From the interior-exterior theorem, we know that a projective transfromation t
maps an interior (exterior) p-point of the p-conic C to an interior (exterior) p-point
of t(C). It turns out that this is the only restriction on the image of a p-point.
the two-point theorem for projective conics
Suppose that C and C' are p-conics, and that A lies on C, and A' on C'.
(a) If P lies inside C, and P' inside C',
then there are precisely two projective transformations which map
C to C', A to A', and P to P'.
(b) If P lies outside C, but not on the tangent to C at A, and
P' lies outside C', but not on the tangent to C' at A',
then there are precisely two projective transformations which map
C to C', A to A', and P to P'.
proof
The restriction in (b) is necessary. If P is on the tangent to C at A, then any
projective transformation t will map P to P' on the tangent to t(C) at t(A).
If the polar of P meets C again at B, then B' = t(B) will be the second
intersection of t(C) with the polar of P'. By the three point theorem, we
can map C to a p-conic C' with A,B and any other p-point on C mapping to
designated points on C'. Hence there will be infinitely many maps taking
P to P'
and A to A'.
Note that this is weaker than the result for p-points on the p-conics, where we can
specify the images of three points (see the three point theorem).
The theorem has some fascinating consequences.
First, observe that, if we take C = C', then we get results about those projective
transformations which map C to itself, i.e. about S(C,P(2)), the group of projective
symmetries of C.
By the interior-exterior theorem,
tεS(C,P(2)) maps D, the
interior of C to itself.
Thus S(C,P(2)) is a group of transformations of
D and so,
according to the Klein
philosophy, defines a geometry on D. Since S(C,P(2))
is a subgroup of P(2), this
new geometry will be related to projective geometry.
As an example of a theorem in such a geometry, we have
A transitivity theorem
If p-points P,Q lie in D the interior of a p-conic C, then
there is
an element
of S(C,P(2))
which maps P to Q.
proof
Take a p-point A on C.
By the two point theorem, there are two projective transformations
which map C to C - so are in S(C,P(2)) - and map A to A and P to Q.
Either of these will do.
Although each p-conic gives a geometry in this way, the geometries from different
p-conics are isomorphic since any two p-conics are projective congruent, and their
projective symmetry groups are conjugate in P(2). Thus, we have essentially one
geometry. In fact, we have already met this geometry - it is hyperbolic
geometry!
The connection is explained on the projective and hyperbolic geometry page.
As a second application, recall that we have met one and two point theorems for the
ellipse, hyperbola and parabola in affine geometry. The proofs involved study of the
structure of the elements of the respective affine symmetry groups.
We can prove the results for the ellipse and hyperbola in a
unified way,
as well as that
for the parabola. The proofs do not involve detailed knowledge of the
relevant groups.
The details are given on the affine and projective symmetry page.
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