ABSTRACT
This chapter focuses on the special case of planar shape deformation, where the barycentric map can be interpreted as a complex-valued function. It generalizes barycentric coordinates from real to complex-valued functions and introduces the notion of complex barycentric coordinates. Complex coordinates can lead to planar barycentric maps with unique shape preserving properties. The chapter provides a general construction for complex barycentric coordinates and shows how to design custom-made complex coordinates. It derives holomorphic coordinates, relying on the rich theory of complex analysis, and explores their intimate relation to conformal maps. Interactive shape deformation is a fundamental problem in computer graphics and geometry processing. It is essential for designing and modeling variations of shapes and for generating believable animation sequences. Another requirement is that the underlying algorithms need to be relatively efficient to compute. This allows the users to explore the space of valuable deformations interactively and to reach their envisioned artistic design by trial and error.
