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I've created what I think is an interesting gif.

It involves a kind of voodoo head that moves around the frame.

What I would like to know is, is there a way that I can encode this so that- rather than moving around on a pre-specified path, the head will move all around the frame in a pseudo-random way?

Maybe that would take us out of gif land, but- that's ok.

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No, this is not possible to do in a gif because a gif is by nature a short clip that is repeated (therefore it can be random, but only until it repeats). You can use various software to create a bunch of gifs and generate different random path for each gif if you'd like, but the same gif can't move in different random paths.

If you want this type of behavior you need to use some format that allows dynamic changing of the content. The most cross-device way to do this would be to use web technology. It can easily be done by laying a repeating gif background like the spiral and abstract lines you have as one layer and by putting the head on a separate layer to be moved around. For something like this you could use regular HTML, CSS, and JavaScript or you could do it on something called Canvas. As for the particulars of implementation, there are posts you can find that show how to move an element randomly within some bounds already.

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