势函数的构造及基于Metaball的过渡曲线

Construction of Potential Function and Transition Curve Based on Metaball Technique

针对原多项式势函数次数较高且其构造的过渡曲线在端点处的连续性较低的问题,为满足过渡曲线设计的更高要求,从提高连续性的角度构造了使过渡曲线在端点处达到kC连续的最低次多项式势函数.首先根据高阶求导公式得到使过渡曲线在端点处达到kC连续的势函数条件,然后根据这些条件来构造含k+1个未知数的2k+1次多项式势函数,最后通过线性方程组的求解得到kC连续的最低次多项式势函数.考虑到kC连续下过渡曲线的形状无法调整问题,又提出了含形状参数的混合三角势函数,其构造的过渡曲线特点是在保持端点处1C连续的情况下能与被过渡曲线的一侧达到很高的重合性.实验结果表明,文中构造的2类势函数是有效的,构造的过渡曲线可实际应用于凸轮廓线的平滑处理等.

Traditional polynomial potential functions have the following problems： their degree is high and the constructed transition curve has low continuity at endpoints. To provide a better transition curve design, a new minimal-degree polynomial potential function that hasCk continuity at endpoints of transition curve is proposed in this paper. Firstly, we analyze the potential function conditions by means of higher derivative formulae that can make transition curve reachingCk continuity at endpoints. Then a polynomial potential function of 2k＋1 degree withk＋1 unknowns is constructed according to the analyzed conditions. Finally, we obtained the minimal degree polynomial potential function that hasCk continuity at the endpoints by solving a system of linear equations. Furthermore, noting that the shape of transition curve cannot be changed based on the conditions ofCk continuity, another new mixed triangular potential function with a shape parameter is constructed, which can reach high coincidence at one side of transited curves based on conditions of keeping C1 continuity at endpoints. Experimental results show the effectiveness of these two new potential functions and the constructed transition curves can be applied practically in smoothing convex cam profile.