Editing Homogeneous coordinates (section)

==Introduction==
The [[projective plane#Extended Euclidean plane|real projective plane]] can be thought of as the [[Euclidean geometry|Euclidean plane]] with additional points added, which are called [[point at infinity|points at infinity]], and are considered to lie on a new line, the [[line at infinity]]. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point {{nowrap|<math>(x, y)</math>}} on the Euclidean plane, for any non-zero real number <math>Z</math>, the triple {{nowrap|<math>(xZ, yZ, Z)</math>}} is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. In particular, {{nowrap|<math>(x, y, 1)</math>}} is such a system of homogeneous coordinates for the point {{nowrap|<math>(x, y)</math>}}.
For example, the Cartesian point {{nowrap|<math>(1, 2)</math>}} can be represented in homogeneous coordinates as {{nowrap|<math>(1, 2, 1)</math>}} or {{nowrap|<math>(2, 4, 2)</math>}}. The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.

The equation of a line through the origin {{nowrap|<math>(0, 0)</math>}} may be written {{nowrap|<math>nx + my = 0</math>}} where <math>n</math> and <math>m</math> are not both <math>0</math>. In [[Parametric equation|parametric]] form this can be written <math>x = mt,  y = -nt</math>. Let <math>Z = 1/t</math>, so the coordinates of a point on the line may be written {{nowrap|<math>(m/Z,  -n/Z)</math>}}. In homogeneous coordinates this becomes {{nowrap|<math>(m,  -n, Z)</math>}}. In the limit, as <math>t</math> approaches infinity, in other words, as the point moves away from the origin, <math>Z</math> approaches <math>0</math> and the homogeneous coordinates of the point become {{nowrap|<math>(m, -n, 0)</math>}}. Thus we define {{nowrap|<math>(m, -n, 0)</math>}} as the homogeneous coordinates of the point at infinity corresponding to the direction of the line {{nowrap|<math>nx + my = 0</math>}}. As any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates.

To summarize:
*Any point in the projective plane is represented by a triple {{nowrap|<math>(X, Y, Z)</math>}}, called 'homogeneous coordinates' or 'projective coordinates' of the point, where <math>X</math>, <math>Y</math> and <math>Z</math> are not all <math>0</math>. 
*The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
*Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same non-zero constant.
*When <math>Z</math> is not <math>0</math> the point represented is the point {{nowrap|<math>( X/Z, Y/Z)</math>}} in the Euclidean plane.
*When <math>Z</math> is <math>0</math> the point represented is a point at infinity.
The triple {{nowrap|<math>(0, 0, 0)</math>}} is omitted and does not represent any point. The [[Origin (mathematics)|origin]] of the Euclidean plane is represented by {{nowrap|<math>(0, 0, 1)</math>}}.<ref>For the section: {{harvnb|Jones|1912| pages= 120&ndash;122}}</ref>

===Notation===
Some authors use different notations for homogeneous coordinates which help distinguish them from Cartesian coordinates. The use of colons instead of commas, for example <math>(x:y:z)</math> instead of {{nowrap|<math>(x, y, z)</math>}}, emphasizes that the coordinates are to be considered ratios.<ref>{{harvnb|Woods|1922}}</ref> Square brackets, as in {{nowrap|<math>[x, y, z]</math>}} emphasize that multiple sets of coordinates are associated with a single point.<ref>{{harvnb|Garner|1981}}</ref> Some authors use a combination of colons and square brackets, as in <math>[x:y:z]</math>.<ref>{{harvnb|Miranda|1995}}</ref>