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Suppose $R$ is some planar region, bounded by a curve. Let $C_1$ be the centroid of $R$, and let $C_2$ be the center of the "circumcircle" (the smallest circle enclosing $R$). Intuitively, it seems that $C_1$ and $C_2$ ought to be fairly close together.

But how close? Can we find a bound $d$ on the distance between $C_1$ and $C_2$?

Assume that $R$ is a polygon, if that helps (I can always replace the boundary curve by an approximating polygon).

Assume that $R$ is convex, if that allows us to obtain a smaller bound (I can replace R by its convex hull).

Why does this matter? I really want the circumcircle, but computing it is quite difficult (yes, I know algorithms exist, but they're fairly complicated). I'm looking for a decent approximation that's easy to compute. So, the idea is:

  1. Estimate $d$ (hence the question above)
  2. Find the centroid $C_1$ (easy)
  3. Compute the maximum distance $M$ from $C_1$ to the boundary curve (easy)
  4. The circle with center $C_1$ and radius $M+d$ encloses $R$. Done.

Here is a somewhat related question (with no answers).

Edit
If $R$ is a semi-circular region of radius $r$, then $d=4r/3\pi \approx 0.42*r$. I'm guessing that's the worst case for convex shapes.

If we don't assume that $R$ is convex, then I can put a fat blob of $R$ over on one side of the circumcircle, and an infinitely thin tentacle snaking over to the other side. By doing this, maybe I can make $d$ arbitrarily close to the radius of the circumcircle..

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bubba
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    I think that approximation would be quite bad if the convex region is half a circle or has quite a similar shape... – Jack D'Aurizio Feb 29 '16 at 13:19
  • @JackD'Aurizio --Thanks. Yes, I just realized that myself. And maybe even worse if we allow $R$ to be non-convex. – bubba Feb 29 '16 at 13:23
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    I think you can "improve" on (or should that be "do worse than") $4r/(3\pi)$: cut segments off the semicircle adjacent to its diameter, thereby "lightening" the portion less than $4r/(3\pi)$ from the diameter and shifting the center of gravity. I'm not sure how much you should cut; it seems the cut should end at least as far from the diameter as the new center of gravity, but clearly you don't want to cut all the way down to a triangle (for which $d = r/3$). – David K Feb 29 '16 at 13:33
  • The slicing should leave the diameter of the semicircle intact (one endpoint of each cut should be at an endpoint of the diameter), so I don't think the circumcenter changes at all. – David K Feb 29 '16 at 14:00

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