Isotropic line

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

Equation

All isotropic lines have equations of the following form:

y = ix + az or y = − ix + az,

or, in matrix-notation,

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

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