an algebraic characterization of interior and exterior

 We know that a projective conic C has equation f(x) = xTMx = 0, with M symmetric. If P =[p] is not on C, then f(p) will be non-zero, and thus either positive or negative. It turns out that all p-points inside C have f(p) with the same sign, and all p-points outside C have f(p) of the opposite sign to that for the interior p-points. the algebraic interior-exterior theorem Suppose that C is the projective conic f(x) = xTMx = 0. Then P = [p] and Q = [q] both lie on the same side of C if and only if f(p) and f(q) have the same sign. Note that we can replace M by -M in the equation. This allows us to choose the equation of C so that the interior corresponds to p-points with f(p) > 0. We know that the cross-ratio of four collinear p-points is a projective invariant. When the p-points lie on a chord of a projective conic, we get interesting results. the cross-ratio theorem for a projective conic Suppose A, B are distinct p-points on the projective conic C, and that P,Q are distinct p-points on AB, but not on C. Then P,Q both lie inside C or both lie outside C if and only if (A,B,P,Q) is positive. This is related to The polar-chord theorem If a p-point P lies on the chord AB of a p-conic C, then the polar of P with respect to C cuts the p-line AB at Q, where (A,B,P,Q) = -1. Combining these, we have the Corollary If a p-point P lies on the chord AB of a p-conic C and the polar of P cuts the p-line AB at Q, then exactly one of P,Q lies inside C . Observe that this is trivial if P is inside C, for then the polar of P does not cut C, so Q must lie outside C. Suppose that P and Q lie inside C. Then the p-line PQ cuts C twice. We can label these A and B in either order. By the theorem, we know that (A,B,P,Q) is positive. From Remarks(1),(3), we see that (B,A,P,Q) = 1/(A,B,P,Q), and the ratios are unequal as neither is 1. It follows that one of the cross-ratios is less than 1. We make the Definition Suppose that P and Q are distinct p-points in the interior of a p-conic C. Then D(P,Q) = (A,B,P,Q), where A and B are the p-points in which PQ meets C, labelled so that (A,B,C,D) < 1. We set D(P,P) = 1. We know that, for a p-conic C, the projective symmetry group S(C,P(2)) maps the interior of C to itself. We also know that any projective transformation preserves cross-ratio. These remarks prove the invariance theorem for D For any projective conic C the function D is an invariant of S(C,P(2)). In the geometry defined on the interior of C by the symmetry group d = -log(D) is essentially a distance function since we have d(P,P) = 0, d(P,Q) > 0 if P ≠ Q, d(P,Q) = d(Q,P). Only the last needs comment. The proof uses Remark(1). If PQ meets C in A and B with D(P,Q) = (A,B,P,Q) < 1, then (A,B,Q,P) = 1/(A,B,P,Q). But this is greater than 1, so that D(Q,P) = (B,A,Q,P) = 1/(A,B,Q,P) = D(P,Q). We shall not pursue the triangle inequality. We already know that the geometry is a model of hyperbolic geometry. As a final example of the use of projective geometry to understand this model of hyperbolic geometry, we will look at the projective analogue of the interchange lemma, and some related ideas.