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

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

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.