Invariance Theorem 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).
proof
(1)
(2)
Now t(P) = [Ap], so its polar with respect to t(C) is
(Ap)TNx = 0,
We leave it as an exercise (along the lines of (1)) to show that, if K is
Thus t(L) is pTMA-1x = 0, so is the required polar.
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