The three point theorem involves three ppoints on a pconic.
The two point theorem involves one ppoint on a pconic and one not on the pconic.
This suggests that controlling a ppoint not on the pconic is more demanding. it is no
surprise, then, that if we take two pairs of ppoints inside a pconic, 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 hinversion mapping P to Q and Q to P.
If P ≠ Q, then the hinversion is unique.
An hinversion has order 2. In fact, there is a second hyperbolic transformation
which interchanges P and Q, obtained by composing the hinversion of the theorem
with hinversion in the hline 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 ppoints in the interior of a pconic 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!
