Dual projective plane ($(\mathbb{P}(V){)}^{\ast }=\mathbb{P}(V^{\ast })$)

The dual projective plane of $\mathbb{P}(V)$ for a three-dimensional vector space $V$ is $\mathbb{P}(V^{\ast })$. A nonzero linear functional $\ell$ defines a line $\mathbb{P}(\ker \ell )$ in the original plane, and scalar multiples define the same line. Thus points of the dual plane parametrize lines of the original plane.

In coordinates, the original line $ax+by+cz=0$ corresponds to the dual point $[a:b:c]$. The incidence equation $ax+by+cz=0$ is symmetric between a point coordinate triple and a line coefficient triple. For example, all original lines through $[1:0:0]$ have $a=0$, and their dual points form a line in the dual plane.

Projective duality interchanges collinear original points with concurrent dual lines and vice versa. An identification of the plane with its dual needs additional data, such as a nondegenerate bilinear form; the canonical construction is the dual space, not a preferred coordinate identification.