  Simple drawings are drawings of graphs in which the edges are Jordan arcs and
each pair of edges share at most one point (a proper crossing or a common
endpoint). We introduce a special kind of simple drawings that we call
generalized twisted drawings. A simple drawing is generalized twisted if there
is a point $O$ such that every ray emanating from $O$ crosses every edge of the
drawing at most once and there is a ray emanating from $O$ which crosses every
edge exactly once.
  Via this new class of simple drawings, we show that every simple drawing of
the complete graph with $n$ vertices contains $\Omega(n^{\frac{1}{2}})$
pairwise disjoint edges and a plane path of length $\Omega(\frac{\log n }{\log
\log n})$. Both results improve over previously known best lower bounds. On the
way we show several structural results about and properties of generalized
twisted drawings. We further present different characterizations of generalized
twisted drawings, which might be of independent interest.
