Group classification of the eikonal equation for a 3-dimensional inhomogeneous medium

The equation $(\nabla\psi)^2=1/v^2(x,y,z)$, known as the eikonal equation, is studied. This is the characteristic equation for the wave equations in an inhomogeneous medium, which plays a central role in the description of the geometry of the rays and the wave fronts. A full geometric classification of the family of eikonal equations is carried out (an equation is determined by the function $v(x,y,z)$, which has the meaning of the propagation velocity of a perturbation in the medium). In the cases of equations with linear or quadratic velocity function $v(x,y,z)$, explicit solutions – point source eikonals – are presented and the geometry of the rays is completely described.