Spline functions are a form of piecewise polynomials with many diverse uses and applications across several fields of study. A particular application concerns approximation of other continuous functions. Although many such approximation methods have already been described and analyzed, this thesis describes a previously less-studied method of approximation via the control polygon of the spline. By placing interpolation conditions on the control polygon, one can produce several well known approximation methods, as well as some possibly uncovered methods described within this thesis. These methods are also evaluated based on their accuracy and convergence rate, with several numerical experiments.