Full plane of temp and density for the O[III] 5007+4959 / 4363 line ratio (shown as contours, with values written on). Note that the line ratio is independent of density up to the critical density of the 5007 line, and the behavior changes yet again above the critical density for the 4363 line.
Full plane of temp and density for the S[II] 6717/6731 line ratio (shown as contours, with values written on). The vertical lines remind us that the ratio is, in fact, very insensitive to temperature.
Figures from Simon Morris's PhD thesis, Cambridge 1983.