  We provide an algebraic framework to compute smallest enclosing and smallest
circumscribing cylinders of simplices in Euclidean space $\E^n$. Explicitly,
the computation of a smallest enclosing cylinder in $\mathbb{E}^3$ is reduced
to the computation of a smallest circumscribing cylinder. We improve existing
polynomial formulations to compute the locally extreme circumscribing cylinders
in $\E^3$ and exhibit subclasses of simplices where the algebraic degrees can
be further reduced. Moreover, we generalize these efficient formulations to the
$n$-dimensional case and provide bounds on the number of local extrema. Using
elementary invariant theory, we prove structural results on the direction
vectors of any locally extreme circumscribing cylinder for regular simplices.
