  We investigate the problem of constructing planar drawings with few bends for
two related problems, the partially embedded graph problem---to extend a
straight-line planar drawing of a subgraph to a planar drawing of the whole
graph---and the simultaneous planarity problem---to find planar drawings of two
graphs that coincide on shared vertices and edges. In both cases we show that
if the required planar drawings exist, then there are planar drawings with a
linear number of bends per edge and, in the case of simultaneous planarity, a
constant number of crossings between every pair of edges. Our proofs provide
efficient algorithms if the combinatorial embedding of the drawing is given.
Our result on partially embedded graph drawing generalizes a classic result by
Pach and Wenger which shows that any planar graph can be drawn with a linear
number of bends per edge if the location of each vertex is fixed.
