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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.

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  • $\begingroup$ Hey, welcome back to MSE (+1) $\endgroup$ – imranfat Mar 12 '16 at 4:27
  • $\begingroup$ Welcome again, Please don't quit entirely,keep a low key on topics of your interest, my 2 cents. $\endgroup$ – Narasimham Mar 28 '16 at 14:20
  • $\begingroup$ Since there were no answers here, I asked the same question at MathOverflow, and got some useful info: mathoverflow.net/questions/235094/… $\endgroup$ – bubba Apr 3 '16 at 4:13

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