In projective geometry, all conics are congruent, so the projective symmetry
groups are conjugate subgroups of P(2). In fact, these groups are related to
the hyperbolic group. This is done in the models page.
Here, we prove a stronger form of the theorem that all conics are projective
congruent. This is the analogue of the one- and two-point theorems
for
affine geometry.
We will choose as our standard projective conic the conic C0 : xy+yz+xz = 0.
This has the advantage that it is symmetric in x, y and z. It also contains the
standard points X[1,0,0], Y[0,1,0] and Z[0,0,1], but not U[1,1,1].
The Three Point Theorem
If P,Q and R are p-points on the projective conic C, then there is a unique
projective transformation which maps C to C0 and maps the points P,Q,R
to X,Y,Z respectively.
proof
Of course, since all conics are projective congruent to C0, we have the
Theorem All conics are projective congruent.
This result has the consequence that any theorem in projective geometry
involving three (or more) points on a conic may be reduced to a problem
about the standard conic and the points X,Y,Z. We simly apply the map v
to get a figure in which the conic is the standard conic, do any necessary
calculations for this, and then recover the result in general
by applying v-1.
Some examples :
The Three Tangents Theorem
Suppose that C is a conic. If A,B,C are p-points such that
AB touches C at P,
BC touches C at R and CA touches C at Q,
then AR, BQ and CP are concurrent.
proof
The Inscribed Quadrilateral Theorem
If A,B,C,D are distinct p-points on a p-conic C, and
AB meets CD in P, BC meets AD in Q, and CA meets BD in R,
then QR is the polar of P with respect to C.
proof
Note that this allows us to draw the polar of a point P not on a conic C :
- Draw two chords AB and CD through P
- Find the intersections Q of BC, AD, and R of CA, BD.
- The required polar is QR.
If the problem involves further points on the conic, the following result
allows us to describe the corresponding points on the standard conic.
The Parametrisation Theorem
If the p-point P lies on the conic C0 : xy+yz+zx = 0, and P ≠ Z,
then P has the form (t,1-t,t(t-1)) for some real t.
proof
A similar result holds for any conic C. We know that C is projective congruent
to C0, so there is a projective transformation t with
C = t(C0). Suppose that
t has matrix M. Each p-point on C has the form t(P) for some P on
C0. If P ≠ Z,
then P =[t,1-t,t(t-1)] for some t, and t(P) has the form M(t,1-t,t(t-1))T. Thus,
t(P) has the form (f(t),g(t),h(t)), where f,g,h are quadratic in t.
This leads to the description of the symmetry group of a projective conic.
Pascal's Theorem
If A,B,C.A',B',C' are p-points on a conic C, and AB', A'B meet at P
BC',B'C at Q, and CA',C'A at R, then P,Q,R are collinear.
proof
A related result, useful in proving other theorems is the five point theorem.
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