Domains with curvature and sharp corners (cusps) are prone to geometric approximation errors, due to which they are sensitive to the order and type of fitting polynomials. Approximating such geometries using lower-order polynomials can lead to errors in the solutions. In this article, we present a methodology to approximate complex boundaries. This method uses various types and orders of shape functions (Lagrange, B-Splines, and NURBS) along with the associated adaptive mesh size to optimise and approximate a boundary within the desired tolerance. We have adopted projection based curve fitting by minimizing of H1-norm and solving a system of linear equations to get the desired fit. A comparative study has been presented on curve fitting by projection using different basis functions.