  In this paper, we introduce new generalized barycentric coordinates (coined
as {\em moment coordinates}) on nonconvex quadrilaterals and convex hexahedra
with planar faces. This work draws on recent advances in constructing
interpolants to describe the motion of the Filippov sliding vector field in
nonsmooth dynamical systems, in which nonnegative solutions of signed matrices
based on (partial) distances are studied. For a finite element with $n$
vertices (nodes) in $\mathbb{R}^2$, the constant and linear reproducing
conditions are supplemented with additional linear moment equations to set up a
linear system of equations of full rank $n$, whose solution results in the
nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral,
moment coordinates using signed distances are identical to mean value
coordinates. For signed weights that are based on the product of distances to
edges that are incident to a vertex and their edge lengths, we recover
Wachspress coordinates on a convex quadrilateral. Moment coordinates are also
constructed on a convex hexahedra with planar faces. We present proofs in
support of the construction and plots of the shape functions that affirm its
properties.
