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I have two equally sized sets of 3D positions. All of the positions are unique. Is there a way to calculate the plane that best separates the two sets from each other?

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It depends on what do you mean by "best separate".

One possible way is as follows:

Label one of the set with $y_i=1$ and the other class with $y_i=-1$.

The goal is to separate the two class with a plane (assuming it exists)

You can solve the following optimization problem using quadratic programming

$$\min_{w,b} \|w \|^2$$

subject to $$y_i(w^Tx_i-b) \geq 1$$

This is a famous algorithm called support vector machine (SVM). It maximizes the margin between $2$ classes of points.

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