Blended Coordinated Based On Derived Polygons And Its Promotion

The generalized barycentric coordinates have many excellent properties,and provide a simple representation method for linear interpolation at the vertices of triangles.Therefore,it is widely used in graphics deformation,graphics rendering,mesh editing,image interpolation and other fields.It is an important mathematical tool in computer graphics and computer-aided geometric design.With the development of science and technology and the needs of engineering practice,more and more scholars have studied the barycentric coordinates and achieved many results.Triangulating the initial polygon,and using the triangular mesh to derive the point polygons,the edge polygons and the face polygons of the triangle,and according to the existing barycentric coordinates,blended barycentric coordinates based on derived polygons is proposed.First,deriving a triangle within mesh to obtain the derived polygons.Then,derived polygon's vertex which is about the initial polygons'vertices can be calculated refer to one of harmonic coordinates,local barycentric coordinates,iterative coordinates.Next,using iterative coordinates to represent a point as an affine combination of the derived polygon's vertices.The most important step is to use a suited mixing coefficient to represent the point as an affine combination of the initial polygon's vertices.The derived barycentric coordinates based on the derived polygons in the plane are extended to three-dimensional space,and the tetrahedron is derived into point polyhedrons,edge polyhedrons,face polyhedrons and volume polyhedrons using tetrahedral meshes,and iterative coordinates are used to calculate the barycentric coordinates of the vertices of the derived polyhedrons relative to the initial polyhedral vertices Then,use iterative coordinates to calculate the barycentric coordinates of a point inside the initial polyhedron with respect to the vertex of the derived polyhedron,and finally use a suitable blending function to calculate the barycentric coordinates of the point with respect to the vertex of the initial polyhedron.The derived barycentric coordinates based on the derived polygons and polydrons satisfy non-negativity and locality,and a large number of examples have proved that the appropriate mixing coefficients can make this new barycentric coordinates satisfy the smoothness.