利用距离与内能极小平滑链接Bézier曲线

目的 对于满足低阶连续的链接Bézier曲线，提高曲线之间的连续性以达到平滑的目的，需要对曲线的控制顶点进行相应调整。因此，可根据具体的目标对需要调整的控制顶点进行优化选取，使得平滑后的链接曲线满足相应的要求。针对这一问题，给出了3种目标下优化调整控制顶点的方法。方法 首先对讨论的问题进行描述，分别指出链接Bézier曲线从C⁰连续平滑为C¹连续和从C¹连续平滑为C²连续两种情形需调整的控制顶点；然后分别给出两种情形下，以新旧控制顶点距离极小为目标、曲线内能极小为目标、新旧控制顶点距离与曲线内能同时极小为目标，对链接Bézier曲线进行平滑的方法，最后对3种极小化方法进行对比，并指出了不同方法的适用场合。结果 数值算例表明，距离极小化方法调整后的控制顶点偏离原控制顶点的距离相对较小，适合于控制顶点取自于实物时的应用场合；内能极小化方法获得的链接曲线内能相对较小，适合于要求曲线能量尽可能小的应用场合；距离与内能同时极小化方法兼顾了新旧控制顶点的距离和链接曲线的内能，适合于对两个目标都有要求的应用场合。结论 提出的方法为链接Bézier曲线的平滑提供了3种有效手段，且易于实现，对其他类型链接曲线的平滑具有参考价值。

Smoothing linked Bézier curves by distance and internal energy minimization

Objective The Bézier curve is a widely used tool in the representation of parametric curves. At present, most computer aided design (CAD) systems take the Bézier curve as a basic module. When using the Bézier curve for geometric modeling, it is often necessary to link multiple curves to meet the needs of designing complex curves. In order to satisfy the continuity between the linked Bézier curves, the control points that satisfy the corresponding continuity conditions need to be selected in advance. For linked Bézier curves with low order continuity, the control points of the curves can be adjusted to improve the continuity so as to smooth the linked curves. In theory, the continuity between linked Bézier curves can be improved by arbitrarily adjusting the control points of the curves to make the corresponding continuity conditions hold. However, this kind of adjustment of control points without specific objectives is often unable to meet the needs of practical applications. Given that the continuity can be improved by adjusting the control points of the linked Bézier curves, the control points that need to be adjusted can be optimized according to some specific targets, so that the smooth linked curves meet the corresponding requirements. In practical applications, if the control points of the curves are taken from the real objects, it is often hoped that the distance between the new control points and the original control points is as small as possible. Hence, the minimum distance between the new control points and the original control points can be used as the target to optimize the control points that need to be adjusted. In addition, energy minimization has become a common method for constructing curves and surfaces in CAD and related fields. Thus, the minimum energy can be used as a target to optimize the control points that need to be adjusted. In this study, three methods for smoothing the linked Bézier curves from C⁰ to C¹ and from C¹ to C² by distance and internal energy minimization are given. Method First, the problems to be discussed are described, and the control points to be adjusted are pointed out when smoothing the linked Bézier curves from C⁰ to C¹ and from C¹ to C². Two control points need to be adjusted when smoothing the linked Bézier curves from C⁰ to C¹ or from C¹ to C². However, only one of them needs to be optimized due to the relationship between the two control points that need to be adjusted. Then, the distance minimizations for smoothing the linked Bézier curves from C⁰ to C¹ and from C¹ to C² are presented. Next, the internal energy minimizations for smoothing the linked Bézier curves from C⁰ to C¹ and from C¹ to C² are given. Then, the simultaneous minimization of the distance and the internal energy for smoothing the linked Bézier curves from C⁰ to C¹ and from C¹ to C² are provided. The optimal solutions of the control points that need to be adjusted can be easily obtained by solving the corresponding unconstrained optimization problems. Finally, the comparison of the three minimization methods is given, and the applicable occasions of different methods are pointed out. The distance between the new control points and the original control points by the distance minimization is relatively small, which is suitable for the application when the control points are taken from the real object; the internal energy of the linked curves obtained by the internal energy minimization is relatively small, which is suitable for the application where the energy of the curve is required to be as small as possible; the distance and internal energy minimization simultaneously takes into account the distance between the new control points and the original control points and the internal energy of the linked curves, which is suitable for applications where both targets are required. Some numerical examples are presented to illustrate the effectiveness of the proposed methods. Result By using the distance minimization, the total distance between the new control points and the original control points is obviously smaller than that of the other two methods. When the control points of Bézier curves are all taken from the real object, the distance between the new control points and the original control points should not be too large, so the distance minimization method is more suitable. By using the internal energy minimization, the total internal energy of the curves is obviously smaller than that of the other two methods. When the energy of the curve is required to be as small as possible, the internal energy minimization is more suitable. In addition, given that the stretch energy, strain energy, and curvature variation energy correspond to the arc length, curvature, and curvature variation of the curve, the corresponding minimization method can be selected according to the specific requirements. By using the simultaneous minimization of distance and internal energy, the total distance between the new control points and the original control points and the total internal energy of the curves are between the other two methods. When the distance between the new control points and the original control points should not be too large and the energy of the curve should be as small as possible, the method of simultaneous minimization of distance and internal energy is a better choice. Conclusion The proposed methods provide three effective means for smoothing the linked Bézier curves and are easy to implement. The linked Bézier curves can be effectively smoothed from C⁰ to C¹ and from C¹ to C² by using the proposed methods according to the demand of minimum distance or internal energy. When smoothing other types of link curves, the distance minimization, internal energy minimization, and simultaneous minimization of distance and internal energy proposed in this study maybe useful.