sets of fixed points. These results are
then combined into a "geometric technique" used to prove several results about
countable non-standard models of set theory. In particular, back-and-forth
constructions are carried out to establish various generalizations and
refinements of Friedman's theorem on the existence of rank-initial embeddings
between countable non-standard models of the fragment $\mathrm{KP}^\mathcal{P}$
+ $\Sigma_1^\mathcal{P}$-Separation of $\mathrm{ZF}$; and Gaifman's technique
of iterated ultrapowers is employed to show that any countable model of
$\mathrm{GBC}$ + "$\mathrm{Ord}$ is weakly compact" can be elementarily
rank-end-extended to models with well-behaved automorphisms whose sets of fixed
points equal the original model. These theoretical developments are then
utilized to prove various results relating self-embeddings, automorphisms,
their sets of fixed points, strong rank-cuts, and set theories of different
strengths. Two examples: The notion of "strong rank-cut" is characterized (i)
in terms of the theory $\mathrm{GBC}$ + "$\mathrm{Ord}$ is weakly compact", and
(ii) in terms of fixed-point sets of self-embeddings.
