Characteristic surface data for the eikonal equation

A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe the wave-front singularities that are associated with their level surfaces (the characteristic surfaces or "big wave fronts"). (C) 1999 American Institute of Physics. [0022-2488(99)03301-0].