Bézier Curve as a Generalization of the Easing Function in Computer Animation

Abstract

The description of movement has always been one of the basic problems in traditional and computer animation. A person watching an animated film may accept simplifications in the appearance of the characters, but will not accept unnatural, unexpected motion. The transition problem is an example of movement description which is successfully solved by Penner’s easing functions. But, the scope of an easing function is limited to known examples of specific functions proposed by Penner or other developers. However, there is often a need to describe the transition problem by a function resulting from the interpolation of points that approximate the trajectory. A convenient way to describe the shape, in such a situation, would be a Bézier curve. The article is an attempt to generalize the problem of interpolation of transition trajectories using the Bézier curve. By analyzing various cases of the cubic polynomial equations in the context of transition problems in animation, we can limit the solution to several families of the transcendental functions.