Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace (a geometric line) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.
Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace (a geometric line) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.