I would like to describe you a technique that I found in a physics paper, for which I cannot give a mathematical interpretation.
Let $f(q,g,e,r)=\frac{r}{2}(g+q) + q^{\frac{p}{2}} e - \frac{T}{2} \ln{g} - \frac{\beta}{4} ( (g+q)^{p} - q^{p} - pgq^{p-1} ) - \frac{r}{2}$ where $T, \beta$ and $p$ are constants.
Then while we compute the stationary (critical) point, from the equation $0=\frac{df}{dr}$ we get that
\begin{equation} g + q -1 =0 \end{equation}, which implies that $g=1-q$.
Then put this formula in the initial formula for $f$. You will get $f=f(q,e) = q^{\frac{p}{2}}e - \frac{T}{2} \ln(1-q) -\frac{\beta}{4} ( 1 - (p-1)q^{p} - pq^{p-1})$
What geometrical explanation does it have ? Why in general would someone do this ? What do we gain ?
One can get a good understanding of this issue in terms of shadows' limits i.e., limits of projections of a certain surface ; but in a first step, we will need to understand how envelopes of curves (or surfaces, etc.) are involved here.
Let us take an example.
Consider the following expression :
$$f(x,y,z):=(x-z)^2+y^2-z=0. \tag{1}$$
Let us interpret (1) as a family of circles :
$$(x-z)^2+y^2=\sqrt{z}^2$$
with center $(z,0)$ and radius $\sqrt{z}$ (see Fig. 1).
Fig. 1.
$z$ is now considered as a parameter.
If we differentiate (1) with respect to parameter $z$ and equate the result to $0$, we get :
$$-2(x-z)=1 \ \ \iff \ \ z=x+\frac12\tag{2}$$
If we plug (2) into (1), we get :
$$\frac14+y^2-(x+\frac12)=0$$
i.e.
$$y^2=x+\frac14 \ \ \iff \ \ y=\pm \sqrt{x+\frac14},$$
which is the equation of the envelope (the parabola in red on Fig. 1).
Why that ? Because it is known (https://en.wikipedia.org/wiki/Envelope_theorem) that the envelope of a family of curves $f(x,y,m)=0$ depending on a parameter $m$, under differentiability conditions that are fulfilled here, is obtained by eliminating $m$ from the two equations $$\begin{cases}f(x,y,m)&=&0\\\dfrac{\partial f}{\partial m}(x,y,m)&=&0\end{cases}\tag{3}$$
There is more to say. In fact, we can reinterpret Fig. 1 by taking $z$, not as any parameter but as a plain third coordinate ; in this way $f(x,y,z)=0$ is the implicit equation of a certain surface (here a paraboloid with oblique axis) ; the curve we have obtained is thus the boundary of the shadow of the projection on plane $xOy$ of this surface. This is clear from Fig. 2 which is an "elevation" of Fig. 1.
Fig. 2. : The paraboloid with equation (1) materialized by certain level lines (in black) and their projections onto the horizontal plane, giving back Fig. 1.
Remarks :
1) The concept of envelopes of curves and the associated technique (see (3)) for computing them can be extended in a straightforward manner to surfaces which are envelopes of families of surfaces.
2) A simple explanation for the case of straight lines envelopes I gave as an appendix to an answer : Loci of intersection of lines with positive intercepts??
3) Many examples of envelopes can be found on the net :
https://www.math24.net/envelope-family-curves-page-2/
https://people.duke.edu/~dgraham/handouts/EnvelopeTheorem.pdf
etc.
4) Red curve in Figure 1 can be interpreted as the moving "sound wave" of an aircraft.
5) In the example above, out of an equation with 3 variables $x,y,z$, the elimination process has generated an equation with 2 variables $x,y$. This kind of "dimension reduction by 1" is the most usual. In the example you give, one jumps from $4$ to $2$ variables : we are in a case of degeneracy.
6) See a slightly different explanation in gave to a cousin query : https://math.stackexchange.com/q/3095569
7) Related : https://www.math24.net/singular-solutions-differential-equations/