# Duality In Projective Geometry
In projective geometry, a mind-blowing concept called Duality exis
ts. Before explaining, a real projective space is just an Euclidea
n space with additional points at infinity, where parallel lines m
eet. So if you are not familiar with projective spaces, you can th
ink of them as Euclidean spaces with few exceptions.
In a n-dimensional projective space, a r-dimensional object corres
ponds to an object of dimension n-1-r.
In a projective plane:
- A point corresponds to a line, and vice versa.
In 3-dimensional:
- A point corresponds to a plane, and vice versa.
- A Line is self-dual.
&c.
In a projective space, a statement is equivalent to its dual.
In a projective plane:
- Two points determine a line, because two lines determine a point
In 3-dimensional:
- Three points determine a plane, because three planes determine a
point.
- Two lines determine a plane, because two lines determine a point
&c.
If they are parallel, they intersect on points at infinity.
Many results by duality exist. I will introduce Desargues' theorem
, because it is important.
## Desargues' theorem
Desargues' theorem is self-dual. It is important, because of its g
enerality. For any arithmetically defined projective space over a
division ring, it is true.
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Theorem. Two triangles are in perspective axially if and only if t
hey are in perspective centrally.
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Perspective axially means the intersection points (ABab, ACac, BCb
c) all lie on a common line. Perspective centrally means the lines
determined by the corresponding points of the objects (Aa, Bb, Cc)
all meet at a point.
It is self-dual. You can interpret the points A, B, C as the lines
AB, AC, BC, and vice versa.
Proof.
By duality,
[Two lines determine a point] corresponds to (Two points determine
a line).
[The lines AB and AC determine the point A] corresponds to (The po
ints A and B determine the line AB).
[The points A and a determine the line Aa] corresponds to (The lin
es AB and ab determine the point ABab).
Therefore [The lines Aa, Bb, Cc all intersect at a point] correspo
nds to (The points ABab, ACac, BCbc all lie on a line).
Q.E.D.
To prove, you need only basic geometric structures and duality. Th
at's why the theorem is generally true.
## Other dual theorems
Ceva's theorem <-> Menelaus' theorem
Pascal's theorem <-> Brianchon's theorem
are famous. Since Ceva's and Menelaus' theorem are useful for plan
e geometry, you might be familiar.