5 Interpolate Data ! onto Surface ! Data on 3-D Grid from ! Electromagnetic Code ! Compute On-Axis ! Gradients Using ! Surface Data ! and Inverse ! Laplacian Kernels ! Compute Vector  ! or Scalar Potential  Expansion from ! On-Axis Gradients ! Compute Design ! Trajectory  z d ! and Transfer ! Map  M ! FIG. 2: Calculation of realistic design orbit z d and its associated realistic transfer map M based on data provided on a 3-dimensional grid for a real beam-line element. Only a few points on the 3-dimensional grid are shown. In this illustration, data from the 3-dimensional grid is interpolated onto the surface of a cylinder with circular cross section, and this surface data is then processed to compute the design orbit (trajectory) and the associated transfer map about the design orbit. The use of other surfaces is also possible, and may offer various advantages.