CHAPTER III: CONICS AND QUADRICS - OCW UPM

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
4.3.1 Tangent variety to a quadric
Definition. The tangent variety to a quadric Q in a point P ∈ P 3 , is the set of
points X ∈ P 3 such that the line that joins P and X is tangent to the quadric
Q; this is,
Observations.
T P Q = {X ∈ P 3 | line XP is tangent to Q}
= {X ∈ P 3 | ∆ = f(P, X) 2 − ω(P )ω(X) = 0}
= {X ∈ P 3 | f(P, X) 2 = ω(P )ω(X)}.
1. T P Q is a degenerate quadric which has P as singular point.
2. If P ∈ Q is a regular point, then
T P Q = {X ∈ P 3 | f(P, X) 2 = 0}
= {X ∈ P 3 | P t AX = 0}
is a plane, called the tangent plane to Q in P . In fact, it is the polar plane
of the point P ; this is, T P Q = π p .