Animation techniques for producing physically-realistic animation while providing the interactivity and control desired by animators. Techniques are provided specifying animation for motion problems that give rise to quadratic optimization functions solvable with linear systems of equations. The resultant splines generalize traditional splines to encompass oscillatory solutions. These problems can be solved at full frame rates, giving animators a keyframe animation tool. Such a formulation is able to address a wide range of oscillatory phenomena while retaining the interactive performance and predictability of traditional splines. The splines may be complex-valued.
