  Optical diagnostic diagrams are powerful tools to separate different ionizing
sources in galaxies. However, the model-constraining power of the most
widely-used diagrams is very limited and challenging to visualize. In addition,
there have always been classification inconsistencies between diagrams based on
different line ratios, and ambiguities between regions purely ionized by active
galactic nuclei (AGNs) and composite regions. We present a simple reprojection
of the 3D line ratio space composed of [N II]$\lambda 6583$/H$\alpha$, [S
II]$\lambda \lambda$6716, 6731/H$\alpha$, and [O III]$\lambda$ 5007/H$\beta$,
which reveals its model-constraining power and removes the ambiguity for the
true composite objects. It highlights the discrepancy between many theoretical
models and the data loci. With this reprojection, we can put strong constraints
on the photoionization models and the secondary nitrogen abundance
prescription. We find that a single nitrogen prescription cannot fit both the
star-forming locus and AGN locus simultaneously, with the latter requiring
higher N/O ratios. The true composite regions stand separately from both
models. We can compute the fractional AGN contributions for the composite
regions, and define demarcations with specific upper limits on contamination
from AGN or star formation. When the discrepancy about nitrogen prescriptions
gets resolved in the future, it would also be possible to make robust
metallicity measurements for composite regions and AGNs.
