  An $st$-path in a drawing of a graph is self-approaching if during the
traversal of the corresponding curve from $s$ to any point $t'$ on the curve
the distance to $t'$ is non-increasing. A path has increasing chords if it is
self-approaching in both directions. A drawing is self-approaching
(increasing-chord) if any pair of vertices is connected by a self-approaching
(increasing-chord) path.
  We study self-approaching and increasing-chord drawings of triangulations and
3-connected planar graphs. We show that in the Euclidean plane, triangulations
admit increasing-chord drawings, and for planar 3-trees we can ensure
planarity. We prove that strongly monotone (and thus increasing-chord) drawings
of trees and binary cactuses require exponential resolution in the worst case,
answering an open question by Kindermann et al. [GD'14]. Moreover, we provide a
binary cactus that does not admit a self-approaching drawing. Finally, we show
that 3-connected planar graphs admit increasing-chord drawings in the
hyperbolic plane and characterize the trees that admit such drawings.
