There Types Of Generalized Barycentric Coordinates With Non-negativity And Smoothness On Planar Polygons

The construction and analysis of generalized barycentric coordinates have al-ways been the attracting topics of geometric modeling in the field of computer-aided geometric design.Currently,there are a variety of construction methods for gener-alized barycentric coordinates.Unfortunately,for arbitrary simple planar polygons,the construction of generalized barycentric coordinates is still an open and chal-lenging task.To the best of our knowledge,the non-negativity and smoothness are the most desirable properties when measure a type of coordinates.In this pa-per,we mainly present the three construction methods of generalized barycentric coordinates with non-negativity and smoothness in the plane.a)Recently,although the construction method with the specific expression as the starting point has been widely proposed,it has not been very satisfactory.By changing the polygon topology,we propose a new way to transform a complex polygon into a visible polygon,and then calculate the coordinates using mean value coordinates and the mapping relationship.The calculation of our coordinates is simple,stable and robust.And it satisfies the good properties of non-negativity and smoothness.b)For polygonal domain,positive Gordon Wixom coordinates extend Gordon Wixom coordinates interpolation,which is non-negative but not smooth.We con-struct a new and smooth coordinate which the given polygon is replaced with an active Bezier curve or B-spline curve for each inner point.Thus the coordinates can achieve required continuity by adjusting the continuity of the active curves.c)Positive mean value coordinates are non-negative without the smoothness which are same as positive Gordon Wixom coordinates.We present a new construc-tion of coordinates for arbitrary simple polygons based on visible curves and mean value coordinates,which are non-negative,smooth and locally supported.Then numerical examples verify the locality of the coordinates.