§ Proof of projective duality
In projective geometry, we can interchange any statement with "points" and
"lines" and continue to get a true statement. For example, we have the
The proof of the duality principle is simple. Recall that any point in
projective geometry is of the from . A projective
equation is of the form for coefficients .
- Two non-equal lines intersect in a unique point (including a point at infinty for a parallel line).
- Two non-equal points define a unique line.
- if we have a fixed point , we can trade this to get a line .
- If we have a line , we can trade this to get a point .
- The reason we need projectivity is because this correspondence is only well defined upto scaling: the line is the same as the line .
- In using our dictionary, we would get , . Luckily for us, projectivity, these two points are the same! .
- The "projective" condition allows us to set points and lines on equal footing: lines can be scaled, as can points in this setting.