Transfinite barycentric coordinates

In this chapter we study properties of transfinite barycentric interpolation schemes, which can be considered as continuous counterparts of generalized barycentric coordinates and are currently a subject of intensive research due to their fascinating mathematical properties and numerous applications in computational mechanics [170, 344], computer graphics [247, 248], and geometric modeling [93, 145, 227, 413] (see also references therein). We start from a general construction of transfinite barycentric coordinates, which is obtained as a simple and natural generalization of Floater’s mean value coordinates [144, 217], and investigate the properties of the continuous analogues of three-point coordinates [149]. We discuss the Gordon-Wixom interpolation approach [171] and consider its generalizations and modifications [37, 38]. Finally we introduce generalized mean value potentials and demonstrate how they can be used for distance function approximation purposes [40].