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.

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.

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.

• 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) ∩ {\displaystyle \cap } S = A
• When K is the finite field of order q, then | R A | {\displaystyle |RA|} = q r + 1 {\displaystyle q^{r+1}} | A | {\displaystyle |A|} + q r + 1 − 1 q − 1 {\displaystyle {\frac {q^{r+1}-1}{q-1}}} , where r = dim(R).

• Cone (geometry)
• Cone (algebraic geometry)
• Cone (topology)
• Cone (linear algebra)
• Conic section
• Ruled surface
• Hyperboloid