Projective quadric

Orthogonality by a quadric

Let be a non-singular quadric in belonging to the quadratic form and let

be the corresponding bilinear form.1 Then for #m/thm/geo

  1. Let be an arbitrary point. Then iff the line is a secant of , i.e. .
  2. Let . Then iff the line is a tangent of at , i.e. .
  3. Let . Then iff the line is a line of , i.e. completely contained in .
Proof

Any point other than on the line has homogenous coördinates .

So if then , giving cases ^2 and ^3. If is arbitrary and , then iff , giving case ^1.

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2020. Finite geometries, ¶4.50, pp. 104–105 ↩