ABSTRACT
In this chapter introduces mesh parameterization, an application of generalized barycentric coordinates. By “unfolding” a surface onto a 2D space, mesh parameterization has many possible applications, such as texture mapping. In this chapter we present some basic notions of topology that characterize the class of surfaces that admit a parameterization. Then we focus on the specific case of a topological disk with its boundary mapped to a convex polygon. In this setting, Tutte’s barycentric mapping theorem not only gives sufficient conditions, but also a practical algorithm to compute a parameterization. We outline the main argument of the simple and elegant proof of Tutte’s theorem by Gortler, Gotsman, and Thurston [172]. It is remarkable that their proof solely uses basic topological notions together with a counting argument. Finally, we mention the importance of the weights in the quality of the result, and demonstrate how mean value coordinates can be used to reduce the distortions.
