Computation And Applications For μ-bases Of Rational Curves With Lower Degree

As a special base of syzygy model of rational curves and surfaces,μ-bases of rational curves and surfaces is widely used in computing implicit equations and singular points of rational curves and surfaces. There are fast algorithms to compute μ-bases for rational curves based mainly on Gaussian Elimination, which means given a rational curve, we can get the μ-bases of the curve efficiently. However, we do not know what the μ-bases look like before we run the algorithms. Thus, it is necessary for us to discuss and find explicit formulas for the μ-bases of some familiar rational curves. In this thesis, we give explicit formulas for the μ-bases of low order curves and the special algorithm to compute μ-bases of some special curve, which enrich the μ-bases theory and make contribution to the application of μ-bases in geometric modeling.As fundamental objects in Computer Aided Geometric Design and Computer Graphics, conic sections and planar rational cubic curves have wide range applications. By the property of outer product, we give explicit formulas for the μ-bases of conic sections and planar rational cubic curves. Based on the explicit formulas for the μ-bases, we also provide a simple implicit matrix, i.e., implicit equation can be derived from the determinant of the matrix. In addition, based on the μ-bases of planar rational cubic curves, we give explicit formulas for singularities of the planar rational cubic curve. Moreover, We extend the explicit formula for the μ-bases of conic sections to μ-bases for rational curves of degree n in n-dimensions.To study canal surface, we define a special rational space curve -quaternion rational curves. The special algorithm to compute μ-bases of quaternion rational curve is also presented.