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The centroid of an object is defined as the arithmetic mean of all points of the object. For non-convex objects, the centroid is often not a part of the object itself:

Polygon with centroid outside the object

Is there a definition of a centroid-like point which always lies within the object?

Polygon with centroid within the object

A definition that tackles 2 dimensional polygon objects is sufficient. I could think of something like the following, however, I would prefer a well-known definition if there is one:

  1. Let $s_p(a, b)$ be the shortest path from $a \in p$ to $b \in p$ such that all points of the path are within $p$.

  2. Let $S_p := \{s_p(a, b) | a, b \in p\}$.

  3. Let $d$ be the longest path in $S_p$.

  4. The mid point of $d$ is a centroid-like point and it always lies within $p$.

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    $\begingroup$ But which properties should such a point have? You cannot expect it to retain all properties of the centroid. For instance: how would you choose this centroid-like point in the case of an annulus? $\endgroup$ – Intelligenti pauca Jan 27 '19 at 16:56
  • $\begingroup$ The point should be some kind of central point that lies within the object. I dont know the exact properties because I am not that familiar with geometry. Are there any useful definitions of "special" points that always lie within the object? $\endgroup$ – pschill Jan 27 '19 at 18:38
  • $\begingroup$ In the case of an annulus, for instance, any point you choose inside the object will not respect its rotational symmetry. Would that be fine for you? $\endgroup$ – Intelligenti pauca Jan 27 '19 at 19:15
  • $\begingroup$ Yes, I think any point is fine in case of an annulus. However, if the annulus is sliced, I expect the point to be directly opposite to the slice. Maybe some mid point of some skeleton would do fine. $\endgroup$ – pschill Jan 27 '19 at 19:18
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In geography there is an interesting point called Pole of inaccessibility: the most distant internal point from the polygon outline.

It is found by an iterative method, described in the Methodology section of this paper.

Links of interest:

https://sites.google.com/site/polesofinaccessibility/

https://github.com/mapbox/polylabel

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  • $\begingroup$ This definition is short, easy to understand, and seems to fit my use case. Thank you :) $\endgroup$ – pschill Jan 29 '19 at 7:52

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