Polars is a high-performance DataFrame library for Python and Rust. Built on Apache Arrow, it's the fastest way to work with tabular data at any scale.

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 _R_ ² or _R_ _n_ , we work in _P_ _n_ over a field _F_ which could be _R_ but doesn’t need to be [1]. The coordinates on _P_ _n_ 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 _R_ _n_ in _P_ _n_ 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 _P_ _n_ : there are no special points in _P_ _n_. A point that is “at infinity” in one embedding of _R_ _n_ in _P_ _n_ 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_ : _x_ ᵀ _A_ _x_ = 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_ : _p_ ᵀ _A_ _x_ = 0} and the **pole** of a hyperplane _H_ = {_x_ : _u_ ᵀ _x_ = 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.

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