Plane at infinity
Encyclopedia
In projective geometry
, the plane at infinity is a projective plane
which is added to the affine 3-space in order to give it closure of incidence
properties. The result of the addition is the projective 3-space,
. If the affine 3-space is real, 
, then the addition of a real projective plane
at infinity produces the real projective 3-space 
.
s are equivalent: S2/{1,-1} where the quotient is understood as a quotient by a group action (see quotient space
).
Using homogeneous coordinates
, any point on affine 3-space can be represented as (X:Y:Z:1). Then, any point on the plane at infinity can be represented as (X:Y:Z:0). The points on the plane at infinity seem to have three degrees of freedom, but homogeneous coordinates are equivalent up to
any rescaling:
so that the coordinates (X:Y:Z:0) can be normalized, thus reducing the degrees of freedom to two (thus, a surface, namely the real projective plane).
Proposition: Any line which passes through the origin
(0:0:0:1) and through a point (X:Y:Z:1) will pass the plane at infinity through point (X:Y:Z:0).
Proof: A line which passes through points (0:0:0:1) and (X:Y:Z:1) will consist of points which are linear combinations of the two given points:
Since all of these points belong to the line, for any n, then letting
(infinity belongs to the projective number line) yields 
, as required. Q.E.D.
Any pair of parallel lines in 3-space will intersect each other at a point on the plane at infinity. Also, every line in 3-space intersects the plane at infinity at a unique point. This point is determined by the direction—and only by the direction—of the line. To determine this point, "draw" a line parallel to the given line, but passing through the origin. Then choose any point, other than the origin, on this second line. If the homogeneous coordinates of this point are (X:Y:Z:1), then the homogeneous coordinates of the point at infinity through which the first and second line both pass is (X:Y:Z:0).
Example: the first line passes through points (0:0:1:1) and (3:0:1:1). The second line passes through points (0:0:0:1) and (3:0:0:1). The second line passes the plane at infinity through point (3:0:0:0). But the first line also passes through this point:
Letting
,
since points at infinity are dominant. ■
Any pair of parallel planes in 3-space will intersect each other at a projective line (a line at infinity) on the plane at infinity. Also, every plane in 3-space intersects the plane at infinity at a unique line. This line is determined by the direction—and only by the direction—of the plane.
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, the plane at infinity is a projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
which is added to the affine 3-space in order to give it closure of incidence
Incidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
properties. The result of the addition is the projective 3-space,
. If the affine 3-space is real, , then the addition of a real projective planeReal projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
at infinity produces the real projective 3-space .Discussion
Note that since the (real) plane at infinity is a projective plane, it is homeomorphic to the surface of a "sphere modulo antipodes", i.e. a sphere in which antipodal pointAntipodal point
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....
s are equivalent: S2/{1,-1} where the quotient is understood as a quotient by a group action (see quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
).
Using homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
, any point on affine 3-space can be represented as (X:Y:Z:1). Then, any point on the plane at infinity can be represented as (X:Y:Z:0). The points on the plane at infinity seem to have three degrees of freedom, but homogeneous coordinates are equivalent up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
any rescaling:
-
,
so that the coordinates (X:Y:Z:0) can be normalized, thus reducing the degrees of freedom to two (thus, a surface, namely the real projective plane).
Proposition: Any line which passes through the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
(0:0:0:1) and through a point (X:Y:Z:1) will pass the plane at infinity through point (X:Y:Z:0).
Proof: A line which passes through points (0:0:0:1) and (X:Y:Z:1) will consist of points which are linear combinations of the two given points:
Since all of these points belong to the line, for any n, then letting
(infinity belongs to the projective number line) yields , as required. Q.E.D.Q.E.D.
Q.E.D. is an initialism of the Latin phrase , which translates as "which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out —...
Any pair of parallel lines in 3-space will intersect each other at a point on the plane at infinity. Also, every line in 3-space intersects the plane at infinity at a unique point. This point is determined by the direction—and only by the direction—of the line. To determine this point, "draw" a line parallel to the given line, but passing through the origin. Then choose any point, other than the origin, on this second line. If the homogeneous coordinates of this point are (X:Y:Z:1), then the homogeneous coordinates of the point at infinity through which the first and second line both pass is (X:Y:Z:0).
Example: the first line passes through points (0:0:1:1) and (3:0:1:1). The second line passes through points (0:0:0:1) and (3:0:0:1). The second line passes the plane at infinity through point (3:0:0:0). But the first line also passes through this point:
Letting
,
since points at infinity are dominant. ■
Any pair of parallel planes in 3-space will intersect each other at a projective line (a line at infinity) on the plane at infinity. Also, every plane in 3-space intersects the plane at infinity at a unique line. This line is determined by the direction—and only by the direction—of the plane.