A cycle is particularly useful in animation when you have a repetitive movement or sequence. A good cycle easily runs in a smooth loop where the viewer is unable to tell when the cycle starts or finishes; the animation is seamless. An example of a cycle put to use can be seen in most character animation if the character is walking/running from A to B:
Various Animation cycles
Wile.E.Coyote and Road Runner - Lickety Splat
Cycles can be made up of many different sequences, or just one: 1-1-1-1-1-1 or 1-2-3-1-2-3-1-2-3-1-2-3. In the above video, it's a sequence of repeated frames that a run together to create the illusion of walking/running.
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