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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*)*S*=*A* - When
*K*is the finite field of order*q*, then = + , where*r*= dim(*R*).

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