Research On Approximation Of Conic Section By PH Curve

In Computer Aided Geometric Design,the conics have important research significance.But the conic cannot be accurately represented by polynomials in explicit form in addition to parabola.So the approximation problem of conic section has been concerned by many scholars.The arc lengths or the offsets of conic section(except circular arcs)cannot be represented by polynomials or rational polynomials.It is not compatible with NURBS systems.The main feature of PH curve is that arc lengths and the offsets can be expressed by polynomials and rational polynomials.In this thesis,the PH curve is introduced as the approximation curve.Based on the upper bound on the Hausdorff distance between the conic section and the approximation curves,the geometric characteristic conditions of the PH curve is used to solve the upper bound of error function.The optimal parameter values are determined.Then the approximation curve is presented.And the approximation of circular arcs by quartic PH curve is studied.In chapter 1,the research background and the development are introduced.And some related studies of the approximation of conic section are introduced.In chapter 2,the basic knowledge and methods are introduced,including the concept and properties of Bézier curves and PH curves.In chapter 3,the approximation method of conic section by quartic PH curves is represented.The upper bound on the Hausdorff distance between the conic section and the approximation curves is optimized.Then the geometric characteristic conditions of the PH curve is used to get the approximation curve.In chapter 4,a method for approximation of circular arcs by cubic PH curves is studied.In chapter 5,we summarize the thesis and put forward the outlook.