the projective analogue of the interchange lemma

 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. 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!