Optimal G2 Hermite interpolation for 3D curves. (December 2019) - Herzog, Raoul; Blanc, Philippe
Abstract: We consider a Hermite interpolation problem for a 3D curve where the functional to be minimized is defined as the integral of squared norm of the third parametric derivative, subject to G 2 continuity constraints at the end points. The first order necessary optimality condition of the variational problem leads to a parametric transition curve with quintic polynomials. The determination of coefficients is given by a polynomial system with 2 unknowns. Stationary points correspond to positive roots of the resultant which is a degree 9 polynomial. Although the formulated variational problem is non-convex, the proposed approach leads to the global solution, which can be computed in a reliable and fast manner. Graphical abstract: Highlights: The problem of G2 curve rounding or curve completion in 3D space is important in CAD–CAM and CNC. Nonconvex variational problem solved analytically. Stationary points given by the positive roots of a degree 9 polynomial. Comparisons with other methods are given. The proposed algorithm is well tested in many different situations.