model. The purpose of this project was to develop a versatile model (from here on referred to as the *map model*) to accurately represent all three components of an arbitrary physical LGB magnet, and to compare the new model to the stack representations by means of charged particle tracking simulations.

Developing a Versatile Model
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Choosing a Class of Model
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Figures 1 and 2 depict CAD models for two different physical LGB prototypes along with their complete magnetic field profiles and isolated on-axis fields. It was our task to accurately model the field throughout the entire volume of the beam pipe, taken as a $1$ cm radius cylinder along the z-axis. There are a myriad of ways to model a vector valued function. The simplest method might be to plot each component, assign it a functional form by eye, and adjust the parameters of that functional form until the best fit is found. For example, looking to Figure 1(b), it might make sense to model the $z$ dependence of the $B_y$ component with a Gaussian function. While this model may be adequate for this particular magnetic field map, it might be entirely inappropriate for another. For instance, consider the 