Polars of algebraic curves and surfaces 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 form of  $F(\mathbf{x})$ at the pole $\mathbf{a} = [a,b,c,h]$ 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 polar with respect to the pole $x_1$ 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.
 A: Geometrically, one first meets polar hypersurfaces when studying tangent lines from a point to a hypersurface. In fact, this concept generalizes the classical polarity of a point with respect to a conic.
More precisely, let $X \subset \mathbb{P}^k$ be a hypersurface of degree $n$ given by the zero locus of a homogeneous polynomial $F$, namely $$F(x_0, \ldots, x_k)=0,$$
and let $\mathbf{a}=[a_0: \ldots:a_k] \in \mathbb{P}^k$ be any point. We define the  first polar of the point $\mathbf{a}$ with respect to $X$ as the hypersurface $P_{\mathbf{a}}(X)$ of degree $n-1$ defined by $$\sum_{i=0}^k a_i\frac{\partial F}{\partial x_i}=0$$ 
(dividing by $n$ is not really important here).
Then the locus $X \cap P_\mathbf{a}(X)$ is given by the points $\mathbf{p}$ such that the tangent space to $X$ at $\mathbf{p}$ contains $\mathbf{a}$. 
In the particular case where $X \subset \mathbb{P}^2$ is a smooth plane curve of degree $n$, the first polar is a curve of degree $n-1$. Therefore we deduce from the Bézout theorem that for a general point $\mathbf{a} \in \mathbb{P}^2$ pass exactly $n(n-1)$ distinct tangent lines to $X$. This number is classically called the class of $X$. When $X$ is singular the formula for the class must be modified according to the number and the local type of the singular points; in the case of nodes and cusps, this leads to the so-called  Plücker formulas.
Dolgachev's excellent book Classical Algebraic Geometry probably contains all you need about this subject and much more.
