Non-overshooting Hermite Cubic Splines for Keyframe Interpolation

e o A technique for limiting the knot slopes of hermite cubic splines in order to eliminat vershoot is proposed. It is proven that constraining each knot’s slope to lie between 0 and l three times the slope to the knot on either side forces all extrema to occur at knots. This al ows overshoot to be eliminated without sacrificing slope continuity. The technique has appli1 cations in keyframe interpolation of motion parameters. . The Problem Hermite cubic splines are often used for keyframe interpolation of motion parameters. Our BBOP n p program, for instance, uses local, independent non-uniform hermite cubics for each of the nine motio arameters (move, scale, and rot in x, y, and z) at each joint. Each parameter’s spline is a univariate , function x (t ) with first derivative (C ) continuity but no guarantees of geometric continuity (see [Bohm 1 . Farin, and Kahmann] for a survey of splines) Users often complain of the overshoot which spline interpolation causes. Sudden changes in n i velocity force the curve to overshoot an extremum (a maximum or minimum) of the function as show n figure 1. Animators often intend their key frames to be motion extremes, but the animation program e u usually ignores this implicit information. Is it possible to constrain a spline so that it doesn’t hav ndesirable overshoots?