  We report new rogue wave patterns whose wave crests form closed or open
curves in the spatial plane, which we call rogue curves, in the
Davey-Stewartson I equation. These rogue curves come in various striking
shapes, such as rings, double rings, and many others. They emerge from a
uniform background (possibly with a few lumps on it), reach high amplitude in
such striking shapes, and then disappear into the same background again. We
reveal that these rogue curves would arise when an internal parameter in
bilinear expressions of the rogue waves is real and large. Analytically, we
show that these rogue curves are predicted by root curves of certain types of
double-real-variable polynomials. We compare analytical predictions of rogue
curves to true solutions and demonstrate good agreement between them.
