The Theory And Applications Of μ-Bases Of Rational Curves And Surfaces

The technique ofμ-bases is originated from the theory of moving lines and moving planes,μ-bases provide a connection between the parametric forms and implicit forms of curves and surfaces due to their special algebraic and geometric properties.We shall explore the theory and computational methods of the singularities of rational planar curves and rational space curves fromμ-bases.For rational planar curves,we shall use the Smith form of the Bezout matrix constructed from aμ-basis for the rational planar curve to compute the singularity trees of the rational planar curve.For rational space curves,we shall introduce the concept of axial moving planes, and examine the relationship between the singularities andμ-bases of rational space curves through all the moving planes that,follow the rational space curves.We also give analysis for the types and upper bounds of the numbers of singularities on low degree rational space curves and some special type of high degree rational space curves: corresponding algorithms are also presented.Later on the wide studied problem of implicitizing rational space curves is discussed.We shall first apply the technique ofμ-bases and local cohomology to study the minimal generators for the moving surface ideals of rational cubic space curves and rational quartic space curves,and then study the implicit equations of rational space curves under different definitions of implicit equations.Finally we shall explore algorithms for computing self-intersection curves of rational ruled surfaces.We use the principal stbresultant sequences ofμ-bases to compute the parametric equations of the self-intersection curves of rational ruled surfaces.