  Due to its many applications, \emph{curve simplification} is a long-studied
problem in computational geometry and adjacent disciplines, such as graphics,
geographical information science, etc. Given a polygonal curve $P$ with $n$
vertices, the goal is to find another polygonal curve $P'$ with a smaller
number of vertices such that $P'$ is sufficiently similar to $P$. Quality
guarantees of a simplification are usually given in a \emph{local} sense,
bounding the distance between a shortcut and its corresponding section of the
curve. In this work, we aim to provide a systematic overview of curve
simplification problems under \emph{global} distance measures that bound the
distance between $P$ and $P'$. We consider six different curve distance
measures: three variants of the \emph{Hausdorff} distance and three variants of
the \emph{Fr\'echet} distance. And we study different restrictions on the
choice of vertices for $P'$. We provide polynomial-time algorithms for some
variants of the global curve simplification problem and show NP-hardness for
other variants. Through this systematic study we observe, for the first time,
some surprising patterns, and suggest directions for future research in this
important area.
