Efficient Implicitization Of Rational Surfaces Without Base Points

The implicitization problem of rational surface is a classical algebraic geometry problem,which has important theoretical value and wide application prospect in computation geometry,computer aided geometric design and computer aided manufacturing.Sederberg and Chen's moving surfaces method,which appeared in 1995,is a new implicitize rational parametric surface method.In 2000,Cox et al.solved the validity problem of using the moving surfaces method to implicitize rational surface without base points and without low degree moving planes.In 2016,based on the research of Cox et al.,Lai and Chen proposed an algorithm for constructing moving quadrics from moving planes,which greatly improved the efficiency of implicitization.At present,no one has solved the problem of efficient implicitize rational surfaces with low degree moving planes without base points.Based on the research results of Lai,Chen and Shi in 2019,that is,the implicit equation of rational surface with low degree moving planes and no base points can be constructed by moving planes together with moving quadrics.this paper studied the efficient implicitization problem of rational surface which with low degree moving planes and without base points.The concrete content is research the minimal basis of the moving plane of a rational surface;The supporting theory of moving quadrics and moving planes needed to generate for implicitization efficiently and constructively from the minimal basis of the moving plane is established;Proved that the determinant of the matrix formed by the generated moving planes and moving quadrics is exactly the implicit equation of the rational surface;Based on the above results,an efficient implicit algorithm for rational surface is presented,and the complexity of the algorithm is analyzed,experimental results and complexity of the algorithm shows that compared with the algorithm proposed by Lai,Chen and Shi in 2019,the algorithm in this paper greatly improves the efficiency of implicit rational surfaces without base points.