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 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.

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tidy | en | sembr

Footnotes

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