We saw in the previous tutorial that projective geometry arose out of perspective painting by the addition of new elements, first noticed by Renaissance painters, where parallel lines and planes meet.
The addition of these new elements in projective geometry (PG) had unexpected and far-reaching consequences. To simplify the discussion, we focus now on 2D projective geometry, the projective plane $P(\mathbb{R}^2)$. It arises from the euclidean plane $\mathbf{E}^2$ by adding an ideal line along with all its points, the ideal points of the projective plane.
Duality #
The innocent-looking addition of these ideal elements brings far-reaching consequences for projective geometry. Consider the following two statements in plane geometry.
Notice that the second statement is not true in euclidean geometry since the two lines can be parallel. But with the addition of ideal points, it is always true in projective geometry.
Dictionary of duality #
Dualizing means to use a dictionary of duality to replace dual terms with their dual partners. Some of the pairs in the dictionary are nouns (point/line) while others are verbs (join/intersect). Any term not in the dictionary remains unchanged by dualizing.
| Term | Dual Term |
|---|---|
| line | point |
| join | intersect |
| lies on | passes through |
| rotate around | move along |
Notice that the dual pairs are symmetric; for each pair, you can switch the left and right terms without changing the meaning. So it’s not like an ordinary dictionary where one of the partner terms belongs to, say, English language and the other is German language. There is only the one language of projective geometry, and within this language there are pairs of dual partners.
Example #
The dual of
isA point moving along a line and joined to a point not lying on the line.
A line rotating around a point and intersected with a line not passing through the point.
From this follows the Principle of duality:
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