  In this paper we introduce self-approaching graph drawings. A straight-line
drawing of a graph is self-approaching if, for any origin vertex s and any
destination vertex t, there is an st-path in the graph such that, for any point
q on the path, as a point p moves continuously along the path from the origin
to q, the Euclidean distance from p to q is always decreasing. This is a more
stringent condition than a greedy drawing (where only the distance between
vertices on the path and the destination vertex must decrease), and guarantees
that the drawing is a 5.33-spanner. We study three topics: (1) recognizing
self-approaching drawings; (2) constructing self-approaching drawings of a
given graph; (3) constructing a self-approaching Steiner network connecting a
given set of points. We show that: (1) there are efficient algorithms to test
if a polygonal path is self-approaching in R^2 and R^3, but it is NP-hard to
test if a given graph drawing in R^3 has a self-approaching uv-path; (2) we can
characterize the trees that have self-approaching drawings; (3) for any given
set of terminal points in the plane, we can find a linear sized network that
has a self-approaching path between any ordered pair of terminals.
