Two optical design scenarios—imaging and illumination—were investigated for their use of Cartesian- and polar-based functions to generate freeform optical surfaces. The imaging scenario investigated a single-element, refracting freeform surface that converts an on-axis object field to an off-axis image point. XY polynomials (Cartesian but not orthogonal) and Zernike polynomials (Polar and orthogonal) were the two different function sets used to manipulate the surfaces to achieve the freeform imaging scenarios. The investigation discovered that the results between both function sets did not differ enough to single out a more effective surface type. However, the results did indicate that the Zernike function set typically required fewer coefficients to converge on an optimal imaging solution. The illumination scenario utilized an architectural lighting situation surrounding the Rothko exhibit for Green on Blue at the University of Arizona Museum of Art. The source location was fixed to the light track in the exhibit space and pointed in many different orientations towards the painting. For each orientation, a point cloud of a freeform optical surface was generated such that the painting surface was illuminated with uniform and low-level light. For each of these generated point clouds, a Legendre (Cartesian and orthogonal) and a Zernike (polar and orthogonal) fitting function was applied, and the convergence results were compared. In general, it was found that, after the 20th included fit term, the Legendre function resulted in a smaller RMS fit error than the Zernike function. However, if the light source was pointed near the center of the painting, the Zernike function converged on a solution with fewer fit terms than Legendre. Amidst the imaging scenario, a definition for the extent to which a surface was freeform, or the "freeformity", was given. This definition proved to be an effective solution when the image size was compared for an F/3.33, F/4, F/5, and F/6.67 system for a range of different image focusing heights: the image size trends for each F-number overlapped, indicating a universal freeform term. In addition, a recursive formula for Cartesian Zernike polynomials was defined, which was used to generate an infinite number of Zernike terms using one single recursive expression.