Editing Homogeneous coordinates (section)

==Change of coordinate systems==
Just as the selection of axes in the Cartesian coordinate system is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore, it is useful to know how the different systems are related to each other.

Let {{nowrap|<math>(x, y, z</math>)}} be homogeneous coordinates of a point in the projective plane. A fixed matrix
<math display="block">A=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix},</math>
with nonzero [[determinant]], defines a new system of coordinates {{nowrap|<math>(X, Y, Z)</math>}} by the equation
<math display="block">\begin{pmatrix}X\\Y\\ Z\end{pmatrix}=A\begin{pmatrix}x\\y\\z\end{pmatrix}.</math>
Multiplication of {{nowrap|<math>(x, y, z)</math>}} by a scalar results in the multiplication of {{nowrap|<math>(X, Y, Z)</math>}} by the same scalar, and <math>X</math>, <math>Y</math> and <math>Z</math> cannot be all <math>0</math> unless <math>x</math>, <math>y</math> and <math>z</math> are all zero since <math>A</math> is nonsingular. So {{nowrap|<math>(X, Y, Z)</math>}} are a new system of homogeneous coordinates for the same point of the projective plane.