A unified view of poles and polars

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…