Barycentric coordinates have become a standard interpolation technique in Computer Graphics. These coordinates solve a boundary value interpolation problem and can be used to interpolate discrete scalar fields, vector fields or even multidimensional fields over irregular tessellations. While barycentric coordinates were first generalized in 1975 by Wachspress for Finite Element Analysis, the Graphics community has made heavy use of these coordinates for applications such as texturing mapping, polygonal rasterization, ray-intersection calculations and spline surfaces. More recently, new research in barycentric coordinates has led to additional applications in surface parameterization, solid texturing and surface deformation. Despite these advances, very few generalizations of barycentric coordinates exist today and most contain restrictions. For instance, many barycentric coordinate constructions require that the input shapes must be convex and, even then, the coordinates may not be well-defined everywhere. Currently, there is only one type of barycentric coordinates, called Mean Value Coordinates, defined for arbitrary, closed shapes. However, these coordinates still leave much to be desired as they can be negative (problematic for interpolation applications) and lack local control, thus producing artifacts.
Ideally, barycentric coordinates should have constant and linear precision (needed for applications such as deformation), produce smooth functions, have local influence and contain only positive values. No current analytic formulation contains all of these properties. This research is generalizing barycentric coordinate construction by investigating a novel approach to building barycentric coordinates that directly solves for coordinates that have these desirable properties. Once the theory is completed, the new barycentric coordinates will be applied to several application domains including: Boundary Value Interpolation, Surface Deformation, and Example-based Deformation Synthesis.
