The Geometric Meaning Of Planar Rational Bilinear Coordinates And Its Generalization In Space

Barycentric coordinates of triangle are unique,but there are more than one choice of barycentric coordinates when the number of side of polygons is larger than three.Firstly,this article introduces the research background,current situation and significance of barycentric coordinates.Secondly,we introduce the definition of barycentric coordinates for planar polygons and a general construction and some properties of barycentric coordinates for convex polygons.Then this article introduce the research about planar bilinear coordinates and rational bilinear coordinates in particular.Thirdly,on one hand,this article study the sufficient and necessary conditions of unique trilinear coordinates for hexahedron.We represent four vertexes of hexahedron as barycentric coordinates of tetrahedron made up of the other vertexes of the hexahedron.We know that barycentric coordinates of tetrahedron are unique and using this we get a ternary equation set which contains three equations.The solution of the equation set corresponds to trilinear coordinates for hexahedron,and therefore the conditions that make the solution of the equation set unique are what we want.On the other hand,we study the “characteristic conic curve” of rational bilinear coordinates for a given planar quadrilateral and give its parametric form.It is the set of all points which make the rational bilinear coordinates unique and thus it is a conic curve.For the convenience of description,we call it the “characteristic conic curve” of a planar quadrilateral.This article concludes that the “characteristic conic curve” is a rational quadratic Bézier curve.Any point in the interior of “characteristic conic curve” doesn't have rational bilinear coordinates,and any point in the outside of “characteristic conic curve” has two choices of rational bilinear coordinates.Besides,this article presents some properties of “characteristic conic curve”.