a geometric-based numerical solution of eikonal equation over a closed level curve

this paper presents a new numerical method for solution of eikonal equation in two dimensions.in contrast to the previously developed methods which try to define the solution surface by its level sets(contour curves), the developed methodology identifies the solution surface by resorting to its characteristics. the suggested procedure is based on the geometric properties of the solution surface and does not require any mesh for computation. it works well in finding the ridge of the solution surface as well. in addition, the area of the surface and its corresponding volume can be easily determined via this method. three examples have been provided to demonstrate the capability of the suggested method in presenting these important features of the solution. the issue of convergence has also been investigated. it has been concluded that the suggested method works well in solving the eikonal equation in problems for which the direction of characteristics of the solution surface, and its area or volume underneath are quite important