  We present a new diagnostic diagram for local ultraluminous infrared galaxies
(ULIRGs) and quasars, analysing particularly the Spitzer Space Telescope's
Infrared Spectrograph (IRS) spectra of 102 local ULIRGs and 37 Palomar Green
quasars. Our diagram is based on a special non-linear mapping of these data,
employing the Kernel Principal Component Analysis method. The novelty of this
map lies in the fact that it distributes the galaxies under study on the
surface of a well-defined ellipsoid, which, in turn, links basic concepts from
geometry to physical properties of the galaxies. Particularly, we have found
that the equatorial direction of the ellipsoid corresponds to the evolution of
the power source of ULIRGs, starting from the pre-merger phase, moving through
the starburst-dominated coalescing stage towards the active galactic nucleus
(AGN)-dominated phase, and finally terminating with the post-merger quasar
phase. On the other hand, the meridian directions distinguish deeply obscured
power sources of the galaxies from unobscured ones. These observations have
also been verified by comparison with simulated ULIRGs and quasars using
radiative transfer models. The diagram correctly identifies unique galaxies
with extreme features that lie distinctly away from the main distribution of
the galaxies. Furthermore, special two-dimensional projections of the ellipsoid
recover almost monotonic variations of the two main physical properties of the
galaxies, the silicate and PAH features. This suggests that our diagram
naturally extends the well-known Spoon diagram and it can serve as a diagnostic
tool for existing and future infrared spectroscopic data, such as those
provided by the James Webb Space Telescope.
