The Geometric Meaning Of The Bilinear Coordinates Of Planar Quadrilateral

The barycentric coordinates can complete the geometric transformation of the figure,including the reduction,enlargement,translation,rotation,curve fitting,curve splicing,splitting and local curve modification and so on.So in recent years,the barycentric coordinates are developed as an independent research field.This paper focuses on a new barycentric coordinates-bilinear coordinates.First this paper introduces the research background and development of the barycentric coordinates,and describes the advantages and disadvantages of the existing barycentric coordinates.The second part introduces the definition of Bezier curve,the geometric mapping algorithm of Bezier curve and the definition of the barycentric coordinates.Bezier curves are presented on the basis of the Coons bicubic patches and parametric spline curves.The appearance of the Bezier curve brings great convenience to the designer and makes the designer to directly observe the relationship between the control vertices and the curves.Then we introduce Bezier curve geometric mapping algorithm-de Casteljau algorithm.The initial definition of barycentric coordinates is defined by the weight coefficients of the vertices of the triangle,and the basic properties of the barycentric coordinates are introduced.In the third part,we introduce the quadratic Bezier curve,and a quadratic Bezier curve is a parabola,and then the parabola three tangent theorem is extended to the n-tangent theorem of the quadratic Bezier curve.And proved by algebraic method,theorem and its inverse theorem are established.Given an example,there is only one quadratic Bezier curve to tangent to four edges for an arbitrary quadrilateral.In the fourth part,the bilinear coordinates proposed by Floater are introduced firstly.Based on the n-tangent theorem of quadratic Bezier curves,the geometrical meaning of inverse of bilinear mapping of planar quadrilateral is studied from the perspective of mapping.For the planar bilinear mapping defined by the four vertices of the trapezoid,there are infinite numbers inverses on the intersection point between two trapezoid waists.For the line which is parallel to bottom edges of the trapezoid and passes the intersection point between two trapezoid waists except the intersection,the inverse does not exist.And there is the unique inverse of the bilinear mapping for other points.For the planar bilinear mapping defined by the four vertices of the planar quadrilateral,there is a parabola which is tangent to the lines on the four edges.For the points on the parabola,there is the unique inverse.For the points out of the parabola,there are two inverses.And for the points inside of the parabola,the inverse doesn't exist.In the fifth part,based on the study of the geometric meaning of bilinear coordinates,we can study the rational bilinear coordinates and its good properties.Or on this basis,we can study the geometric significance of the rational bilinear coordinates on the space.