Clinical dose calculations are often performed by scaling distances from a dose distribution measured in one medium to calculate the dose in another. These perturbation calculations have the mathematical form of a mapping. In this paper we identify five conditions required for particle transport to reduce to this form and develop a new mapping for electrons which approximately satisfies these conditions. This continuous scattering mapping is based on two parameters, the scattering power of the medium which determines the shape of the scaling paths, and the stopping power of the medium which determines where the energy is deposited along these paths. Pencil beam dose distributions are calculated with EGS4 in one medium and mapped to other media. The resultant distributions are compared with EGS4 calculations done directly in the second medium. The accuracy of the mapping algorithm is shown to be superior to both linear density scaling and the MDAH electron pencil beam algorithm [Kenneth R. Hogstrom, Michael D. Mills, and Peter R. Almond, “Electron beam dose calculations,” Phys. Med. Biol. 26, 445–459 (1981)] for pencil beams in homogeneous media and inhomogeneous phantoms (both slab and nonslab geometries) for a variety of materials of clinical interest.