  We propose to describe the variety of galaxies from SDSS by using only one
affine parameter. To this aim, we build the Principal Curve (P-curve) passing
through the spine of the data point cloud, considering the eigenspace derived
from Principal Component Analysis of morphological, physical and photometric
galaxy properties. Thus, galaxies can be labeled, ranked and classified by a
single arc length value of the curve, measured at the unique closest projection
of the data points on the P-curve. We find that the P-curve has a "W" letter
shape with 3 turning points, defining 4 branches that represent distinct galaxy
populations. This behavior is controlled mainly by 2 properties, namely u-r and
SFR. We further present the variations of several galaxy properties as a
function of arc length. Luminosity functions variate from steep Schechter fits
at low arc length, to double power law and ending in Log-normal fits at high
arc length. Galaxy clustering shows increasing autocorrelation power at large
scales as arc length increases. PCA analysis allowed to find peculiar galaxy
populations located apart from the main cloud of data points, such as small red
galaxies dominated by a disk, of relatively high stellar mass-to-light ratio
and surface mass density. The P-curve allows not only dimensionality reduction,
but also provides supporting evidence for relevant physical models and
scenarios in extragalactic astronomy: 1) Evidence for the hierarchical merging
scenario in the formation of a selected group of red massive galaxies. These
galaxies present a log-normal r-band luminosity function, which might arise
from multiplicative processes involved in this scenario. 2) Connection between
the onset of AGN activity and star formation quenching, which appears in green
galaxies when transitioning from blue to red populations. (Full abstract in
downloadable version)
