The criterion that has mostly been discussed up to now I want to re-christen "EXTRINSIC", since it considers the surface z = f(x, y) embedded in 3-space: Given a plane curve C defined by f = 0 with f(x, y) polynomial, and a simply-connected compact region R of the plane, C avoids R if C avoids the boundary of R , and there is no point (x, y) in R where df/dx = df/dy = 0 . I asked earlier whether "if" could be strengthened to "if and only if"; a counter-example is easy to find. However, the following "INTRINSIC" criterion -- which is anyway what I have actually been experimenting with -- seems more robust in this respect, besides (like the rane in Spane?) remaining firmly in 2-space: Given a plane curve C defined by f = 0 with f(x, y) polynomial, and a simply-connected compact region R of the plane, C avoids R if and only if C avoids the boundary of R , and there is no point (x, y) in R where df/dx = f = 0 . Counter-examples in either direction please? If the weak ("if") or strengthened ("if and only if") version stands up, can Warren's proof -- if it stands up -- be modified to apply to either? I keep promising a worked example, but have tripped over a technical detail: it's easy to compute one component of the coordinate of a critical point -- say y above -- but rather trickier to pair with the corresponding x . For my application I haven't needed to take that final step; but omitting it is didactically rather unsatisfactory. Fred Lunnon