Talk:Circumcenter of mass

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24.4.106.160 (talk) 11:17, 12 October 2020 (UTC)Don Hatch This page makes several statements about circumcenter of mass (ccm) that aren't true.


 * 1) "The ccm satisfies [...] if a polygon is decomposed into two smaller polygons, then the ccm of that polygon is a weighted sum of the ccms of the two smaller polygons".  This isn't true in general.  Counterexample: let P = an isosceles right triangle (45-45-90).  Its ccm is the midpoint of its diagonal.  Bisect P into P1,P2, both smaller isosceles right triangles.  The area-weighted avg of their ccms is halfway between the midpoints of their respective diagonals; that's not the same as the ccm of P.
 * 2) "[...] any triangulation with nondegenerate triangles may be used to define the ccm." This is true only if no vertices are added or removed along the boundary.
 * 3) "The ccm is invariant under the operation of "recutting" of polygons."  I'm not sure what exactly "recutting" means (haven't seen the referenced paper), but I suspect this is a generalization of 1. and 2. above, which are false already.
 * 4) "The ccm can be extended to smooth curves via a limiting procedure. This continuous limit coincides with the center of mass of the homogeneous lamina bounded by the curve."  This isn't necessarily false, but it's not very useful as a definition, since nailing it down would require some more work (we'd need to clarify we're talking about adding continuously many vertices, even along the "straight parts" of the boundary, which means this definition is *not* really an extension of the definition of ccm for polygons) and the result is simply the center of mass of the interior anyway, which doesn't need a new name.  The interesting true statement here is that the limit of the ccm, as the boundary is subdivided more and more finely by adding more vertices, is the center of mass of the interior.
 * 1) "The ccm is invariant under the operation of "recutting" of polygons."  I'm not sure what exactly "recutting" means (haven't seen the referenced paper), but I suspect this is a generalization of 1. and 2. above, which are false already.
 * 2) "The ccm can be extended to smooth curves via a limiting procedure. This continuous limit coincides with the center of mass of the homogeneous lamina bounded by the curve."  This isn't necessarily false, but it's not very useful as a definition, since nailing it down would require some more work (we'd need to clarify we're talking about adding continuously many vertices, even along the "straight parts" of the boundary, which means this definition is *not* really an extension of the definition of ccm for polygons) and the result is simply the center of mass of the interior anyway, which doesn't need a new name.  The interesting true statement here is that the limit of the ccm, as the boundary is subdivided more and more finely by adding more vertices, is the center of mass of the interior.
 * 1) "The ccm can be extended to smooth curves via a limiting procedure. This continuous limit coincides with the center of mass of the homogeneous lamina bounded by the curve."  This isn't necessarily false, but it's not very useful as a definition, since nailing it down would require some more work (we'd need to clarify we're talking about adding continuously many vertices, even along the "straight parts" of the boundary, which means this definition is *not* really an extension of the definition of ccm for polygons) and the result is simply the center of mass of the interior anyway, which doesn't need a new name.  The interesting true statement here is that the limit of the ccm, as the boundary is subdivided more and more finely by adding more vertices, is the center of mass of the interior.