Extension Of Parametric Curves

The topic of this paper is to make a study of algorithms for extending parametric curves in CAD.The essential part of this thesis discusses a new problem of extending one given cubic Bezier curve to another nonadjacent one. One cubic Bezier curve is inserted between them with G~1 continuity in the simple case. Two shape parameters are provided for the user to adjust the shape of the extended curve. Then it is constructed by the geometric continuous conditions by minimizing the some fairing functions based on the shortest arc-length, minimum energy and minimum curvature variation of the curve respectively. Geometric relations of the given control points are used to describe necessary and sufficient conditions for existence of the three optimization curves.As extension using one cubic Bezier curve sometimes can't meet with the user's demand that the extended curve should pass a given point. We solve that using two cubic Bezier curves which connect each other at the given point, where the user can supply a directional tangent vector to control the curve shape more easily. Necessary and sufficient conditions for three optimization curves are also presented.G~2 continuity is used to improve the smoothness when G~1 is not enough to ensure the quality of the curve shape in some situations. To simplify the optimization functions and provide free shape parameters, we construct the extended curve with two cubic Bezier curves C -continuous.