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.
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) ∩ {\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).
See also Cone (geometry) Cone (algebraic geometry) Cone (topology) Cone (linear algebra) Conic section Ruled surface Hyperboloid
References