Given a surface $f(x,y,z)=0$, how could you determine that it's symmetric about some plane, and, if so, how would you find this plane.
The special case where $f$ is a polynomial is of some interest.
The question is somewhat related to this one: How to identify surfaces of revolution
Here's a simple 2D example to experiment with: $$ 27 x^3 + 108 x^2 y + 144 x y^2 + 64 y^3 - 80 x^2 + 120 x y - 45 y^2 - 200 x + 150 y - 125 = 0$$ This one is symmetric, as the following picture suggests:
In fact, if we do the translation/rotation described by the substitution $u = \tfrac15(3x+4y)$, $v = \tfrac15(4x−3y) + 1$, then the curve is just $u^3 - v^2 = 0$, which is obviously symmetric about the line $v=0$.
But how would you discover this translation/rotation if I didn't tell you, and how would you do similar things in the 3D surface case?
Added after a few days of thought:
We can consider the surface as an object made of thin sheet metal. As such, it has a center of mass, provided it's bounded, and any plane(s) of symmetry must pass though this center of mass. The plane then has only two remained degrees of freedom, so may be easier to find.
The same sort of reasoning applies in the case of a bounded 2D curve. Again, any line of symmetry must pass through the curve's centroid, so it has only one remaining degree of freedom, namely its slope/angle.
For curves and surfaces given by implicit equations, I don't really know how to calculate centroids, but I expect this can be done.
Fabricated from Comments Below:
Several people suggested looking at highest-degree terms only. So, in my example, we just look at the equation $$ 27 x^3 + 108 x^2 y + 144 x y^2 + 64 y^3 = 0 $$ Putting $w = y/x$, this is roughly equivalent to $$ 64w^3 + 144w^2 + 108 w + 27 = 0 $$ But the polynomial on the left is just $(4w + 3)^3$, so we have a root $w = -3/4$, with multiplicity three. Is the repeated root an accident that happens only in this case, or will it always happen?? Anyway, the vector $(-4,3)$ gives us the normal to the line of symmetry, and that surely can not be an accident.
I don't really understand why this magic process works, but it looks very promising for the 2D curve case.
I don't know how to generalize to the 3D surface case, or to non-polynomial cases.