# What are the primal and dual planes in the context of the point-line duality?

In computational geometry, we can define a duality between points and lines. The line is the primal (or dual) object of a point, or a point is the primal (or dual) object of a line. However, the exact definition of the primal and dual planes is often not given in the literature. It's a little bit confusing which one is which. What are exactly the primal and dual planes in this context?

In short, the original $(x, y)$-plane is the primal plane and the new $(a, b)$-plane the dual plane.

More specifically, in a $(x, y)$-plane (or coordinate system)1, a non-vertical line can be represented as $\ell \colon y = ax - b$, where $a \in \mathbb{R}$ is the slope and $b \in \mathbb{R}$ the $y$-intercept. Hence, to describe a line, we just need to scalars, $a$ and $b$. We can state that a line has two degrees of freedom.

The idea of the duality between lines and points is that lines in the $(x, y)$-plane can be thought of or represented as points in a new $2$-dimensional space (or coordinate system), in which the coordinate axes are labeled $(a, b)$, rather than $(x, y)$. This new plane is denoted or labeled as $(a, b)$ to emphasize the fact that the values that $a$ and $b$ may assume respectively refer to the slope and $y$-intercept of the line $\ell$ in the original $(x, y)$-plane.

For example, the line $\ell : y = 2x + 1$, in the $(x, y)$-plane, corresponds to the point $\ell^* = (2, −1)$ in this new $(a, b)$-space 2, 3; that is, $\ell^*$ is a $2$-dimensional point with $2$ as the value of the $a$-coordinate and $-1$ as the value of this $b$-coordinate. A point $p=(5, 2)$ in the $(x, y)$-plane corresponds to the line $p^* \colon b = 5 a - 2$ in the $(a, b)$-plane 4. Similarly, a point $q = (1, 6)$ in the $(a, b)$-space, corresponds to a non-vertical line, $q^* \colon y = 1x − 6$ in the "original" plane $(x, y)$. And a line $t \colon \frac{1}{2} a - 5$ in the $(a, b)$-plane corresponds to the point $t^*=(\frac{1}{2}, 5)$ in the $(x, y)$-plane.

A convention is to call the original $(x, y)$-plane the primal plane, and the new (a, b)-plane the dual plane.

Why is $(x, y)$ the "original plane"? What's "original" about it? It's simply the plane where our initial problem is formulated. For example, we may have a problem where we are given a set of points. Then, we say that those points are given in the primal plane. We may convert our points to lines in the dual plane, because maybe we can formulate our problem in terms of lines, and the new problem is easier to solve.

Of course, we could have labeled the "original" plane with other letters, e.g. $(g, h)$, but we often label it $(x, y)$, so the new system is labeled differently, e.g. $(a, b)$. We could have labeled the new plane or coordinate system $(o, z)$: it really does not matter, i.e. it's just notation!

Here's a picture which summarizes all these words

Notes

1. Note that we label this plane $(x, y)$, but could have labeled it $(u, v)$ or any other tuple of two letters: it does not matter (as long as we are consistent).

2. Note the $-$ (i.e. the minus) in front of the $1$ in the point $\ell^*$.

3. We use an asteristik, $*$, to denote the dual of some object. For example, $u^*$ is the dual (in the dual plane) of $u$ (in the $(x, y)$ or primal plane). $u$ can either be a line or a point. If $u$ is a line, then $u^*$ is a point. If $u$ is a point, then $u^*$ is a line.

4. Note the $-$ (i.e. the minus) in front of the $-2$ in the equation of the line $p^*$.