  The concept of reduced variables is revisited with regard to van der Waals'
theory and an application is made to polytropic spheres, where the reduced
radial coordinate is ${\rm red}(r)=r/R=\xi/\Xi$, $R$ radius, and the reduced
density is ${\rm red}(\rho)=\rho/\lambda=\theta^n$, $\lambda$ central density.
Reduced density profiles are plotted for several polytropic indexes within the
range, $0\le n\le5$, disclosing two noticeable features. First, any point of
coordinates, $({\rm red}(r),{\rm red}(\rho))$, $0\le{\rm red}(r)\le1$,
$0\le{\rm red}(\rho)\le1$, belongs to a reduced density profile of the kind
considered. Second, sufficiently steep i.e. large $n$ reduced density profiles
exhibit an oblique inflection point, where the threshold is found to be located
at $n=n_{\rm th}=0.888715$. Reduced pressure profiles, ${\rm
red}(P)=P/\varpi=\theta^{n+1}$, $\varpi$ central pressure, Lane-Emden
fucntions, $\theta=(\rho/\lambda)^{1/n}$, and polytropic curves, ${\rm
red}(P)={\rm red}(P)({\rm red}(\rho))$, are also plotted. The method can be
extended to nonspherical polytropes with regard to a selected direction, ${\rm
red}(r)(\mu)=r(\mu)/R(\mu)=\xi(\mu)/\Xi(\mu)$. The results can be extended to
polytropic spheres made of collisionless particles, for polytropic index within
a more restricted range, $1/2\le n\le5$.
