We define the visual complexity of a plane graph drawing to be the number of geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g. you need only one line segment to draw two collinear edges of the same vertex). We show that trees can be drawn with 3n/4 straight-line segments on a polynomial grid, and with n/2 straight-line segments on a quasi-polynomial grid. We also study the problem of drawing maximal triangulations with circular arcs and provide an algorithm to draw such graphs using only (5n - 11)/3 arcs. This provides a significant improvement over the lower bound of 2n for line segments for a nontrivial graph class.