Hermite G 1 rational spline motion of degree six

Construction of G 3 rational motion of degree eight

“…Since the final motion would be of degree eight, one can choose also w as polynomial curves of degree eight which gives additional parameters of freedom that can be used to adjust a shape of the motion. More details on a construction of trajectory c can be found in [16].…”

“…Suppose that (6) holds true and let α 1,2 = 0, α 0,3 = 0. If there exists at least one positive solution u of (21) for which also V (u) in (20) is positive, then there exists an admissible quartic polynomial q satisfying (3) with λ 0,0 = λ 1,0 = 1, where λ j,i and ϕ j,i , j = 0, 1, i = 1, 2, 3, are determined by (16) and 19, where in all unknowns the superscript is omitted.…”

“…Later the generalization of this approach to cubic interpolation, which leads to G 2 rational spline motions of degree six, was considered in [14]. Other geometrically continuous motions of degree six can be found in [15] and [16]. The difficulty when using geometric interpolation methods is that nonlinear equations are involved and the existence and uniqueness of an interpolant are not assured for all given data.…”

“…In practical applications, parameter γ could be determined using some optimization or minimization approach. Perhaps, one could minimize the average of the stretch and the band energies of the quaternion curves (see [16], e.g.). Alternatively, one could try to minimize the perturbation of the computed and the user-given second order curvature quaternions.…”

Bézier motions with end-constraints on speed

“…In particular, transforms can be combined additively so that free-form motions can be created using the Bézier approach as an alternative to the slerp. In specifying free-form motions, one approach is to deal separately with the (translational) motion of a reference point in the moving body and the (rotational) motion of the body with respect to it [20,21,22]. An alternative is to handle the motion of the body as a single variable transform [5,23,24].…”