chapter  6
Mesh Parameterization
ByBruno Lévy
Pages 12

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. The chapter presents some basic notions of topology that characterize the class of surfaces that admit a parameterization. It focuses 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. The chapter outlines the main argument of the simple and elegant proof of Tutte's theorem by Gortler, Gotsman, and Thurston. It is remarkable that their proof solely uses basic topological notions together with a counting argument. The chapter mentions the importance of the weights in the quality of the result, and demonstrates how mean value coordinates can be used to reduce the distortions.