Plane at Infinity
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High Quality Content by WIKIPEDIA articles! 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, P3. If the affine 3-space is real, mathbb{R}^3, then the addition of a real projective plane mathbb{R}P^2 at infinity produces the real projective 3-space mathbb{R}P^3. 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 points a...
High Quality Content by WIKIPEDIA articles! 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, P3. If the affine 3-space is real, mathbb{R}^3, then the addition of a real projective plane mathbb{R}P^2 at infinity produces the real projective 3-space mathbb{R}P^3. 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 points are equivalent: S2/{1,-1} (see quotient space). This spherical plane at infinity in a sense surrounds our usual affine 3-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).
