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In https://codepen.io/winkerVSbecks/pen/NazWxg, there's a "hitch" as the discs touch due to a first-derivative discontinuity. Here's a version which extrapolates the u1 and u2 variables, making the transition much smoother: https://codepen.io/panic_/pen/BwvjmK.

This is just calling out for a propagation wave effect through the parent ball at the moment of separation.

Much better. There's definitely many improvements that could be done, but that was the main one. The other big one is how the bigger disc doesn't change size.

The approximation OP does is a good start but still far from being real metaballs.

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