Isotropic line

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}1&i&aend{bmatrix}.begin{bmatrix}xyzend{bmatrix} = 0 or begin{bmatrix}1&-i&aend{bmatrix}.begin{bmatrix}xyzend{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
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