Do planes in space have centers like surfaces and circles do?

Question

A circular has a center, so can a plane have a center? A plane extends infinitely in all directions parallel to itself, so it doesn't make sense for a plane to have a center me thinks. If a plane doesn't have a center, does it not make sense to shift a plane right or left to the effect that the equation of the plane z=a is unchanged for some a$\in$ $\mathbb{R}$?. I know it doesn't make sense for a plane to be centered at some point in space, but I think it makes sense for a surface to be centered at some point in space. If we can change where a torus is centered in space by editing its equation, can we change where a plane is centered in space or does it not make sense?

Answer 1

From a purely geometric point of view, I think you're right: there is no such thing as the center of a plane. But in a way that means that you can treat any point of the plane as the center (this idea can be useful in certain Physics problems that have a certain symmetry; e.g. computing the electric field across a parallel plane capacitor), and in a way we can select a point to be the center by imposing a coordinate system on the space.

For example, let's consider 2D space for a minute and consider the line defined by $y = 2x$. As a geometric object, this line also has no center. But with our usual coordinate system, we are tempted to view it as "centered" about the origin. We could imagine "shifting" the line along itself two units up and one unit to the right, and the "center" would now be located at the point $(1,2)$.

Geometrically, of course, we haven't changed anything. That's the symmetry of a line. But I think it's enlightening to see what happens to the equation of the line when we think about performing this "shift" operation. To shift the graph of some equation involving $x$ and $y$, we subtract the horizontal and vertical shift amounts from the corresponding variables. In this case, $x \to x-1$ and $y \to y-2$. Thus our line equation becomes $$ y-2 = 2(x-1) $$ which, of course, simplifies into $y = 2x$ upon algebraic manipulation, which corresponds to the geometric symmetry that shifting a line along itself doesn't really change it. Also note that the equation above is the point-slope form of the equation of a line! And for that matter, when we construct an equation for a line using point-slope form, the point we pick could be thought of as the "center".

We can even play this game in 3D with your flat plane example of $z = a$, except, let's rewrite this equation in a way that reflects a coordinate system. I can rewrite it as $$ z-a = 0(x-h) + 0(y-k) $$ which is something like the point-slope form for a line, but "centered" at the point $(h,k,a)$ in space. And you see that, no matter what $h$ and $k$ we choose, the equation trivially simplifies into $z=a$, which corresponds to the plane's symmetry.

So if you want to think of it this way, to "shift" a plane (or a line, or any other such symmetrical object) doesn't change the plane itself, but it does change the equation of the plane--although it remains algebraically equivalent to the original equation just as, geometrically, a "shifted" plane, though it is shifted, looks identical to the original.