Isotropic line

In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form.

Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point (α, β) that depend on the imaginary unit i:

Laguerre then interpreted these lines as geodesics:

In the complex projective plane, points are represented by homogeneous coordinates ${\displaystyle (x_{1},x_{2},x_{3})}$ and lines by homogeneous coordinates ${\displaystyle (a_{1},a_{2},a_{3})}$. An isotropic line in the complex projective plane satisfies the equation:

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