I wrote several posts on poles and polars a few days ago. First, I wrote about the dual of a spherical triangle, called the polar triangle. Then I wrote about poles and polars in plane geometry and briefly mentioned the extension to projective geometry.
Poles and polars are a little awkward in the plane because you have to pay attention to special cases. Poles and polars are relative to a conic section, and are defined/computed differently depending on whether you’re taking the dual with respect to an ellipse, parabola, or hyperbola. Also, the polar of the center of a conic section isn’t defined.
Projective space
If you move from the Euclidean plane to the projective plane, the exceptions go away, though it may not seem so if you’re talking about points at infinity or lines at infinity. If you fully commit to working in projective space, there aren’t any hand-wavy points or lines “at infinity.” There’s not even a distinction between ellipses, parabolas, and hyperbolas.
There’s no such thing as a free lunch, and the cost of this elegant unification is moving to a bigger geometric space with a different coordinate system. Instead of working in the Euclidean spaces R² or Rn, we work in Pn over a field F which could be R but doesn’t need to be [1].
The coordinates on Pn are homogeneous coordinates, (n + 1)-tuples of field elements, not all equal to zero, with two (n + 1)-tuples representing the same point if they are proportional, i.e. one is a non-zero multiple of the other.
We can embed Rn in Pn by adding an extra coordinate on the end, set equal to 1. You could think of this extra coordinate as a switch, set to 1 for finite points and set to 0 for infinite points. But this distinction between finite and infinite points is an artifact of the embedding, not a property of Pn: there are no special points in Pn. A point that is “at infinity” in one embedding of Rn in Pn need not be in another embedding.
Poles and polars
Once you accept the need to work in homogeneous coordinates, everything becomes simple and consistent.
A quadric (a conic section when n = 2) is the set of points
{x : xᵀAx = 0}
where A is a non-singular (n + 1) by (n + 1) symmetric matrix over F.
The polar of a point p is the hyperplane
{x : pᵀAx = 0}
and the pole of a hyperplane
H = {x : uᵀx = 0}
is the point
p = A−1 u.
Notice there’s no mention of “infinity” in this section. All the calculations take place in the field F with no points at infinity added.
[1] The only requirement is that the field F cannot have characteristic 2. So F could be the reals, the rationals, the complex numbers, etc. F could even be a finite field as long as the number of elements in F is not a power of 2.