An Approximation Of Conic Section By Bezier Curve

Computer Aided Geometric Design, referred to as CAGD, is widely used in the shape design of modern industrial products. The conies have important applications in geometric modeling design, unfortunately, the conic cannot be accurately represented by polynomials in explicit form in addition to parabola. So the approximation problem of conic sections has been concerned by many scholars. The main work includes the following two aspects in this thesis. One is the approximation conic sections by quartic Bezier curves, the other is a highly accurate approximation of ellipse arcs by cubic Bezier curves. This thesis is organized as follows:The background and the development of CAGD, and some related studies of the approximation of conic sections are introduced in the first chapter.In the second chapter, the basic knowledge and methods, including the concept and properties of Bezier curves, a highly accurate approximation of arcs by quartic Bezier curves, are introduced.We give a highly accurate approximation of conic sections by quartic Bezier curves in the third chapter. The approximation order is 8. We also obtain Hausdorff distance between conic sections and quartic Bezier curves. Compared with the previous methods, our method has a smaller error bound. Using the subdivision scheme, the composite curves of the quartic Bezier approximation are globally G2 continuous.In the fourth chapter, a highly accurate method for approximation of ellipse arcs by cubic Bezier curves is studied. The explicit representations of control vertex coordinates are given. The error of the approximation in Hausdorff distance is optimized. The cubic Bezier spline approximation of elliptic curve is obtained and satisfies C1 continuity.In the finally chapter, we summarize this thesis. And the future research direction is prospected.