We study the classic graph drawing problem of drawing a planar graph using
straight-line edges with a prescribed convex polygon as the outer face. Unlike
previous algorithms for this problem, which may produce drawings with
exponential area, our method produces drawings with polynomial area. In
addition, we allow for collinear points on the boundary, provided such vertices
do not create overlapping edges. Thus, we solve an open problem of Duncan et
al., which, when combined with their work, implies that we can produce a planar
straight-line drawing of a combinatorially-embedded genus-g graph with the
graph's canonical polygonal schema drawn as a convex polygonal external face.