Free-boundary linear parameterization of 3D meshes in the presence of constraints

Linear parameterization of 3D meshes with disk topology is usually performed using the method of barycentrie coordinates pioneered by Tutte and Floater. This imposes a convex boundary on the parameterization, which can significantly distort the result. Recently, several methods showed how to relax the convex boundary requirement 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 shortcomings 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.