Circumcenter of mass

More generally, the circumcenter of mass may be defined for simplicial polytopes and also in the spherical and hyperbolic geometries.

In the special case when the polytope is a quadrilateral or hexagon, the circumcenter of mass has been called the "quasicircumcenter" and has been used to define an Euler line of a quadrilateral.

[1][2] The circumcenter of mass allows us to define an Euler line for simplicial polytopes.

be an arbitrary point not lying on the sides (or their extensions).

with weight equal to its oriented area (positive if its sequence of vertices is countercyclical; negative otherwise).

As a consequence, any triangulation with nondegenerate triangles may be used to define the circumcenter of mass.

More generally, the circumcenter of mass and center of mass coincide for a simplicial polytope for which each face has the sum of squares of its edges a constant.

[4] The circumcenter of mass is invariant under the operation of "recutting" of polygons.

The generalized Euler line makes other appearances in the theory of integrable systems.

is given by the formula The circumcenter of mass 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.

Under natural assumptions, the centers of polygons which satisfy Archimedes' Lemma are precisely the points of its Euler line.

In other words, the only "well-behaved" centers which satisfy Archimedes' Lemma are the affine combinations of the circumcenter of mass and center of mass.

The circumcenter of mass allows an Euler line to be defined for any polygon (and more generally, for a simplicial polytope).