A plane conic can be obtained as (the nonideal points) of an
embedding of a pconic.
Suppose that C is a pconic, and that L is a pline. In RP^{2}, L
is represented by a plane
Π_{0} through the origin. If Π is any plane parallel to Π_{0}, then C
embeds in the extended
plane Π as the plane conic C(Π) = C_{n}Π together with at most two
ideal points. The latter
correspond to the intersections of C with L = Π_{0}. 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 pline.
If Π_{0} 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 ppoints 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.

