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The three point theorem involves three p-points on a p-conic.
The two point theorem involves one p-point on a p-conic and one not on the p-conic.
This suggests that controlling a p-point not on the p-conic is more demanding. it is
no
surprise, then, that if we take two pairs of p-points inside a p-conic, there is usually
no
projective transformation mapping one pair to the other.
We begin with a positive result, when the pairs are (P,Q) and (Q,P).
In hyperbolic geometry, we met the origin lemma and the interchange lemma.
The latter states that
If P and Q ε D, there is an h-inversion mapping P to Q and Q to P.
If P ≠ Q, then the h-inversion is unique.
An h-inversion has order 2. In fact, there is a second hyperbolic transformation
which interchanges P and Q, obtained by composing the h-inversion of the theorem
with h-inversion in the h-line PQ. This also has order 2.
As we shall now see how these results arise from projective geometry.
The interchange theorem for projective conics
If P and Q are distinct p-points in the interior of a p-conic C, then
there are precisely two projective transformations t which satisfy
(a) t(C) = C, and
(b) t interchanges P and Q.
proof
What seems much harder to prove is that both transformations have order 2.
Our proof involves many earlier results - it could well be skipped at first!
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