Monotonicity Of Generalized Barycentric Coordinates On Two Types Of Convex Polygons

In recent years,with the need of engineering practice,The property of generalized barycentric coordinates is gradually concerned.For example,the generalized barycentric coordinates have locality,which makes the control vertices only affect the change of the points nearby,and the smoothness can make the coordinates change smoothly.In order to apply the generalized barycentric coordinates more widely,we need a deeper understanding of the generalized barycentric coordinates.This paper mainly studies the new properties of generalized barycentric coordinates,namely monotonicity.With the monotonous generalized barycentric coordinates,it can make the contour line without local extremum point.When applied to the deformation of the graphic image,the processing effect will not appear human trace.Firstly,this paper expounds the significance of the research on the generalized barycentric coordinates and the current research status in this field at home and abroad.Then the concept and basic properties of centroid coordinates are described,and the concept and properties of generalized barycentric coordinates on the polygons from polygonal coordinates on the triangle are introduced.Secondly,the concept of a kind of generalized barycentric coordinates in a convex quadrilateral is studied,and prove that the generalized barycentric coordinates on such convex quadrilateral have monotonicity through rigorous theoretical deduction.Namely,for any convex quadrilateral,the generalized barycentric coordinate is monotonically decreasing on the line of any vertex to the point on any boundary,that is,the generalized barycentric coordinate has monotonicity.Then,the definition of five point coordinates on convex polygons on the plane is described,and the numerical simulation is obtained by a large number of numerical simulations to show that the two kinds of five point coordinates are not monotonous.By finding a special convex polygon,it is proved that the two kinds of five point coordinates do not have monotonicity on the convex polygon.In other words,there is a convex polygon,which does not have monotonicity along the line of the specified vertex to the top of the fixed edge.Generally,the eneralized barycentric coordinate are different in different application scenarios,and then whether these generalized barycentric coordinates are still monotonous!For example,three point coordinates,Green coordinates,Poisson coordinates,etc,these problems will be further study in the future.