I asked this on Math.StackExchange, but received no response, so trying here ...

A paper I'm reading says the following ...

With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equation of a surface of degree $n$. The

first polar formof $F(\mathbf{x})$ at thepole$\mathbf{a} = [a,b,c,h]$ is defined as $$ F^1(\mathbf{x}) = \frac1n(aF_x + bF_y + cF_z + hF_w) $$

And then, later, there's a similar definition ...

Let $\mathbf{p}(x)$ be a polynomial curve of degree $n$ and let $\mathbf{p}(x,w)$ denote its homogeneous form. Its

first polarwith respect to thepole$x_1$ is defined by $$ \mathbf{p}^1(x_1 \,|\,x) = \frac1n(x_1\mathbf{p}_x + \mathbf{p}_w) $$

The paper says these concepts are "well-known in algebraic geometry". I'm having trouble understanding what these things mean geometrically. Could someone explain, or provide a reference, please. I'd like the explanation or reference to be something simple and concrete in 2D or 3D space, please; abstraction and generality probably won't help me.

I recall polar lines of circles and conics from high school analytic geometry. That's the case $n=2$, and it makes sense. It's the $n > 2$ case that I'm having trouble grasping.

I'm guessing that there is some relationship with "polar forms" (aka blossoms in the CAGD field). In this context, the polar form of a polynomial $f:\mathbb{R} \to \mathbb{R}$ of degree $n$ is a symmetric multi-affine function $F:\mathbb{R}^n \to \mathbb{R}$ such that $F(x, \ldots,x) = f(x)$. But I can't see that relationship, either.