Supriya Ghosh (Editor)

Projective cone

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A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R.

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In the special case that R is a single point, S is a plane, and A is a conic section on S, the projective cone is a conical surface; hence the name.

Definition

Let X be a projective space over some field K, and R, S be disjoint subspaces of X. Let A be an arbitrary subset of S. Then we define RA, the cone with top R and basis A, as follows :

  • When A is empty, RA = A.
  • When A is not empty, RA consists of all those points on a line connecting a point on R and a point on A.

    • Properties

  • As R and S are disjoint, one may deduce from linear algebra and the definition of a projective space that every point on RA not in R or A is on exactly one line connecting a point in R and a point in A.
  • (RA) ∩ S = A
  • When K = GF(q), | R A | = q r + 1 | A | + q r + 1 − 1 q − 1 .

    • References

    Projective cone Wikipedia
    (Text) CC BY-SA


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