Research On The Problems Related To DP Curve And Approximation Of Conic Sections

In Computer Aided Geometric Design(CAGD),curves and surfaces design are becoming more and more mature.And the construction of the new type of curves is widely concerned by researchers.Delgado and Pe?a introduced a new type of DP-NTP parametric curves,which have the properties of end-point interpolation,numerical stability and linear complexity,etc.So the DP-NTP curves have promising application prospect in CAGD.In this thesis,the construction of cubic trigonometric DP curve with shape parameters,the construction of quartic DP curve with shape parameters and approximation of conic sections are studied.The main work is as follows:1.The research results and development history of DP curves,trigonometric polynomial curves and approximation of conic sections including circular arcs by polynomial curves are introduced.And the basic framework of this thesis is showed.2.A set of cubic trigonometric DP curve with shape parameters is presented,which has good properties of the cubic DP curve.The basis is a set of normalized totally positive basis.It has two parameters.The introduction of shape parameters enhances the performance and shape control ability of the curve,which is better for curve design.The geometrical meanings of the shape parameters are given.G~1 andG~2 continuous conditions of the cubic trigonometric DP curve are discussed.And the examples of accurate representation for the ellipse,circle and parabola by the cubic trigonometric DP curve with shape parameters are given,which show the effectiveness of the theory.It is non-rational form.3.By extending the quartic DP basis functions,the quartic DP basis functions with shape parameters are constructed.Based on the basis functions,the definition and matrix representation of the quartic DP curve with shape parameters are given.By using the least squares method,the approximation of conic sections are presented.The introduction of shape parameters enhances the shape control ability.Finally,numerical examples are given.The error distance function in the₂L norm is calculated.The approximation error between conic sections and the quartic DP curve with shape parameters is given.