CHAPTER III: CONICS AND QUADRICS - OCW UPM

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Definition. Given a plane π of the space P 3 , we call pole of the plane π with
respect to the quadric Q to the point whose polar plane is π; this is, π P = π.
If the equation of the plane π is
then π P = π if and only if
π ≡ u 0 x 0 + u 1 x 1 + u 2 x 2 + u 3 x 3 = U T X = 0,
with U = (u 0 , u 1 , u 2 , u 3 ) and X = (x 0 , x 1 , x 2 , x 3 ),
P T AX = U T X, for every X ∈ P 3
equivalently,
P T A = U T ⇐⇒ AP = U.
And if the quadric Q is not degenerate (therefore, det A ≠ 0), then P =
A −1 U.