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:^[1]

First system: ${\displaystyle (y-\beta )=(x-\alpha )i,}$

Second system: ${\displaystyle (y-\beta )=-i(x-\alpha ).}$

Laguerre then interpreted these lines as geodesics:

An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the differential equation ds² = 0. On an arbitrary surface one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines.^[1]:90

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

$a_{3}(x_{2}\pm ix_{1})=(a_{2}\pm ia_{1})x_{2}.$

In terms of the affine subspace x₃ = 1, an isotropic line through the origin is

$x_{2}=\pm ix_{1}.$

In projective geometry, the isotropic lines are the ones passing through the circular points at infinity.

In geology, isotropic lines "separate mutually orthogonal principle trajectories on each side. In a plane-strain field, the strain is zero at isotropic points and lines, and they can be termed neutral points and neutral lines."^[3]

See also

• Light cone
• Null vector

References

1. Edmond Laguerre (1870) "Sur l’emploi des imaginaires en la géométrie", Oeuvres de Laguerre 2: 89
2. C. E. Springer (1964) Geometry and Analysis of Projective Spaces, page 141, W. H. Freeman and Company
3. Jean-Pierre Brun (1983) "Isotropic points and lines in strain fields", Journal of Structural Geology 5(3):321–7

• Pete L. Clark, Quadratic forms chapter I: Witts theory from University of Miami in Coral Gables, Florida.
• O. Timothy O'Meara (1963,2000) Introduction to Quadratic Forms, page 94