A transformation property of the mapping between points on a ray path


  • Harry Gelman


In some applications of ray tracing, it is useful to know the behavior of certain ray-related quantities under a transformation of the variable of path parameterization. One such quantity is the Jacobian of the mapping between small regions surrounding two points on a ray path. A narrow bundle of rays emanating from a point source in an inhomogeneous, anisotropic propagation environment is considered. The mapping between a small region surrounding a reference point on an interior ray of the bundle, and the corresponding region surrounding an observation point on the same ray, is investigated. The Jacobian of the mapping is studied as a function of an arbitrary path parameter, and a new relation expressing the behavior of the Jacobian under a transformation of the parameter is derived.