  We consider several natural sets of curves associated to a given
Teichm\"uller disc, such as the systole set or cylinder set, and study their
coarse geometry inside the curve graph. We prove that these sets are
quasiconvex and agree up to uniformly bounded Hausdorff distance. Furthermore,
we describe two operations on curves and show that they approximate nearest
point projections to their respective targets. Our techniques can be used to
prove a bounded geodesic image theorem for a natural map from the curve graph
to the filling multi-arc graph associated to a Teichm\"uller disc.
