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(also characteristic function, characteristic), in geometrical optics, a function that specifies the optical path length of a ray of light between two arbitrary points, one (point A) in object space and the other (point A’) in image space (seeIMAGE, OPTICAL).

Depending on the choice of parameters, several different types of eikonals are distinguished. The point characteristic, which is also referred to as Hamilton’s characteristic or the characteristic function of Hamilton, is the eikonal of an optical system with respect to the coordinates x, y, and z of A and x’, y’, and z’ of A’. The angle characteristic, which was discovered by H. E. Bruns, is the eikonal of an optical system with respect to a ray’s direction cosines μ, v and μ′, v′. An example of a more complex eikonal is the characteristic function that was discovered by K. Schwarzschild.

In the designing of an optical system, the use of an eikonal makes it possible to obtain expressions for the system’s transverse aberrations by differentiating the eikonal with respect to certain parameters. Functions called eikonals are widely used in charged-particle optics in the framework of the general analogy that exists between charged-particle and classical optics. Such functions are also used to describe particle and wave scattering processes in cases where analogies with optics arise. Examples of the techniques employed in such cases include the Hamiltonian method and the eikonal, or slowly varying amplitude, approximation in quantum mechanics and quantum field theory.


Born, M., and E. Wolf. Osnovy optiki. Moscow, 1973. (Translated from English.)
Kel’man, V. M., and S. Ia. Iavor. Elektronnaia optika, 3rd ed. Leningrad, 1968.
Goldberger, M., and K. Watson. Teoriia stolknovenii. Moscow, 1967. (Translated from English.)
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For example, in order to propagate half discs or half spheres in the increasing Ox direction (noted Ox+), the Eikonal equation solved into the Fast Marching (algorithm 12 and 13) is modified as Eq.
the eikonal corresponding to a translation, as expected.
The most general treatment for point source in 3D is given by the eikonal equation [6, p.