Editing Homogeneous coordinates (section)

==Alternative definition==
Another definition of the real projective plane can be given in terms of [[equivalence class]]es. For non-zero elements
of <math>\mathbb{R}^3</math>, define {{nowrap|<math>(x_1, y_1, z_1) \sim (x_2, y_2, z_2)</math>}} to mean there is a
non-zero <math>\lambda</math> so that {{nowrap|<math>(x_1, y_1, z_1) = (\lambda x_2, \lambda y_2, \lambda
    z_2)</math>}}. Then <math>\sim</math> is an [[equivalence relation]] and the projective plane can be defined as the
equivalence classes of {{nowrap|<math>\mathbb{R}^3 \setminus \left\{0\right\}.</math>}} If {{nowrap|<math>(x, y, z)</math>}} is
one of the elements of the equivalence class <math>p</math> then these are taken to be homogeneous coordinates of <math>p</math>.

Lines in this space are defined to be sets of solutions of equations of the form {{nowrap|<math>ax + by + cz =
    0</math>}} where not all of <math>a</math>, <math>b</math> and <math>c</math> are zero. Satisfaction of the condition {{nowrap|<math>ax + by + cz =
    0</math>}} depends only on the equivalence class of {{nowrap|<math>(x, y, z),</math>}} so the equation defines a set
of points in the projective plane. The mapping {{nowrap|<math>(x, y) \rightarrow (x, y, 1)</math>}} defines an inclusion from the
Euclidean plane to the projective plane and the complement of the image is the set of points with {{nowrap|<math>z =
    0</math>}}. The equation {{nowrap|<math>z = 0</math>}} is an equation of a line in the projective plane
([[Homogeneous coordinates#Line coordinates and duality|see definition of a line in the projective plane]]), and is
called the line at infinity.

The equivalence classes, <math>p</math>, are the lines through the origin with the origin removed. The origin does not really play an
essential part in the previous discussion so it can be added back in without changing the properties of the projective
plane. This produces a variation on the definition, namely the projective plane is defined as the set of lines in
<math>\mathbb{R}^3</math> that pass through the origin and the coordinates of a non-zero element {{nowrap|<math>(x, y, z)</math>}}
of a line are taken to be homogeneous coordinates of the line. These lines are now interpreted as points in the
projective plane.

Again, this discussion applies analogously to other dimensions. So the projective space of dimension n can be defined as
the set of lines through the origin in <math>\mathbb{R}^{n+1}</math>.<ref>For the section:
    {{harvnb|Cox|Little|O'Shea|2007|pp=360&ndash;362}}</ref>