Least Square Coordinates

Generalized barycentric coordinates provide a convenient and efficient way to rep-resent a point inside the domain as an affine combination of control points and then are used to interpolate data values on domain bounded by polygons(polyhedra).Nowadays a few kinds of generalized barycentric coordinates have been proposed for polygon and polyhedra.In this paper,we introduce least square coordinates(LSC)based on t2 regu-larization model,which are well defined for polytopes as well as polytopes attached with lower-dimensional elements,which makes LSC extremely useful in shape deformation.We solve LSC by the Alternating Direction Method of Multipliers(ADMM)and make LSC non-negative.Furthermore,we do not partition the polytope and so LSC can be extended to higher dimensions easily.we present the details of LSC for 2D and 3D.For the 2D case,given a polygon and a interior point,the vertices of the polygon are projected onto the unit circle centred at the interior point and then the angle bisector between the two adjacent points is calculated.By ADMM solving the l2 regularization model satisfying the constraints of reproduction,partition of unity and non-negativity,the coordinate functions with non-negativity are obtained.For the 3D case,given a polyhedron with triangular facets and a interior point,the vertices of the polyhedron are projected onto the unit sphere centred at the interior point and the average unit normal vector of each spherical triangle is calculated.By ADMM solving the constrained variational model based on l2 regularization in 3D,the coordinate functions with non-negativity are obtained.Extensive experiments on polygons and polyhedra demonstrate that LSC are smooth.