Is there a rule for "kinks" in boundaries of point sets?

Question

Consider sets $A$, $B$, $C$ each being a subset of $\mathbb{R}^2$ and a set $D := (A \cap B) \cup C$. For the given examples the following obervation can be made: each of those marked red points $p_{R}:=\partial A \cap \partial B$ is has "kink" in $\partial D$ (the boundary of D) iff $p_{R} \in C$. Analogous observatsions can be made for the green and blue points.

Is this a coincidence, or is there there a theory behind this?

Answer 1

It might help to look up Gauss Codes:

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          ^{Image by Jeff Erickson: PDF download.}

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