affine and projective symmetry groups of conics

A plane conic can be obtained as (the non-ideal points) of an embedding of a p-conic.
Suppose that C is a p-conic, and that L is a p-line. In RP2, L is represented by a plane
Π0 through the origin. If Π is any plane parallel to Π0, then C embeds in the extended
plane Π as the plane conic C(Π) = CnΠ together with at most two ideal points. The latter
correspond to the intersections of C with L = Π0. The plane conic is an ellipse if there
are no ideal points, a parabola if there is one - so L is a tangent to C, and a hyperbola
if there are two. Different choices of Π give similar C(Π), and these have conjugate
symmetry groups. See embedding of projective conics.

Affine transformations can be obtained from projective transformations fixing a p-line.
If Π0 and Π are as above, then the affine group for the plane Π is (isomorphic to) the
projective symmetry group S(L,P(2)). See the affine and projective group page.

Suppose that t is an affine symmetry of C(Π). Then t corresponds to an element t* of
S(L,P(2)). Since C can be recovered from C(Π), t* maps C to itself, i.e. t*εS(C,P(2)).
Thus, the affine symmetry group S(C(Π),A(2)) is isomorphic to the intersection of the
projective symmetry groups S(L,P(2)) and S(C,P(2)).
Now suppose that sεS(C,P(2) ), so s maps C to C. If P is the pole of L with respect to C,
then, by the invariance theorem, s maps L to L if and only if it maps P to P. Thus s
corresponds to an element of S(C(Π),A(2)) if and only if it maps C to C and P to P.

The one point theorem for a central conic
Suppose that C, Π and P are such that C(Π) is a central conic.
If A, A' are points of C(Π) then there are precisely two affine
transformations which map C(Π) to C(Π) and A to A'.

proof
We know that C(Π) is central if and only if P is not on C.
By the above discussion, the required affine transformations
arise from elements of S(C,P(2)) which fix P and map A to A'.
If P lies outside C, the polar of P meets C in two p-points R,S .
Then C(Π) is a hyperbola, with R,S embedding as the ideal
points corresponding to the asymptotes. As A is on C(Π) A ≠ R,S.
By the two point theorem, there are precisely two maps.

The two point theorem for a parabola
Suppose that C, Π and P are such that C(Π) is a parabola.
If A, B and A', B' are pairs of distinct points of C(Π) then
there are precisely two affine transformations which map
C(Π) to C(Π), A to A', and B to B'.

proof
We know that C(Π) is a parabola if and only if P is on C.
By the above discussion, the required affine transformations
arise from elements of S(C,P(2)) which fix P, map A to A' and
B to B'. By the three point theorem, there is exactly one
element of S(C,P(2)) which maps the list (P,A,B) to (P,A',B').
Thus, there is exactly one affine transformation of the required
type.

two point theorem page