  Since its introduction to computational geometry by Alt and Godau in 1992,
the Fr\'echet distance has been a mainstay of algorithmic research on curve
similarity computations. The focus of the research has been on comparing
polygonal curves, with the notable exception of an algorithm for the decision
problem for planar piecewise smooth curves due to Rote (2007). We present an
algorithm for the decision problem for piecewise smooth curves that is both
conceptually simpler and naturally extends to the first algorithm for the
problem for piecewise smooth curves in $\mathbb{R}^d$. We assume that the
algorithm is given two continuous curves, each consisting of a sequence of $m$,
resp.\ $n$, smooth pieces, where each piece belongs to a sufficiently
well-behaved class of curves, such as the set of algebraic curves of bounded
degree. We introduce a decomposition of the free space diagram into a
controlled number of pieces that can be used to solve the decision problem
similarly to the polygonal case, in $O(mn)$ time, leading to a computation of
the Fr\'echet distance that runs in $O(mn\log(mn))$ time. Furthermore, we study
approximation algorithms for piecewise smooth curves that are also $c$-packed
for some fixed value $c$. We adapt the existing framework for
$(1+\epsilon)$-approximations and show that an approximate decision can be
computed in $O(cn/\epsilon)$ time for any $\epsilon > 0$.
