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 equationds2 = 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 real orthogonal geometry of Emil Artin, isotropic lines occur in pairs:
A non-singular plane which contains an isotropic vector shall be called a hyperbolic plane. It can always be spanned by a pair n, m of vectors which satisfy
We shall call any such ordered pair n, m a hyperbolic pair. If V is a non-singular plane with orthogonal geometry and n ≠ 0 is an isotropic vector of V, then there exists precisely one m in V such that n, m is a hyperbolic pair. The vectors xn and ym are then the only isotropic vectors of V.[3]
Relativity
Isotropic lines have been used in cosmological writing to carry light. For example, in a mathematical encyclopedia, light consists of photons: "The worldline of a zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line."[4] For isotropic lines through the origin, a particular point is a null vector, and the collection of all such isotropic lines forms the light cone at the origin.
O. Timothy O'Meara (1963, 2000) Introduction to Quadratic Forms, page 94
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