Linear parameterization of 3D meshes with disk to-pology is usually performed using the method of barycen-tric coordinates pioneered by Tutte and Floater. This im-poses a convex boundary on the parameterization which can significantly distort the result. Recently, several methods showed how to relax the convex boundary re-quirement while still using the barycentric coordinates formulation. However, this relaxation can result in other artifacts in the parameterization. In this paper we explore these methods and give a general recipe for "natural" boundary conditions for the family of so-called "three point" barycentric coordinates. We discuss the shortcom-ings of these methods and show how they may be rectified using an iterative scheme or a carefully crafted "virtual boundary". Finally, we show how these methods adapt easily to solve the problem of constrained parameterization.