Isotropic line

From Wikipedia, the free encyclopedia

An isotropic line or null line is a line in the complex projective plane with slope $i$ or $- i .$

[edit] Equation

All isotropic lines have equations of the following form:

y = i x + a z

y = - i x + a z,

or, in matrix-notation,

$\begin{bmatrix}1&i&a\end{bmatrix}.\begin{bmatrix}x\\y\\z\end{bmatrix} = 0$ or $\begin{bmatrix}1&-i&a\end{bmatrix}.\begin{bmatrix}x\\y\\z\end{bmatrix} = 0.$

[edit] Properties

An isotropic line is perpendicular to itself
The euclidean distance between two points on an isotropic line is zero (hence null line)
The union of two conjugate isotropic lines is a circle
The point at infinity of an isotropic line is always one of the two circular points at infinity
If an isotropic line is rotated 90 degrees its image is itself