Duality and the Duality Controversy
We have seen that Poncelet used a way (in a sense, discovered by Brianchon) of pairing each point of the plane with a line, called the polar of the point, and each line in the plane with a point, called its pole. All that was needed for this was a conic. Any conic will do, and if a different conic is chosen the details of which point is paired with which line is changed, but nothing else. We also saw that the process of starting from a point (a pole) and producing a line (its polar) is the inverse of the process of starting from a line (a polar) and producing a point (its pole). We also saw that if three points lie on a line then their polars meet in a point and, conversely, if three lines meet in a point then their poles lie on a line. (See Figure 5.1.)
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