Smooth Extension Of Parametric Curves

In CAGD, we often need extend a piece of parametric curve to a known point or a piece of curve, the extended curve is the parametric curve with the same degree, and at the joint of these two curves arrive certain smoothness, which is called the parametric curve extension. There are many discussions and studies on this question, and in the most systems, the method is often used, which is to construct a cubic Bezier curve with GC~1 continuity between the last point of the parametric curve and a given point. But the constructed curve is almost unadjustable. To adjust the shape of the extended curve, a new method of the curve extension is made by Zhou Yuanfeng, by which we can extend a curve to a given point with G~2 continuity, and the extended curve is adjustable.The thesis discusses the smooth extension of cubic and quartic Bezier curves with the shape parameter, which is an expansion form of Bezier curve. Because the expression in the formula has the regulative parameter, the shape of the extended curve with C~1 or C~2 continuity can adjust along with the change of the shape parameter. The thesis discusses the smooth extension of cubic Bezier curve with a shape parameter, which arrive G~1 continuity at the joint of the given curve and the extended curve. The author separately uses the shortest arc-length and minimum energy to set up the objective functions. By minimizing the objective functions, the freedom degree and the shape parameter of the extended curve are determined, which can get the more faring curves. On the smooth extension of quartic Bezier curve with a shape parameter, the relation formulas of the control points are given in the cases of the curve extension with G~1 continuity between two given curves and the curve extension with G~2 continuity to the given point. Because the set of equations, which made by minimizing the objective functions, is non-linear, the operation is too complex and the discussion on the solution of equations is too difficult, the author only analyses the influence of the parameter values on the extended curve's shape qualitatively.