Defining Fillets

I realized after playing around in CAD for a while that I didn't really understand what the formal definition of the filleting operation is. It seemed like a fun exercise to reconstruct what I feel it ought to be from first principles. If I take a collection of cubes and a bunch of edges that are alternatingly convex and concave, and fillet them all, OnShape does something:

What's happening in isolation on each edge is clear to me, but what happens when the edges come together isn't. This is mostly because I can imagine a notion of "positive" and "negative" filleting operations separately.

Positive filleting only adds material, and negative filleting only removes it. If $S$ is a subset of a metric space $M$ with distance function $d$, then I can define the operations of $F^+_r(S)$, "positive fillet to radius $r$", and $F^-_r(S)$, "negative fillet to radius $r$" (which both output a subset of $M$) as follows.

First define \[ B_r(x) = \{ y \in M \st d(y,x) \le r \}\] \[ \square S = \{ x \in M \st \forall y \in B_r(x) . y \in S\}\] \[ \dia S = \{ x \in M \st \exists y \in B_r(x) . y \in S\}\] Then we can define \[ F^+_r(S) = \square \dia S \] \[ F^-_r(S) = \dia \square S \] The trouble arises when I want to consider doing both of these simultaneously. I can't unambiguously sequence them, since they don't commute. Consider the following example of a thin (relative to $r$) square ring with a small (relative to $r$) hole:

Across the top we do a $F^+_r$. The small hole gets filled in. If we did a subsequent $F^-_r$, it would round off the corners of the square. Across the bottom we do a $F^-_r$. This immediately eliminates the ring, because no point of the ring is surrounded by a solid disc of radius $r$. So $F^-_r F^+_r S$ in this case is a rounded rectangle, but $F^+_r F^-_r S$ is the empty set. $F^-_r$ and $F^+_r$ do not commute.

One Idea for Simultaneous Positive and Negative Fillets

I'm still not sure what OnShape is doing precisely, but here's a definition that might be plausible. Instead of working with exclusively subsets of the metric space as intermediate data, let's generalize to maps from the metric space to the reals. Define \[ \red{\square} S = \lambda x . \max_{y \in B_r(x)} \min_{z \not\in S} d(z,y) \] \[ \red{\dia} S = \lambda x . \max_{y \in B_r(x)} \min_{z \in S} d(z,y) \] Both $\red\square$ and $\red\dia$ take a subset $S \subseteq M$ to a function $M \to \R$.

Claim: Although I may be off by some edge cases, I think morally we have \[\square S \qquad =\qquad \{ x \st \red{\square} S(x) \ge r\} \] \[ \dia S \qquad =\qquad \{ x \st \red{\dia} S(x) \le r \} \] This means that it seems plausible to define a filleting operation by comparing these with each other, instead of with $r$: \[ F S = \{ x \st \red\square S(x) \ge \red\dia S(x) \} \] Informally: a point $x$ belongs to the filleting of $S$ if and only the deepest depth you can get into the interior of $S$ (measured by how far away the exterior is) is greater than the farthest you can escape into the exterior of $S$ (measured by how far away $S$ is), if you're only allowed to go $r$ away from $x$. For points far away from any complicated crinkling of the boundary of $S$, this amounts to simply asking whether you're in $S$ or not.