In the algebra pages, we obtained algebraic descriptions of p-conics, poles and polars.
We also have an algebraic description of projective transformations. We now show that
projective transformations preserve each type of object.
Suppose that C is a projective conic, and t a projective transformation, then
(1) t(C) is a projective conic, and
(2) if L is the polar of P with respect to C, then
t(L) is the polar of t(P) with respect to t(C).
The proof establishes the equations of various images, but these should be seen as the
means of establishing the results, and not of interest in themselves.
Of course, if P is a p-point on the p-conic C, then the polar of P with respect to C, then
the polar is the tangent to C at P. This yields the
In the notation of the theorem, if L is the tangent to C at P, then
t(L) is the tangent to t(C) at t(P).
Part (1) can be used, with the three points theorem, to give another proof of the fact that
the p-conics form a congruence class in projective geometry. The three points theorem
shows (among other things) that every p-conic is projective congruent to the standard
p-conic C0 : xy+yz+zx = 0. Thus the p-conics belong to a
single congruence class. The
present theorem shows that any member of the congruence class of a p-conic is a p-conic.
Thus, the p-conics constitute an entire class (and not just part of one).
The three point theorem also shows that, if P and Q lie on a p-conic C, then there exist
projective transformations which map C to C and P to Q. We now discuss the existence
of transformations mapping C to C and a p-point R, not on C to a p-point S. Obviously,
this is impossible if S lies on C. We shall see that just requiring that S is not on C is not
Our investigation of this problem leads to the concept of the interior of a p-conic.
Before embarking on this, we consider the general problem of betweenness and interiors.