A projective cone (or just cone) in
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of order ''q'', then
=
+
, where ''r'' = dim(''R'').
See also
*
Cone (geometry)
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
*
Cone (algebraic geometry) In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme ''X'', the relative Spec
:C = \operatorname_X R
of a quasi-coherent graded ''O'X''-algebra ''R'' is called the cone or affine cone of ''R''. Simil ...
*
Cone (topology)
In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching ''X'' into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by CX or by \operatorname(X).
...
*
Cone (linear algebra)
*
Conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
*
Ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
*
Hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
Projective geometry
{{geometry-stub