Definitions

# 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,

$begin\left\{bmatrix\right\}1&i&aend\left\{bmatrix\right\}.begin\left\{bmatrix\right\}xyzend\left\{bmatrix\right\} = 0$ or $begin\left\{bmatrix\right\}1&-i&aend\left\{bmatrix\right\}.begin\left\{bmatrix\right\}xyzend\left\{bmatrix\right\} = 0.$

## 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
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