I asked this [on Math.StackExchange](https://math.stackexchange.com/questions/1693831/polar-forms-of-algebraic-curves-surfaces), 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)](http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-34.pdf). 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.



Polars of algebraic curves and surfaces
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](https://en.wikipedia.org/wiki/Pl%C3%BCcker_formula).


Dolgachev's excellent book *[Classical Algebraic Geometry](http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/classical-algebraic-geometry-modern-view)* probably contains all you need about this subject and much more.


