## Duality transformation

In this chapter we define and explain the concept of duality transformation in geometry.

#### A mapping

Duality transformation in geometry can be defined as a mapping between points and lines. Several mappings exists based on different transformations and each of these mappings preserves different properties. Duality transformation can be usefull in the process of designing efficient algorithms for important geometric problems.

#### What mapping?

We know that in the cartesian plane:

• Points are defined by two coordinates $x$ and $y$.
• Lines are also defined by two parameters, their slope ($m$) and their y-intercept ($b$).
This relationship is long known to geometers. In our current project we will use the following transformation denoted $D$:

We define:
• a point $p$ by $p \equiv (x_p,y_p)$ its two coordinates in the cartesian plane
• a line $l$ by the following equation $l \equiv y = mx+b$
The dual transformation $D$ of:
• the point $p$ is noted $D(p)$. It is a line defined by $D(p) \equiv y = (-x_p) x + y_p$
• the line $l$ is noted $D(l)$. It is a point defined by $D(l) \equiv (m, b)$

#### Properties of this mapping

Each dual transformation conserves a number of properties wich are usefull to geometers. In this small section we highlight some of these properties.

• Inverse: The inverse property is not respected by the chosen dual transformation $D$. For any point $p$ or line $l$, $D(D(p)) \neq p$ and $D(D(l)) \neq l$. In order to find back the original point or line one needs to cycle twice trough a double dual transformation.
• Incidence: $D$ respects the incedence property. Indeed, given a point $p$ and a line $l$, $I(p, l) \Leftrightarrow I(D(p), D(l))$.
Given a point $p \equiv (x_p, y_p)$ and a line $l \equiv y = mx + b$, we define the boolean function $I$:

$I(p,l) = \begin{cases}true & mx_p + b - y_p = 0 \\ false & \text{otherwise} \end{cases}$

$I$ is a symmetric relation, meaning that $I(p, l) \equiv I(l, p)$
• Ordering: $D$ respectes the ordering property. This means that given a point $p$ and a line $l$, if $p$ is above (below) $l$, $D(p)$ is above (below) $D(l)$.
Given a point $p \equiv (x_p, y_p)$ and a line $l \equiv y = mx+b$:
• $p$ is above $l$ if $y_p > m x_p + b$
• $p$ is below $l$ if $y_p < m x_p + b$

#### Example

Here is a small application where you can add points, lines and get the plot corresponding to the dual transformation $D$ (described above) of all added elements.