| Any point on the plane can be linearly represented by the vertices of a trian-gle,the set of coefficients is called barycentric coordinates.Since the barycentric coordinates have a series of excellent properties such as normality,linear re-construction,Lagrange,and affine invariance,they are widely used in computer graphics,numerical computation and other disciplines or fields.In order to further meet,the needs of different applications,the definition of barycentric coordinates is extended to polygons,polyhedrons,or curved edges,this kind of barycen-tric coordinate is called generalized barycentric coordinates.Most of the existing generalized barycentric coordinates can only reach the first-order approximation accuracy,in applications such as function approximation and equation solving,people prefer generalized barycentric coordinates to have second-order or higher approximation accuracy.For example,barycentric coordinate with second-order approximation order can provide a more accurate approximation or faster con-vergence speed with the same number of slices.At present,only a few works have given methods to improve the approximation accuracy of generalized barycentric coordinates,these methods are generally only applicable to constructing second-order generalized barycentric coordinates on convex polygons.This paper pro-poses a method of constructing second-order generalized barycentric coordinates on general polygons,the second-order generalized barycentric coordinates have higher approximation accuracy on the basis of inheriting the excellent proper-ties of generalized barycentric coordinates.Starting from generalized generalized barycentric coordinates,we obtain high-order function space by multiplying any two of the generalized barycentric coordinates,and then reduce the dimension of the function space by appropriate linear combinations,finally,the property of La-grange interpolation is ensured through Lagrangianization.We have proved that the constructed coordinates satisfies a series of properties such as Lagrangian,normality,linear reconstruction,and quadratic reconstruction.Finally,we study the application in image deformation of the second-order generalized barycentric coordinates constructed in this paper.We studied the conditions for ensuring one-to-one mapping when moving vertices of a polygon,under this condition,we performed several experiments comparing the deformation of mean value coordi-nates with the results of our experiments,the results show that our deformation effect is more abundant. |
