Construction of Potential Function and Transition Curve Based on Metaball Technique

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 has C^k 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 reaching C^k continuity at endpoints. Then a polynomial potential function of 2k+1 degree with k+1 unknowns is constructed according to the analyzed conditions. Finally, we obtained the minimal degree polynomial potential function that has C^k 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 of C^k 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.