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:

This relationship is long known to geometers. In our current project we will use the following transformation denoted $D$:

We define: The dual transformation $D$ of:

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.


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.