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"Circumcenter of mass" is a natural generalization of circumcenter to non-cyclic polygons. CCM(P) can be defined as the weighted average of the circumenters of the triangles in any triangulation of P, the weights being proportional to the triangle areas. It's a surprising fact that that the resulting point is independent of the triangulation chosen.

"Circumcenter of mass" is a natural generalization of circumcenter to non-cyclic polygons. CCM(P) can be defined as the weighted average of the circumcenters of the triangles in any triangulation of P, the weights being proportional to the triangle areas. It's a surprising fact that that the resulting point is independent of the triangulation chosen.

I don't know a simple formula for ICM(P), but we can compute it for any convex polygon P recursively, by taking as an ansatz the assumed property that ICM(P) will be some (mysteriously) weighted average of the ICMs of its parts, where the partitioning is dual, in a sense, to the kind of partitioning down to a triangulation which is done during the computation of CCM. That is, partitioning is done at a chosen pair of sides, rather than a pair of vertices, of P, and repeated partitioning eventually arrives at a fully refined set of side-triples, rather than vertex-triples (triangles).

I don't know a simple formula for ICM(P), but we can compute it for any convex polygon P recursively, by taking as an ansatz the assumed property that ICM(P) will be some (mysteriously) weighted average of the ICMs of its parts, and that it will be the same regardless of which partitioning is chosen.

The appropriate kind of partitioning here is dual, in a sense, to the kind of partitioning down to a triangulation which is done during the computation of CCM. That is, partitioning is done at a chosen pair of sides, rather than a pair of vertices, of P, and repeated partitioning eventually arrives at a fully refined set of side-triples, rather than vertex-triples (triangles). We can see immediately that this makes sense when the polygon is tangential-- all side-triples will agree on a common point which is the incenter of P.

• For subsequence S consisting of only 3 segments, find $\mathrm{ICM}(S)$ as for an incenter of a triangle: that is, find the center of the unique circle having those three sides (directed lines) as tangents, being careful to choose sidedness so that it's in the interior of the original P. That is, it's the unique point equidistant (using signed distance) from the three lines.

• For subsequence S having more than 3 sides, subdivide into two smaller problems, in two different ways:

  (1) Cut the cyclic list of sides $S=[s_0,...s_{n-1}]$ into two (overlapping) parts, at any two non-adjacent entries in it; let's say $s_0$ and $s_2$ for definiteness. Recursively compute the two points $\mathrm{ICM}([s_0,s_1,s_2])$ and $\mathrm{ICM}([s_2,s_3,...,s_{n-1},s_0])$. We know the desired $\mathrm{ICM}(S)$ will be some weighted average of those two points (i.e. it will lie on the line through those two points) but we don't know the weights yet.

  (2) Cut $S$ into two overlapping parts at a different pair of non-adjacent sides; this time at, say, $s_1$ and $s_3$. Recursively compute the two points $\mathrm{ICM}([s_1,s_2,s_3])$ and $\mathrm{ICM}([s_3,...,s_{n-1},s_0,s_1])$. Again, we know $\mathrm{ICM}(S)$ will lie on the line through these two points.

  (3) The point $\mathrm{ICM}(S)$ can now be pinpointed, as the intersection of the two lines computed in (1) and (2).

• Is this a known result?
• Is there a simple formula for ICM(P), as a weighted average of the incenters of the side-triples corresponding to triangles in any given triangulation of the dual of P? (I anticipate something of the flavor of CCM, whose weights are proportional to the triangle areas of a triangulation of P, but I haven't been able to figure out exactly what it is; the above diagram doesn't seem to reveal any obvious formula.)
• Is there a duality relationship between ICM and CCM? E.g. perhaps ICM(P) can be computed as some simple function of CCM(P') for some polygon P' derived from P.

• For subsequence S consisting of only 3 segments, find $\mathrm{ICM}(S)$ directly as for an incenter of a triangle: that is, find the center of the unique circle having those three sides (directed lines) as tangents, being careful to choose sidedness so that it's in the interior of the original P. That is, it's the unique point equidistant (using signed distance) from the three lines.

• For subsequence S having more than 3 sides, subdivide into two smaller problems, in two different ways:

  (1) Cut the cyclic list of sides $S=[s_0,...s_{n-1}]$ into two (overlapping) parts, at any two non-adjacent entries in it; let's say $s_0$ and $s_2$ for definiteness. Recursively compute the two points $\mathrm{ICM}([s_0,s_1,s_2])$ and $\mathrm{ICM}([s_2,s_3,...,s_{n-1},s_0])$. We know the desired $\mathrm{ICM}(S)$ will be some weighted average of those two points (i.e. it will lie on the line through those two points) but we don't know the weights yet.

  (2) Cut $S$ into two overlapping parts at a different pair of non-adjacent sides; this time at, say, $s_1$ and $s_3$. Recursively compute the two points $\mathrm{ICM}([s_1,s_2,s_3])$ and $\mathrm{ICM}([s_3,...,s_{n-1},s_0,s_1])$. Again, we know $\mathrm{ICM}(S)$ will lie on the line through these two points.

  (3) The point $\mathrm{ICM}(S)$ can now be pinpointed, as the intersection of the two lines computed in (1) and (2).

• Is this a known result?
• Is there a simple formula for ICM(P), as a weighted average of the incenters of the side-triples corresponding to triangles in any given triangulation of the dual of P? (I anticipate something of the flavor of CCM, whose weights are proportional to the triangle areas of a triangulation of P, but I haven't been able to figure out exactly what it is; the above diagram doesn't seem to reveal any obvious formula.)
• Is there a duality relationship between ICM and CCM? E.g. perhaps ICM(P) can be computed as some simple function of CCM(P') for some polygon P' derived from P.
• Any other insights about how ICM relates to other properties?