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.