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Aprojective cone (or justcone) inprojective geometry is theunion of alllines thatintersect aprojective subspaceR (the apex of the cone) and an arbitrarysubsetA (thebasis) of some othersubspaceS,disjoint fromR.

In the special case thatR is a singlepoint,S is aplane, andA is aconic section onS, the projective cone is aconical surface; hence the name.

LetX be a projective space over some fieldK, andR,S be disjoint subspaces ofX. LetA be an arbitrary subset ofS. Then we defineRA, the cone with topR and basisA, as follows :

• WhenA is empty,RA =A.
• WhenA is not empty,RA consists of all thosepoints on a line connecting a point onR and a point onA.

• AsR andS are disjoint, one may deduce fromlinear algebra and the definition of a projective space that every point onRA not inR orA is on exactly one line connecting a point inR and a point inA.
• (RA)${\displaystyle \cap }$S =A
• WhenK is thefinite field of orderq, then${\displaystyle |RA|}$ =${\displaystyle q^{r+1}}$${\displaystyle |A|}$ +${\displaystyle \frac{q^{r+1}-1}{q-1}}$, wherer = dim(R).

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

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